#A102 INTEGERS 25 (2025)
DISTRIBUTIONS OF ULAM WORDS UP TO LENGTH 30
Paul Adu wum
Ba es College, Lewis on, Maine
[email p o ec ed]
Hoppe Cla k
Ba es College, Lewis on, Maine
[email p o ec ed]
Ro Eme son
Ba es College, Lewis on, Maine
[email p o ec ed]
Alexand a (Sasha) Sheyd asse
Uni e si y o Massachuse s, Amhe s , Massachuse s
[email p o ec ed]
A seniy (Senia) Sheyd asse
Depa men o Ma hema ics, Ba es College, Lewis on, Maine
[email p o ec ed]
Axelle Tougouma
Ba es College, Lewis on, Maine
[email p o ec ed]
Recei ed: 10/16/24, Re ised: 7/2/25, Accep ed: 10/26/25, Published: 11/5/25
Abs ac
We u he explo e he no ion o Ulam wo ds conside ed by Bade, Cui, Labelle,
and Li, gi ing some lowe bounds on how many he e a e o a gi en leng h. Gaps
be ween wo ds and wo ds o special ype also e eal ema kable s uc u e. By
subs an ially inc easing he numbe o compu ed e ms, we a e also able o sha pen
some o he conjec u es made by Bade e al.
1. In oduc ion
In hei 2020 pape , Bade, Cui, Labelle, and Li [1] in oduced he no ion o Ulam
wo ds, de ined as ollows. Conside he ee semig oup S 0,1us on wo gene a o s
DOI: 10.5281/zenodo.17535327
INTEGERS: 25 (2025) 2
0 and 1. We say ha 0 and 1 a e Ulam and hen de ine all o he Ulam wo ds
induc i ely: a wo d w‰0,1 is Ulam i and only i he e exis s exac ly one pai o
Ulam wo ds u1‰u2such ha w“u1
"u2. (He e, "deno es conca ena ion.) We
shall deno e he en i e se o Ulam wo ds as U, and Ulam wo ds o leng h nby Un.
I is easy o check ha :
U1“ 0,1uU3“ 001,011,100,110u
U2“ 01,10uU4“ 0001,0010,0100,0111,1000,1011,1101,1110u.
All Ulam wo ds up o leng h 24 we e compu ed in [1]; we we e able o compu e
up o leng h 30. While his migh appea as a small imp o emen a i s glance,
because he numbe o Ulam wo ds o leng h nappea s o (almos ) double on each
i e a ion, in eali y, his ep esen s nea ly 60 imes as much da a.
I is an open ques ion whe he |Un|( he size o Un) g ows exponen ially. The
bes lowe bound ha we can p o e is linea , mainly using explici cons uc ions o
wo ds om [1].
Theo em 1. Fo all ně6, we ha e ha |Un| ě 2n`4.
Howe e , we a e able o demons a e ha he e is a subsequence o Ulam wo ds
ha g ows exponen ially, using a comple ely di e en a gumen .
Theo em 2. The e exis s 1ăα0ď2such ha o all 1ăαăα0, we ha e ha
lim sup
nÑ8
|Un|
αn“ 8.
Conc e ely,
α0“ˆ101847671
31 ˙1{5
«1.648996
su ices.
We gi e p oo s o bo h o hese heo ems in Sec ion 3. Un o una ely, bo h o
hese esul s a e s ill qui e a om wha is conjec u ed o hold. To wi , de ine he
densi y
ρpnq:“|Un|
2n;
I was conjec u ed in [1] ha ρpnq Ñ o some 0 ă ă1 (see Conjec u e 3.10 in
[1]); wi h ou enla ged da a se , we ins ead posi some hing a li le s ange .
Conjec u e 1. The densi y o Ulam wo ds ρpnq “ Θpn´3{10q.
This conjec u e is suppo ed by he nume ical e idence—see Figu es 1 and 2 o
an example—bu i also has ies o ano he conjec u e in ol ing he a e age gap
be ween Ulam wo ds, which we shall desc ibe below. In any case, obse e ha i
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5 10 15 20 25 30
n
0.1
0.2
0.3
0.4
δ
Figu e 1: A plo o he densi ies ρpnq o 4 ďnď30, oge he wi h a plo o
pnq “ 0.526n´3{10.
n|Un|
13 1916
14 3812
15 7772
16 14822
17 29368
18 58478
n|Un|
19 114300
20 225166
21 441724
22 876238
23 1717748
24 3406884
n|Un|
25 6720784
26 13303332
27 26273948
28 52010642
29 102933200
30 203695342
Figu e 2: The exac coun s o |Un| o 13 ďnď30.
ei he conjec u e is co ec , he numbe o Ulam wo ds g ows only e y sligh ly
slowe han 2n.
This no ion o Ulam wo ds was buil on he ea lie no ion o Ulam se s due o
K a i z and S eine be ge [8], which was i sel a gene aliza ion o Ulam’s eponymous
in ege sequence, also de ined ecu si ely [13]: he (classical) Ulam sequence begins
wi h 1,2, and hen e e y subsequen e m is he nex smalles in ege ha can be
w i en as he sum o wo dis inc p io e ms in exac ly one way. Gene aliza ions
o Ulam’s classic sequence ha e become an inc easingly popula objec o s udy:
in 1972, Queneau did some p elimina y wo k s udying gene aliza ions whe e he
ini ial wo e ms o he in ege sequence a e a ied [10]; in he 1990s, Cassaigne,
Finch, Shme l, and Spiegel de e mined some o he amilies o such sequences such
ha he consecu i e di e ences a e e en ually pe iodic [2, 3, 4, 5, 11]; in 2017, [8]
conside ed gene alizing he Ulam condi ion o abelian g oups; in 2020, [1] ga e
he a o emen ioned no ion o Ulam wo ds wi h some p elimina y esul s; and in
2021, Sheyd asse showed ha he e is an analogous no ion o Ulam se s o in ege
polynomials [12] by building o ea lie wo k o Hinman, Kuca, Schlesinge , and
Sheyd asse [6, 7].
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S0S1S2S3
Figu e 3: Visual o he i s 4 s eps o cons uc ing he disc e e Sie pi´nski iangle.
Ea lie wo k a ound Ulam wo ds has la gely cen e ed a ound gi ing simple c i-
e ia o when wo ds o some special ype a e Ulam— o example, [1] showed ha a
wo d o he o m 0a10bis Ulam i and only i pa`b
aqis odd. Simila ly, Mandelsh am
[9] conside ed Ulam wo ds o he o m 0a10b10cand demons a ed a connec ion o
he Sie pi´nski gaske . We also p o e a ew such esul s, such as he ollowing.
Theo em 3. Conside he se o poin s px, yq P Z2
ě1such ha 1y0x´yPU. This
is he disc e e Sie pi´nski iangle, union a poin .
We will discuss his cons uc ion mo e p ecisely in Sec ion 4, bu b ie ly, he
disc e e Sie pi´nski iangle is an app oxima ion o he s anda d Sie pi´nski iangle.
I can be cons uc ed ei he i e a i ely (as in Figu e 3) o by colo ing Pascal’s
iangle by pa i y.
On he o he hand, we also ha e a no el way o conside ing Ulam wo ds by in e -
p e ing hem as in ege s. Obse e ha he e exis s a na u al map π:S 0,1us Ñ
Zě0 ia in e p e ing a wo d as he bina y ep esen a ion o an in ege . In gene al,
his map is no injec i e— o example, πp0q “ πp00q “ πp000q “ 0. Howe e , i
we es ic i o wo ds o a ixed leng h, hen i is. In pa icula , he es ic ions
π:UnÑZX 0,2n´1sa e injec i e maps. This gi es a na u al o de ing on Unand
allows us o ask ques ions abou how Ulam wo ds a e dis ibu ed. Fo example,
we migh ask abou he dis ibu ion o he gaps—di e ences be ween consecu i e
Ulam wo ds, in e p e ed as in ege s.
Conjec u e 2. Le u1ău2ă. . . ăuknbe he (o de ed) elemen s o πpUnq.
De ine
pn:Zě1Ñ 0,8q
gÞÑ ˇˇ i|ui`1´ui“guˇˇ
kn´1.
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This has a na u al in e p e a ion as a p obabili y measu e. As nÑ 8, he unc ions
pncon e ge poin wise o a p obabili y measu e p:Zě1Ñ 0,8q. Fu he mo e, le
µgpnqbe he mean o he p obabili y measu e pn. Then µgpnq “ Θpn3{10q—indeed,
i may be ha he e is a cons an c«1.9 such ha µgpnq “ cn3{10 `op1q.
This conjec u e is well-suppo ed by ou a ailable da a—see Sec ion 5 o de ails,
illus a ions, and u he odd p ope ies o he appa en dis ibu ion. Wha is
in e es ing abou his s a emen abou a e age gaps is ha , i ue, i immedia ely
implies Conjec u e 1.
Theo em 4. As nÑ 8, we ha e ha ρpnq´1—µgpnq. Consequen ly, Conjec u e
2 implies Conjec u e 1.
This is salien , since ou nume ical e idence o Conjec u e 2 is a guably much
s onge han o Conjec u e 1! Again, see Sec ion 5 o de ails. Finally, in Sec ion
6, we ask he ques ion o how πpUnqis dis ibu ed modulo N.
Conjec u e 3. Fo any in ege Ną1 and aPZ{NZ, de ine he ela i e densi y
ρa,N pnq:“ˇˇ wPUn|πpwq ” amod Nuˇˇ
|Un|.
Then limnÑ8 ρa,N pnq “ 1{N.
Rema k 1. As we discuss in Sec ion 6, while his conjec u e is consis en wi h
he a ailable da a, i is somewha su p ising. Fo one hing, ρ5,6p1q “ ρ5,6p2q “
ρ5,6p3q “ 0, and i akes some ime be o e i appea s o s a o con e ge o 1{6.
Fo ano he , he e is an appa en bias modulo 6 in he dis ibu ion o he gaps.
Ou code and some o ou da a can be ound on Gi Hub1, bu i is a om
e icien —as was poin ed ou o us Tom´as Oli ei a e Sil a, i is possible o use
bi maps o make hese compu a ions much as e ; a good implemen a ion should
gi e a Op2nlogpnqq unning ime. Howe e , we lea e his as ma e ial o u u e
wo k.
2. De ini ions and Visualiza ions
We s a wi h some basic de ini ions and cons uc ions. Gi en a wo d wPS 0,1us,
we de ine i s complemen ˆw o be he wo d wi h e e y ins ance o 0 eplaced wi h
a 1, and ice e sa. We also de ine he e e se w, which is he wo d ob ained by
e e sing he o de o he le e s. I was shown in [1] ha wPUi and only i
ˆwPU, i and only i wPU.
1h ps://gi hub.com/asheyd a/Ulam-Wo ds.gi
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Figu e 4: All wo ds o leng h 20 beginning wi h a ze o.
To be e isualize he se U, we made use o hea maps, which depic each Ulam
wo d as a colo ed ba and s acks all o he wo ds e ically— ha is, o a gi en
wo d, a 0 co esponds o a ec angle o one colo , and a 1 co esponds o a ec angle
o a second colo . An example is p o ided in Figu e 4. In gene al, we ab idge such
diag ams: we c ea ed igu es only using all he Ulam wo ds ha s a ed wi h ze o,
since Ulam wo ds a e closed unde complemen s. Mo eo e , we impose he o de ing
discussed in he in oduc ion, de ining wďw1i and only i πpwq ď πpw1q. Using
his way o isualizing Ulam wo ds allows us o easily see ha he e is bo h a clea
bina y ee s uc u e ha go e ns he exis ence o Ulam wo ds, as well as a chao ic
elemen o he se whe e he bina y ee b eaks down.
We can be mo e speci ic abou ou meaning ega ding his b eakdown: since Ulam
wo ds a e p ese ed unde he e e se map, his is equi alen o saying ha o any
n he e exis s nąℓną0 such ha all possible subwo ds o leng h ℓnoccu as he
inal ℓncha ac e s o wo ds in Un. In u n, ha is equi alen o saying ha he
quo ien map πpUnq Ñ Z{2ℓnZis su jec i e. Ou obse a ion is ha ℓnappea s
o inc ease as a unc ion o n, albei no e y quickly—see Figu e 5. Assuming
ha Conjec u e 3 is ue, hen i would ollow immedia ely ha ℓnÑ 8 simply
by conside ing he case whe e N“2ℓn—indeed, he hea maps we e he o iginal
impe us o ou equidis ibu ion conjec u es. On he o he hand, he “chao ic”
la e hal o he hea map is mo e o a mys e y.
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n ℓn
1 1
2 1
3 1
4 1
5 3
6 2
n ℓn
7 4
8 4
9 4
10 4
11 4
12 5
n ℓn
13 5
14 5
15 6
16 7
17 7
18 8
n ℓn
19 9
20 9
21 9
22 10
23 10
24 11
n ℓn
25 11
26 11
27 12
28 12
29 13
30 13
Figu e 5: Tables o n e sus ℓn, whe e ℓnis he la ges in ege such ha UnÑ
Z{2ℓnZis su jec i e.
3. Lowe Bounds on G ow h
Ou goal in his sec ion is o p o e ou lowe bounds on |Un|; we begin wi h Theo em
1, o which we need some explici examples o Ulam wo ds. The i s h ee a e due
o [1].
Theo em 5 ([1]).The e a e Gpn´1qUlam wo ds o leng h no he o m 0a10b,
whe e Gpnqis he n- h en y in Gould’s sequence.
Rema k 2. Gould’s sequence Gpnqis he numbe o odd en ies in he n- h ow
o Pascal’s iangle; equi alen ly, Gpnq “ 2#1pnq, whe e #1pnqis he numbe o
non-ze o bi s in he bina y ep esen a ion o n.
Rema k 3. Since wPUi and only i wPU, i and only i ˆwPU, we ge
analogous esul s wi h 1’s eplaced wi h 0’s and he o de o he le e s e e sed.
This is ue o all he esul s ha we p o e he e.
Theo em 6 ([1]).Fo any a, b PZě0, he wo d 0a120bis in Ui and only i he
leng h o he wo d is odd ( ha is, a`b”1pmod 2q).
Theo em 7 ([1]).Fo any a, b PZě0such ha a`bě2, he wo d 0a1010bis in
Ui and only i he leng h o he wo d is e en ( ha is, a`b”1 mod 2).
Lemma 1. Fo any a, b PZě0such ha a`bě1, he wo d 0a140bis in Ui and
only i a`b”1pmod 4q.
P oo . We will use p oo by induc ion on he leng h o he wo d n, whe e he base
cases n“5,6,7,8 can be e i ied di ec ly. Assume he s a emen holds o all wo ds
o leng h s ic ly less han n, and conside he wo d u“0k140lo leng h n, whe e,
since ně9, a leas one o kand lis a leas 3. By applying he e e se map o
swi ch kand li necessa y, we may assume ha kě3.
Case 1: l“0. The only possible ep esen a ions a e 0"0k´114and 0k13"1. By
he induc i e hypo hesis, he i s is alid i and only i n”2pmod 4q. By Lemma
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3, he second is alid i and only i n”1,2pmod 4q. Thus, exac ly one o hese
ep esen a ions is alid i and only i n”1pmod 4q.
Case 2: lě1. The e a e i e po en ial ep esen a ions:
1. 0"0k´1140l,
2. 0k1"130l,
3. 0k12"120l,
4. 0k13"10l, and
5. 0k140l´1"0.
Obse e ha by he induc i e hypo hesis, ep esen a ions (1) and (5) a e alid i
and only i n”2pmod 4q, which is o say ha k`l”2pmod 4q. By Theo em 8
and Lemma 3, ep esen a ion (2) is alid i and only i l”0,3pmod 4q; simila ly,
ep esen a ion (4) is alid i and only i k”0,3pmod 4q. Finally, by Theo em 6,
ep esen a ion (3) is alid i and only i k”l”1pmod 2q. This allows us o coun
he numbe o alid ep esen a ions in e ms o he cong uence classes o kand l
modulo 4, as seen in Figu e 6. In pa icula , he e is a unique ep esen a ion i and
only i n”1pmod 4q.
kzl0123
0 2132
1 1302
2 3001
3 2215
Figu e 6: Table o numbe o ep esen a ions o 0k140l o alues o n“k`l
modulo 4.
Wi h his, we a e eady o gi e a p oo o he gene al linea bound.
P oo o Theo em 1. We conside h ee cases.
Case 1: nis e en. By Theo em 7, we know ha 0a1010n´a´3PUn o all 0 ď
aďn´3— his yields n´2 Ulam wo ds. By Theo em 5, we also know ha he e
a e Gpn´1qUlam wo ds o leng h no he o m 0a10b. No e ha hese wo se s o
Ulam wo ds do no in e sec ( hey ha e di e en numbe s o ones), and since n´1
is odd, Gpn´1q ě 22“4. In o al, his yields n`2 Ulam wo ds.
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No e ha {
0a1010n´a´3“1a0101n´a´3“0a110b1i and only i n“3, so he
e e ses o he cons uc ed Ulam wo ds a e also dis inc Ulam wo ds. The e o e,
we ha e a leas 2n`4 Ulam wo ds in his case.
Case 2: n”3pmod 4q. By Theo em 6, we know ha 0a120n´a´2PUn o
all 0 ďaďn´2— his yields n´1 Ulam wo ds. Since n´1”2pmod 4q,
Gpn´1q ě 22“4, and so we can again use Theo em 5 o conclude ha he e a e
a leas 4 wo ds 0a10bo he igh leng h. In o al, his yields n`3 Ulam wo ds.
No e ha {
0a120n´a´2“1a021n´a´2“0a110b1i and only i a“0 and a1“2.
The e o e, he e e ses o ou wo amilies o cons uc ed Ulam wo ds in e sec , bu
only in wo places; he e o e, we ha e 2pn`3q ´ 2“2n`4 Ulam wo ds.
Case 3: n”1pmod 4q. As in he p e ious case, we ha e n´1 wo ds o he o m
0a120n´a´2, bu i is possible ha Gpn´1q “ 2, so we ha e o a gue di e en ly:
speci ically, we use Lemma 1 o conclude ha 0a140n´4´aPU o all 0 ďaďn´4,
which yields ano he n´3 Ulam wo ds, o a o al o a leas 2n´4.
Obse e ha {
0a120n´a´2‰1a021n´a´2“0a1140n´4´a1e e , so we may simply
double ou coun o Ulam wo ds. In o al, we ha e 4n´8, which is a leas 2n`4
i ně6.
In each case, we ha e iden i ied a leas 2n`4 dis inc Ulam wo ds.
Nex , we ackle he exponen ial bound, which we app oach in a comple ely di -
e en ashion using he ollowing lemma.
Lemma 2. Fo any nPZě1,
|Un|2ď |Un|`|Un`1| ` . . . ` |U2n|.
P oo . Conside he se
X:“␣pw1, w2q P U2
nˇˇw1‰w2(.
Fo any pw1, w2q P X, ei he w1
"w2PU2no he e exis s 1PUk, 2PU2n´ksuch
ha w1
"w2“ 1
" 2, whe e kP 1, n´1sY n`1,2n´1s; o cou se, i kP 1, n´1s,
hen 2n´kP n`1,2n´1s, and so we may conclude ha
|X|ď|Un`1| ` . . . ` |U2n|.
On he o he hand,
|X|“|Un|2´ |Un|.
As a consequence o Lemma 2, we ge he ollowing e y weak lowe bound: o
any nPZě1,
max
nďiď2n|Ui| ě |Un|2
n`1.(1)
This is su icien o ou pu poses.
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Figu e 11: F om le o igh , op o bo om: ba g aphs showing he equency o
gaps o a ious sizes be ween consecu i e wo ds in Un o n“13,...,30, shown
ou o 4 s anda d de ia ions.
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6. Modula Dis ibu ion
Le us now conside he ela i e densi y o Ulam wo ds. As we men ioned ea lie , he
se o Ulam wo ds is p ese ed unde he complemen map. This o ces a symme y
on cong uence classes.
Theo em 11. I wPUn hen π´1`2n`1´1´πpwq˘PUn. Consequen ly, o any
posi i e in ege Nand aPZ{NZ,
ρa,N pnq “ ρ2n`1´1´a,N paq.
P oo . Gi en wPUn, w i e x“πpwq “ an2n`. . . `a0in bina y. Then
πpˆwq “ p1´anq2n`. . . ` p1´a0q “ 2n`1´1´x.
Bu ˆwPUn. Now, obse e ha i πpwq ” amod N, hen 2n`1´1´πpwq ”
2n`1´1´amod N, which o ces he equali y o he ela i e densi ies.
Fo N“2,3, his is pa icula ly simple.
Co olla y 2. Fo any posi i e in ege n, we ha e ha ρ0,2pnq “ ρ1,2pnq. Fu he -
mo e, o any aPZ{3Z,
ρa,3pnq “ #ρ1´a,3pnqi n”0 mod 2
ρ´a,3pnqi n”1 mod 2.
P oo . Obse e ha 2n`1´1´x”x`1 mod 2, om which ρ0,2pnq “ ρ1,2pnq
ollows immedia ely. Fo he second pa , obse e ha
2n`1´1´xmod 3 ”#1´xi n”0 mod 2
´xi n”1 mod 2.
While he e mus always exis o any n wo cong uence classes a, b PZ{3Zsuch
ha ρa,3pnq “ ρb,3pnq, he e is no eason why he las cong uence class cshould be
oughly equal. Indeed, o nď5, we see ha ρc,3pnq “ 0. Howe e , o la ge n,
i does appea o be he case ha ρc,3pnq Ñ ρa,3pnq “ ρb,3pnq; as we will illus a e
p esen ly. To help measu e he ex en o which wo ds a e equidis ibu ing modulo
N, we de ine he modula disc epancy.
De ini ion 1. Fo any posi i e in ege s n, N, he modula disc epancy is
dNpnq:“max
a,bPZ{NZ|ρa,N pnq ´ ρb,N pnq| .
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Figu e 12: A plo o he modula disc epancies o p ime powe moduli pkă30.
T i ially, saying ha Ulam wo ds equidis ibu e modulo Nis equi alen o saying
ha dNpnq Ñ 0 as nÑ 8. Mo eo e , by appealing o he Chinese emainde
heo em, p o ing ha dNpnq Ñ 0 o all Nis educible o p o ing ha dpkpnq Ñ 0
o all p ime powe s pk. To in es iga e Conjec u e 3, we compu ed dpkpnq o all
p ime powe s pkă30—as nea as we can ell, dpkpnqdecays exponen ially as a
unc ion o n(see Figu e 12).
Acknowledgemen s. Ou collabo a ion was unded by ou Ba es College g an s,
all awa ded by he Dean o Facul y’s o ice: a STEM Facul y-S uden Summe
Resea ch G an and h ee Summe Resea ch Fellowships. We would also like o
hank Tom´as Oli ei a e Sil a o con i ming ou compu a ions o |Un|and gi ing
many help ul sugges ions o imp o ing he exposi ion.
Re e ences
[1] T. Bade, K. Cui, A. Labelle, and D. Li, Ulam se s in new se ings, p ep in , a Xi :
2008.02762.
[2] J. Cassaigne and S. R. Finch, A class o 1-addi i e sequences and quad a ic ecu ences, Exp.
Ma h. 4(1995), 49–60.
[3] S. R. Finch, Conjec u es abou s-addi i e sequences, Fibonacci Qua . 29 (1991), 209–214.
[4] S. R. Finch, On he egula i y o ce ain 1-addi i e sequences, J. Combin. Theo y Se . A 60
(1992), 123–130.
[5] S. R. Finch, Pa e ns in 1-addi i e sequences, Exp. Ma h. 1(1992), 57–63.
[6] J. Hinman, B. Kuca, A. Schlesinge , and A. Sheyd asse , Rigidi y o Ulam se s and sequences,
In ol e 12 (2019), 521–539.
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[7] J. Hinman, B. Kuca, A. Schlesinge , and A. Sheyd asse , The un easonable igidi y o Ulam
sequences, J. Numbe Theo y 194 (2019), 409–425.
[8] N. K a i z and S. S eine be ge , Ulam sequences and Ulam se s, In ege s 18 (2018), A80.
[9] A. Mandelsh am, On ac al pa e ns in Ulam wo ds, p ep in , a Xi : 2211.14229.
[10] R. Queneau, Su les sui es s-addi i es, J. Combin. Theo y Se . A 12 (1972), 31–71.
[11] J. Schme l and E. Spiegel, The egula i y o some 1-addi i e sequences, J. Combin. Theo y
Se . A 66 (1994), 172–175.
[12] A. Sheyd asse , The Ulam sequence o linea in ege polynomials, J. In ege Seq. 24 (2021).
[13] S. Ulam, Combina o ial analysis in in ini e se s and some physical heo ies, SIAM Re . 6
(1964), 343–355.
