A034691 -id:A034691 - cp's OEIS frontend

A343351 Expansion of Product_{k>=1} 1 / (1 - x^k)^(6^(k-1)).

1, 1, 7, 43, 280, 1792, 11586, 74550, 479892, 3083640, 19794678, 126908502, 812761299, 5199586119, 33230586285, 212172173565, 1353444677529, 8626044781761, 54931168743703, 349524243121795, 2222294161109422, 14119034725444774, 89639674321304392, 568720801952770012

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A034691, A104460, A144070, A343349, A343350, A343352, A343353, A343354, A343355.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*6^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 12 2021

nmax = 23; CoefficientList[Series[Product[1/(1 - x^k)^(6^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 6^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 23}]

a(n) ~ exp(sqrt(2*n/3) - 1/12 + c/6) * 6^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (6^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

A343352 Expansion of Product_{k>=1} 1 / (1 - x^k)^(7^(k-1)).

Original entry on oeis.org

1, 1, 8, 57, 428, 3172, 23689, 176324, 1312550, 9757798, 72480269, 537854094, 3987751860, 29540543908, 218652961074, 1617159619805, 11951595353413, 88264810625245, 651404299886762, 4804261815210433, 35410065096578748, 260832137791524693, 1920169120639498017, 14127684273966098698

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A034691, A104460, A144071, A343349, A343350, A343351, A343353, A343354, A343355.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*7^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 12 2021

nmax = 23; CoefficientList[Series[Product[1/(1 - x^k)^(7^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 7^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 23}]

a(n) ~ exp(2*sqrt(n/7) - 1/14 + c/7) * 7^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (7^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

A343353 Expansion of Product_{k>=1} 1 / (1 - x^k)^(8^(k-1)).

Original entry on oeis.org

1, 1, 9, 73, 621, 5229, 44293, 374277, 3162447, 26694159, 225163687, 1897751079, 15983278059, 134519816427, 1131395821587, 9509592524371, 79880259426102, 670590654977718, 5626336598011078, 47179486350900358, 395410837699366686, 3312225325409475038, 27731588831310844302

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A034691, A104460, A144072, A343349, A343350, A343351, A343352, A343354, A343355.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*8^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 12 2021

nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^(8^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 8^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 22}]

a(n) ~ exp(sqrt(n/2) - 1/16 + c/8) * 2^(3*n - 7/4) / (sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (8^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

A343354 Expansion of Product_{k>=1} 1 / (1 - x^k)^(9^(k-1)).

Original entry on oeis.org

1, 1, 10, 91, 865, 8155, 77251, 730435, 6905560, 65233120, 615847378, 5810270782, 54784324495, 516250199827, 4862041512625, 45765734635702, 430560567351208, 4048630897384450, 38051334554031551, 357459295903931045, 3356488167698692226, 31503001136703776561

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A034691, A104460, A144073, A343349, A343350, A343351, A343352, A343353, A343355.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*9^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..21);  # Alois P. Heinz, Apr 12 2021

nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(9^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 9^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

a(n) ~ exp(2*sqrt(n/9) - 1/18 + c/9) * 9^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (9^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

A343355 Expansion of Product_{k>=1} 1 / (1 - x^k)^(10^(k-1)).

Original entry on oeis.org

1, 1, 11, 111, 1166, 12166, 127436, 1332936, 13939651, 145683351, 1521743103, 15886781603, 165770328383, 1728861822083, 18022063489023, 187778810866043, 1955660195168328, 20358764860253028, 211849198103034998, 2203562708619192998, 22911457758236641451, 238129937419462634151

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

In general, if m > 1 and g.f. = Product_{k>=1} 1 / (1 - x^k)^(m^(k-1)), then a(n, m) ~ exp(2*sqrt(n/m) - 1/(2*m) + c(m)/m) * m^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c(m) = Sum_{j>=2} 1/(j * (m^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

Cf. A034691, A104460, A292837, A343349, A343350, A343351, A343352, A343353, A343354.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*10^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..21);  # Alois P. Heinz, Apr 12 2021

nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(10^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 10^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

a(n) ~ exp(sqrt(2*n/5) - 1/20 + c/10) * 10^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (10^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

A356937 Number of multisets of intervals whose multiset union is of size n and covers an initial interval of positive integers.

Original entry on oeis.org

1, 1, 3, 9, 29, 94, 310, 1026, 3411, 11360, 37886, 126442, 422203, 1410189, 4711039, 15740098, 52593430, 175742438, 587266782, 1962469721, 6558071499, 21915580437, 73237274083, 244744474601, 817889464220, 2733235019732, 9133973730633, 30524096110942, 102006076541264

Offset: 0

Gus Wiseman, Sep 08 2022

nonn

An interval such as {3,4,5} is a set with all differences of adjacent elements equal to 1.

			The a(1) = 1 through a(3) = 9 set multipartitions (multisets of sets):
  {{1}}  {{1,2}}    {{1,2,3}}
         {{1},{1}}  {{1},{1,2}}
         {{1},{2}}  {{1},{2,3}}
                    {{2},{1,2}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{2}}
                    {{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200
Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

A000041 counts integer partitions, strict A000009.
A000670 counts patterns, ranked by A333217, necklace A019536.
A011782 counts multisets covering an initial interval.
Cf. A055887, A063834, A270995, A304969, A349050, A349055.
Intervals are counted by A000012, A001227, ranked by A073485.
Other conditions: A034691, A116540, A255906, A356933, A356942.
Other types: A107742, A356936, A356938, A356939.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
chQ[y_]:=Or[Length[y]<=1,Union[Differences[y]]=={1}];
Table[Length[Select[Join@@mps/@allnorm[n],And@@chQ/@#&]],{n,0,5}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
R(n,k) = {EulerT(vector(n, j, max(0, 1+k-j)))}
seq(n) = {my(A=1+O(y*y^n)); for(k = 1, n, A += x^k*(1 + y*Ser(R(n,k), y) - polcoef(1/(1 - x*A) + O(x^(k+2)), k+1))); Vec(subst(A,x,1))} \\ Andrew Howroyd, Jan 01 2023

Terms a(10) and beyond from Andrew Howroyd, Jan 01 2023

A356942 Number of multisets of gapless multisets whose multiset union is a size-n multiset covering an initial interval.

Original entry on oeis.org

1, 1, 4, 15, 61, 249, 1040, 4363, 18424, 78014, 331099, 1407080, 5985505, 25477399, 108493103, 462147381, 1969025286, 8390475609, 35757524184, 152398429323, 649555719160, 2768653475487, 11801369554033, 50304231997727, 214428538858889, 914039405714237

Offset: 0

Gus Wiseman, Sep 08 2022

nonn

A multiset is gapless if it covers an interval of positive integers. For example, {2,3,3,4} is gapless but {1,1,3,3} is not.

			The a(1) = 1 through a(3) = 14 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1,2}}    {{1,1,2}}
         {{1},{1}}  {{1,2,2}}
         {{1},{2}}  {{1,2,3}}
                    {{1},{1,1}}
                    {{1},{1,2}}
                    {{1},{2,2}}
                    {{1},{2,3}}
                    {{2},{1,1}}
                    {{2},{1,2}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{2}}
                    {{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200
Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

A000041 counts integer partitions, strict A000009.
A000670 counts patterns, ranked by A333217, necklace A019536.
A011782 counts multisets covering an initial interval.
Cf. A063834, A072233, A270995, A304969, A349050, A349055, A356934.
Gapless multisets are counted by A034296, ranked by A073491.
Other conditions: A034691, A055887, A116540, A255906, A356933, A356937.
Other types of multiset partitions: A356233, A356941, A356943, A356944.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
Table[Length[Select[Join@@mps/@allnorm[n],And@@nogapQ/@#&]],{n,0,5}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
R(n,k) = {EulerT(vector(n, j, sum(i=1, min(k, j), (k-i+1)*binomial(j-1, i-1))))}
seq(n) = {my(A=1+O(y*y^n)); for(k = 1, n, A += x^k*(1 + y*Ser(R(n,k), y) - polcoef(1/(1 - x*A) + O(x^(k+2)), k+1))); Vec(subst(A,x,1))} \\ Andrew Howroyd, Jan 01 2023

Terms a(9) and beyond from Andrew Howroyd, Jan 01 2023

A200544 Number of distinct bags of distinct sequences of 1s and 2s such that the sum of all terms is n.

Original entry on oeis.org

1, 1, 3, 6, 14, 28, 61, 122, 253, 505, 1017, 2008, 3976, 7769, 15169, 29379, 56751, 108993, 208725, 397913, 756385, 1432578, 2705744, 5094749, 9568504, 17922756, 33492061, 62438472, 116151352, 215612548, 399451325, 738612472, 1363261171, 2511748010, 4620024202

Offset: 0

Sarah Nibs, Nov 18 2011

nonn

This is the number of distinct ways to build minimal Jenga towers out of n blocks. The number of distinct ways to build a single minimal Jenga tower out of n blocks is the Fibonacci number F(n+1) (A000045(n+1)).
To calculate this, first create all partitions of n.
An example partition, for n=4, is
  1 1 1 1
  1 1 2
  1 3
  2 2
  4
Then each set of towers of the same size gets a configuration. For 2 2 2, for instance, there are two possibilities for each tower (a single level with two blocks or two levels with one block each) but the total possibilities is not 2*2*2=8, since the configuration "1/1,2,2" is the same as "2,1/1,2". Instead we want to choose three towers with repetition from two possibilities which is 3+2-1 choose 3, aka 4C3 = 4.
Multiply all the sets of towers of the same size and sum over partitions for the result.
For n=4, then, 1 1 2 becomes "1 with multiplicity 2" and "2 with multiplicity 1".
There is f(1+1)=1 way to build a tower of size 1, and f(1+1)+2-1 choose 2 = 2C2 = 1 way to build 2 towers of size 1. f(2+1)=2 ways to build a tower of size 2. 1 1 2 has 1*2=2 ways to be built. Sum over each of the 5 partitions of n=4.
This is apparently the limit of the row-reversed rows of the Multiset transform T(n,k) of the Fibonacci sequence in A337009, a(k) = lim_{n->oo} T(n,n-k). - R. J. Mathar, Aug 10 2020

			For n = 4, a(4)=14 and the bags are: 1/1/1/1; 1/1/1,1; 1/1/2; 1/1,1,1; 1/1,2; 1/2,1; 1,1/1,1; 1,1/2; 2/1,1; 2/2; 1,1,1,1; 1,1,2; 1,2,1; 2,1,1.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
W. S. Gray, K. Ebrahimi-Fard, Affine SISO Feedback Transformation Group and Its Faa di Bruno Hopf Algebra, arXiv:1411.0222 [math.OC], 2014.
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015
Vaclav Kotesovec, Asymptotics of sequence A034691
Sarah Nibs, C# code to generate sequence terms
Wikipedia, Jenga

Cf. A000045, A034691, A166861, A260787, A337009.

with(numtheory):with(combinat):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      fibonacci(d+1), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Nov 05 2013

CoefficientList[Series[Product[1/(1-x^k)^Fibonacci[k+1], {k, 1, 40}], {x, 0, 40}], x] (* Vaclav Kotesovec, Aug 05 2015 *)

# uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(1, 1, 1)
b = EulerTransform(a)
print([b(n) for n in range(35)]) # Peter Luschny, Nov 11 2020

sum{m1*a1+m2*a2+...+mk*ak}(prod{k}(binomial(A000045[ak + 1]+mk-1,mk))).
G.f.: Product_{s>=1}(sum{d>=0}(binomial(F(s+1)+d-1,d)*x^(d*s))). - Sarah Nibs, Oct 21 2013
Euler Transform of A000045 starting at index 2, i.e. EULER(1, 2, 3, 5, 8, 13, ...). - Sarah Nibs, Nov 05 2013
a(n) ~ phi^(n+1/4) / (2 * sqrt(Pi) * 5^(1/8) * n^(3/4)) * exp(phi/10 - 3/5 + 2*5^(-1/4)*sqrt(phi*n) + s), where s = Sum_{k>=2} (1+phi^k) / ((phi^(2*k) - phi^k - 1)*k) = 0.7902214013751085262994702391769374769675268259229550490716908... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 06 2015
a(n) = A337009(2*n,n). - Alois P. Heinz, Apr 30 2023

Corrected terms from n=8 and onwards by Sarah Nibs, Oct 18 2013

C# program corrected and made much more efficient by Sarah Nibs, Oct 18 2013

A299152 Denominators of the positive solution to 2^(n-1) = Sum_{d|n} a(d) * a(n/d).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Feb 03 2018

			Sequence begins: 1, 1, 2, 7/2, 8, 14, 32, 121/2, 126, 248, 512, 1003, 2048, 4064, 8176, 130539/8, 32768.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A000010, A000740, A018804, A023900, A029935, A034691, A046643, A059966, A228369, A257098, A296302, A298971, A299119, A299149, A299151.

nn=50;
sys=Table[2^(n-1)==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
Denominator[Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]]

up_to = 65537;
prepareA299151perA299152(up_to) = { my(vmemo = vector(up_to)); for(n=1,up_to, vmemo[n] = if(1==n,n,(2^(n-1)-sumdiv(n,d,if((d>1)&&(dA299152 = prepareA299151perA299152(up_to);
A299151perA299152(n) = v299151perA299152[n];
\\ Or without memoization as:
A299151perA299152(n) = if(1==n,n,(2^(n-1)-sumdiv(n,d,if((d>1)&&(dA299151perA299152(d)*A299151perA299152(n/d),0)))/2);
A299152(n) = denominator(A299151perA299152(n)); \\ Antti Karttunen, Jul 29 2018

More terms from Antti Karttunen, Jul 29 2018

A303551 Number of aperiodic multisets of compositions of total weight n.

Original entry on oeis.org

1, 2, 6, 15, 41, 95, 243, 567, 1366, 3189, 7532, 17428, 40590, 93465, 215331, 493150, 1127978, 2569049, 5841442, 13240351, 29953601, 67596500, 152258270, 342235866, 767895382, 1719813753, 3845442485, 8584197657, 19133459138, 42583565928, 94641591888

Offset: 1

Gus Wiseman, Apr 26 2018

nonn

A multiset is aperiodic if its multiplicities are relatively prime.

			The a(4) = 15 aperiodic multisets of compositions are:
{4}, {31}, {22}, {211}, {13}, {121}, {112}, {1111},
{1,3}, {1,21}, {1,12}, {1,111}, {2,11},
{1,1,2}, {1,1,11}.
Missing from this list are {1,1,1,1}, {2,2}, and {11,11}.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000740, A000837, A007716, A007916, A034691, A100953, A255906, A269134, A301700, A303386, A303431, A303546, A303552.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*2^(d-1), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> add(mobius(d)*b(n/d), d=divisors(n)):
seq(a(n), n=1..35);  # Alois P. Heinz, Apr 26 2018

nn=20;
ser=Product[1/(1-x^n)^2^(n-1),{n,nn}]
Table[Sum[MoebiusMu[d]*SeriesCoefficient[ser,{x,0,n/d}],{d,Divisors[n]}],{n,1,nn}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(u=EulerT(vector(n, n, 2^(n-1)))); vector(n, n, sumdiv(n, d, moebius(d)*u[n/d]))} \\ Andrew Howroyd, Sep 15 2018

a(n) = Sum_{d|n} mu(d) * A034691(n/d).

cp's OEIS Frontend

A343351 Expansion of Product_{k>=1} 1 / (1 - x^k)^(6^(k-1)).

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A343352 Expansion of Product_{k>=1} 1 / (1 - x^k)^(7^(k-1)).

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A343353 Expansion of Product_{k>=1} 1 / (1 - x^k)^(8^(k-1)).

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A343354 Expansion of Product_{k>=1} 1 / (1 - x^k)^(9^(k-1)).

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A343355 Expansion of Product_{k>=1} 1 / (1 - x^k)^(10^(k-1)).

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A356937 Number of multisets of intervals whose multiset union is of size n and covers an initial interval of positive integers.

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A356942 Number of multisets of gapless multisets whose multiset union is a size-n multiset covering an initial interval.

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Mathematica

PARI

Extensions

A200544 Number of distinct bags of distinct sequences of 1s and 2s such that the sum of all terms is n.

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