cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A337764 Number of compositions (ordered partitions) of the n-th n-gonal number into n-gonal numbers.

Original entry on oeis.org

1, 1, 2, 7, 124, 14371, 12842911, 103590035354, 8621925847489749, 8307493939404888703058, 102488432265617100812550713499, 17706351554929677399562928448484650120, 46435685450659378932235460132506329282776942795
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 19 2020

Keywords

Examples

			a(3) = 7 because the third triangular number is 6 and we have [6], [3, 3], [3, 1, 1, 1], [1, 3, 1, 1], [1, 1, 3, 1], [1, 1, 1, 3] and [1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A011782, A060354, A224366, A224677, A337762, A337763.

Formula

a(n) = [x^p(n,n)] 1 / (1 - Sum_{k=1..n} x^p(n,k)), where p(n,k) = k * (k * (n - 2) - n + 4) / 2 is the k-th n-gonal number.

A224679 Number of compositions of n^2 into sums of positive triangular numbers.

Original entry on oeis.org

1, 1, 3, 25, 546, 28136, 3487153, 1038115443, 742336894991, 1275079195875471, 5260826667789867957, 52137661179700350278531, 1241165848412448464485760897, 70972288312605764017275784402928, 9748291749334923037419108242002717050
Offset: 0

Views

Author

Paul D. Hanna, Apr 14 2013

Keywords

Links

Crossrefs

Cf. A023361, A224366, A224677.

Programs

  • Maple
    b:= proc(n) option remember; local i; if n=0 then 1 else 0;
      for i while i*(i+1)/2<=n do %+b(n-i*(i+1)/2) od; %  fi
    end:
a:= n-> b(n^2):
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 05 2018
  • Mathematica
    b[n_] := b[n] = Module[{i, j = If[n == 0, 1, 0]}, For[i = 1, i(i+1)/2 <= n, i++, j += b[n-i(i+1)/2]]; j];
a[n_] := b[n^2];
a /@ Range[0, 20] (* Jean-François Alcover, Nov 04 2020, after Alois P. Heinz *)
  • PARI
    {a(n)=polcoeff(1/(1-sum(r=1,n+1, x^(r*(r+1)/2)+x*O(x^(n^2)))), n^2)}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = A023361(n^2), where A023361(n) = number of compositions of n into positive triangular numbers.
a(n) = [x^(n^2)] 1/(1 - Sum_{k>=1} x^(k*(k+1)/2)).

A337799 Number of compositions (ordered partitions) of the n-th n-gonal pyramidal number into n-gonal pyramidal numbers.

Original entry on oeis.org

1, 1, 2, 15, 2780, 94947913, 5470124262136760, 5979009355803053742719666641, 1610158754567753309521653012201612266212334009, 1566217729562552701894041200097975651072376485590145959656670312797530
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 22 2020

Keywords

Examples

			a(3) = 15 because the third tetrahedral (or triangular pyramidal) number is 10 and we have [10], [4, 4, 1, 1], [4, 1, 4, 1], [4, 1, 1, 4], [1, 4, 4, 1], [1, 4, 1, 4], [1, 1, 4, 4], [4, 1, 1, 1, 1, 1, 1], [1, 4, 1, 1, 1, 1, 1], [1, 1, 4, 1, 1, 1, 1], [1, 1, 1, 4, 1, 1, 1], [1, 1, 1, 1, 4, 1, 1], [1, 1, 1, 1, 1, 4, 1], [1, 1, 1, 1, 1, 1, 4] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A006484, A224677, A337764, A337797, A337798.

Formula

a(n) = [x^p(n,n)] 1 / (1 - Sum_{k=1..n} x^p(n,k)), where p(n,k) = k * (k + 1) * (k * (n - 2) - n + 5) / 6 is the k-th n-gonal pyramidal number.

A299032 Number of ordered ways of writing n-th triangular number as a sum of n squares of positive integers.

Original entry on oeis.org

1, 1, 0, 3, 6, 0, 12, 106, 420, 2718, 18240, 120879, 694320, 5430438, 40668264, 300401818, 2369504386, 19928714475, 174151735920, 1543284732218, 14224347438876, 135649243229688, 1331658133954940, 13369350846412794, 138122850643702056, 1462610254141337590
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 01 2018

Keywords

Examples

			a(4) = 6 because fourth triangular number is 10 and we have [4, 4, 1, 1], [4, 1, 4, 1], [4, 1, 1, 4], [1, 4, 4, 1], [1, 4, 1, 4] and [1, 1, 4, 4].

Links

Crossrefs

Cf. A000217, A000290, A066535, A072964, A104383, A126683, A196010, A224677, A224679, A278340, A288126, A298330, A298858, A298939, A299031.

Programs

  • Maple
    b:= proc(n, t) option remember; local i; if n=0 then
      `if`(t=0, 1, 0) elif t<1 then 0 else 0;
      for i while i^2<=n do %+b(n-i^2, t-1) od; % fi
    end:
a:= n-> b(n*(n+1)/2, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 05 2018
  • Mathematica
    Table[SeriesCoefficient[(-1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n (n + 1)/2}], {n, 0, 25}]

Formula

a(n) = [x^(n*(n+1)/2)] (Sum_{k>=1} x^(k^2))^n.

A299031 Number of ordered ways of writing n-th triangular number as a sum of n squares of nonnegative integers.

Original entry on oeis.org

1, 1, 0, 3, 18, 60, 252, 1576, 10494, 64152, 458400, 3407019, 27713928, 225193982, 1980444648, 17626414158, 165796077562, 1593587604441, 15985672426992, 163422639872978, 1729188245991060, 18743981599820280, 208963405365941380, 2378065667103672024, 27742569814633730608
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 01 2018

Keywords

Examples

			a(3) = 3 because third triangular number is 6 and we have [4, 1, 1], [1, 4, 1] and [1, 1, 4].

Links

Crossrefs

Cf. A000217, A000290, A066535, A072964, A104383, A126683, A196010, A224677, A224679, A278340, A288126, A298329, A298858, A298938, A299032.

Programs

  • Mathematica
    Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n (n + 1)/2}], {n, 0, 24}]

Formula

a(n) = [x^(n*(n+1)/2)] (Sum_{k>=0} x^(k^2))^n.

A331900 Number of compositions (ordered partitions) of the n-th triangular number into distinct triangular numbers.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 3, 13, 3, 55, 201, 159, 865, 1803, 7093, 43431, 14253, 22903, 130851, 120763, 1099693, 4527293, 4976767, 7516897, 14349685, 72866239, 81946383, 167841291, 897853735, 455799253, 946267825, 5054280915, 3941268001, 17066300985, 49111862599
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 31 2020

Keywords

Examples

			a(6) = 3 because we have [21], [15, 6] and [6, 15].

Links

Crossrefs

Cf. A000217, A072964, A126683, A196010, A224677, A288126, A298858, A307614, A331843, A331884.

Programs

  • Maple
    b:= proc(n, i, p) option remember; (t->
      `if`(t*(i+2)/3n, 0, b(n-t, i-1, p+1)))))((i*(i+1)/2))
    end:
a:= n-> b(n*(n+1)/2, n, 0):
seq(a(n), n=0..37);  # Alois P. Heinz, Jan 31 2020
  • Mathematica
    b[n_, i_, p_] := b[n, i, p] = With[{t = i(i+1)/2}, If[t(i+2)/3 < n, 0, If[n == 0, p!, b[n, i-1, p] + If[t > n, 0, b[n-t, i-1, p+1]]]]];
a[n_] := b[n(n+1)/2, n, 0];
a /@ Range[0, 37] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

Formula

a(n) = A331843(A000217(n)).

Extensions

More terms from Alois P. Heinz, Jan 31 2020
Showing 1-6 of 6 results.

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