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.

Previous Showing 31-33 of 33 results.

A331983 Number of compositions (ordered partitions) of n into distinct squares > 1.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 8, 0, 0, 0, 0, 2, 0, 1, 0, 6, 0, 2, 2, 0, 0, 0, 8, 0, 0, 0, 7, 6, 0, 2, 2, 24, 0, 6, 0, 2, 0, 0, 8, 6, 0, 1, 32, 0, 0, 2, 6, 6, 0, 0, 2, 32, 0, 0, 12, 30, 0, 2
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 03 2020

Keywords

Examples

			a(25) = 3 because we have [25], [16, 9] and [9, 16].

Links

Crossrefs

Cf. A000290, A006456, A032020, A032021, A032022, A033461, A078134, A280129, A280542, A331844, A331918.

Programs

  • Maple
    b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`(i*(i+1)*(2*i+1)/6-1n, 0, b(n-i^2, i-1, p+1))+b(n, i-1, p)))
    end:
a:= n-> b(n, isqrt(n), 0):
seq(a(n), n=0..87);  # Alois P. Heinz, Feb 03 2020
  • Mathematica
    b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i(i+1)(2i+1)/6 - 1 < n, 0, If[i^2 > n, 0, b[n - i^2, i - 1, p + 1]] + b[n, i - 1, p]]];
a[n_] := b[n, Floor@Sqrt[n], 0];
a /@ Range[0, 87] (* Jean-François Alcover, Nov 26 2020, after Alois P. Heinz *)

A294071 Number of ordered ways of writing n^2 as a sum of n squares > 1.

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 6, 7, 288, 262, 13702, 69531, 610567, 5356091, 51724960, 521956086, 5467658641, 59931636545, 690518644584, 8100858045744, 99142980567486, 1246972499954475, 16142015005905558, 215722810653380845, 2955759897694815985, 41614888439136252691
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 07 2018

Keywords

Examples

			a(5) = 5 because we have [9, 4, 4, 4, 4], [4, 9, 4, 4, 4], [4, 4, 9, 4, 4], [4, 4, 4, 9, 4] and [4, 4, 4, 4, 9].

Links

Crossrefs

Cf. A000290, A037444, A066535, A078134, A092362, A232173, A280129, A280542, A281154, A281155, A298329, A298330, A298640, A298642.

Programs

  • Mathematica
    Table[SeriesCoefficient[((-1 - 2 x + EllipticTheta[3, 0, x])/2)^n, {x, 0, n^2}], {n, 0, 25}]

Formula

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

A303909 Expansion of 2*(1 - x)/(3 - theta_3(x)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 2, 4, 5, 6, 8, 13, 19, 26, 36, 51, 74, 105, 148, 208, 296, 421, 597, 846, 1198, 1699, 2409, 3417, 4843, 6865, 9732, 13799, 19566, 27739, 39325, 55749, 79041, 112063, 158877, 225241, 319331, 452734, 641866, 910001, 1290137, 1829079, 2593169, 3676457, 5212266
Offset: 0

Views

Author

Ilya Gutkovskiy, May 02 2018

Keywords

Comments

First differences of A006456.

Links

Crossrefs

Cf. A000290, A006456, A078134, A303667.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n<0, 0,
      `if`(n=0, 1, add(b(n-j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n)-`if`(n=0, 0, b(n-1)):
seq(a(n), n=0..60);  # Alois P. Heinz, May 02 2018
  • Mathematica
    nmax = 50; CoefficientList[Series[2 (1 - x)/(3 - EllipticTheta[3, 0, x]), {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[(1 - x)/(1 - Sum[x^k^2, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Boole[IntegerQ[k^(1/2)]] a[n - k], {k, 1, n}]; Differences[Table[a[n], {n, -1, 50}]]

Formula

G.f.: (1 - x)/(1 - Sum_{k>=1} x^(k^2)).
Previous Showing 31-33 of 33 results.

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