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 1-4 of 4 results.

A092362 Number of partitions of n^2 into squares greater than 1.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 5, 8, 11, 28, 44, 94, 167, 354, 643, 1314, 2412, 4792, 8981, 17374, 32566, 62008, 115702, 217040, 402396, 745795, 1372266, 2517983, 4595652, 8354350, 15125316, 27265107, 48972467, 87584837, 156119631, 277152178, 490437445, 864534950
Offset: 0

Views

Author

Reinhard Zumkeller, Mar 19 2004

Keywords

Comments

a(n) = A078134(A000290(n)).

Examples

			a(6) = 5: 6^2 = 36 = 16+16+4 = 16+4+4+4+4+4 = 9+9+9+9 = 4+4+4+4+4+4+4+4+4.

Links

Crossrefs

Cf. A001156, A037444.
Cf. A093115, A093116.

Programs

  • Maple
    b:=proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<2, 0, b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i))))
   end:
a:= n-> b(n^2, n):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 15 2013
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<2, 0, b[n, i-1] + If[i^2>n, 0, b[n-i^2, i]]]]; a[n_] := b[n^2, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

Formula

a(n) ~ exp(3*Pi^(1/3) * Zeta(3/2)^(2/3) * n^(2/3) / 2^(4/3)) * Zeta(3/2)^(4/3) / (2^(11/3) * sqrt(3) * Pi^(5/6) * n^(11/3)). - Vaclav Kotesovec, Apr 10 2017

Extensions

Corrected a(0) and more terms from Alois P. Heinz, Apr 15 2013

A093115 Number of partitions of n^2 into squares not greater than n.

Original entry on oeis.org

1, 1, 1, 1, 5, 7, 10, 13, 17, 108, 159, 228, 317, 430, 572, 748, 5753, 8125, 11266, 15376, 20672, 27430, 35942, 46575, 59717, 523905, 708028, 946875, 1253880, 1645224, 2140099, 2761318, 3535658, 4494602, 5674753, 7118724, 69766770, 90940578, 117756370
Offset: 0

Views

Author

Reinhard Zumkeller, Mar 21 2004

Keywords

Examples

			n=6: 6^2 = 9*2^2 = 8*2^2+4*1^2 = 7*2^2+8*1^2 = 6*2^2+12*1^2 = 5*2^2+16*1^2 = 4*2^2+20*1^2 = 3*2^2+24*1^2 = 2*2^2+28*1^2 = 1*2^2+32*1^2 = 36*1^2, therefore a(6)=10.

Links

Crossrefs

Cf. A093116, A092362, A001156, A037444, A078134.
Cf. A072925.
Cf. A072213, A161407. [Reinhard Zumkeller, Jun 10 2009]

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i))))
    end:
a:= proc(n) local r; r:= isqrt(n);
      b(n^2, r-`if`(r^2>n, 1, 0))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 15 2013
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i^2 > n, 0, b[n-i^2, i]]]]; a[n_] := (r = Sqrt[n] // Floor; b[n^2, r - If[r^2 > n, 1, 0]]); Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 29 2015, after Alois P. Heinz *)

Formula

Coefficient of x^(n^2) in the series expansion of Product_{k=1..floor(sqrt(n))} 1/(1 - x^(k^2)). - Vladeta Jovovic, Mar 24 2004

Extensions

More terms from Vladeta Jovovic, Mar 24 2004
Corrected a(0) by Alois P. Heinz, Apr 15 2013

A161408 Number of partitions of n^2 into parts greater than n.

Original entry on oeis.org

1, 0, 1, 2, 6, 17, 56, 179, 607, 2076, 7269, 25736, 92360, 334506, 1222463, 4499892, 16673446, 62130710, 232700539, 875483029, 3307244256, 12539455600, 47702381244, 182021195608, 696487788847, 2671877845634, 10274126646175, 39593295985708, 152889766657797
Offset: 0

Views

Author

Reinhard Zumkeller, Jun 10 2009

Keywords

Examples

			a(4) = #{16, 11+5, 10+6, 9+7, 8+8, 6+5+5} = 6.

Links

Crossrefs

Cf. A072213, A093116, A161407.

Programs

  • Maple
    a := proc (n) local G, Gser: G := 1/(product(1-x^j, j = n+1 .. n^2)): Gser := series(G, x = 0, n^2+5): coeff(Gser, x, n^2) end proc: 1, seq(a(n), n = 1 .. 27); # Emeric Deutsch, Jun 22 2009
  • Mathematica
    a[n_] := a[n] = 1/Product[1 - x^j, {j, n + 1, n^2}] + O[x]^(n^2 + 1) // CoefficientList[#, x]& // Last;
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 28}] (* Jean-François Alcover, May 18 2017, adapted from Maple *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 4.076293... and c = 0.187307... - Vaclav Kotesovec, Sep 08 2021

Extensions

Extended by Emeric Deutsch, Jun 22 2009
a(0)=1 from Alois P. Heinz, Dec 21 2014

A298642 Number of partitions of n^2 into distinct squares > 1.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 5, 2, 10, 4, 12, 12, 11, 19, 23, 43, 50, 55, 78, 120, 126, 234, 207, 407, 385, 701, 712, 1090, 1231, 1850, 2102, 3054, 3385, 4988, 5584, 7985, 9746, 12205, 15737, 18968, 25157, 30927, 39043, 47708, 61915, 74592, 99554
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 24 2018

Keywords

Examples

			a(5) = 2 because we have [25] and [16, 9].

Links

Crossrefs

Cf. A000290, A001156, A030273, A033461, A037444, A078134, A092362, A093115, A093116, A280129, A298640.

Formula

a(n) = [x^(n^2)] Product_{k>=2} (1 + x^(k^2)).
a(n) = A280129(A000290(n)).
Showing 1-4 of 4 results.

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