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.

A339365 Number of partitions of n into an odd number of squares.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 1, 1, 1, 3, 1, 3, 2, 3, 3, 3, 4, 5, 4, 6, 5, 7, 7, 7, 8, 10, 9, 12, 10, 14, 13, 14, 15, 18, 17, 21, 20, 24, 24, 25, 27, 31, 30, 35, 34, 40, 40, 42, 45, 50, 50, 56, 55, 64, 65, 68, 72, 78, 79, 88, 85, 99, 99, 105, 110, 118, 122, 131, 132, 146, 149
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 01 2020

Keywords

Examples

			a(9) = 3 because we have [9], [4, 4, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000290, A001156, A027193, A292520, A339364, A339366, A339367.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[(1/2) (Product[1/(1 - x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}] - Product[1/(1 + x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (Product_{k>=1} 1 / (1 - x^(k^2)) - Product_{k>=1} 1 / (1 + x^(k^2))).
a(n) = (A001156(n) - A292520(n)) / 2.

A339367 Number of partitions of n into an odd number of distinct squares.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 2, 0, 1, 1, 2, 0, 0, 2, 1, 0, 0, 0, 2, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 0, 2, 1, 1, 2, 3
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 01 2020

Keywords

Examples

			a(49) = 2 because we have [49] and [36, 9, 4].

Links

Crossrefs

Cf. A000290, A033461, A067659, A276516, A339364, A339365, A339366.

Programs

  • Mathematica
    nmax = 90; CoefficientList[Series[(1/2) (Product[(1 + x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}] - Product[(1 - x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (Product_{k>=1} (1 + x^(k^2)) - Product_{k>=1} (1 - x^(k^2))).
a(n) = (A033461(n) - A276516(n)) / 2.

A339366 Number of partitions of n into an even number of distinct squares.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 3, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 3, 1, 1, 1, 1, 0, 1, 3, 1, 1, 0, 1, 3
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 01 2020

Keywords

Examples

			a(50) = 2 because we have [49, 1] and [36, 9, 4, 1].

Links

Crossrefs

Cf. A000290, A033461, A067661, A276516, A339364, A339365, A339367.

Programs

  • Mathematica
    nmax = 90; CoefficientList[Series[(1/2) (Product[(1 + x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}] + Product[(1 - x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (Product_{k>=1} (1 + x^(k^2)) + Product_{k>=1} (1 - x^(k^2))).
a(n) = (A033461(n) + A276516(n)) / 2.

A339373 Number of partitions of n into an even number of triangular numbers.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 1, 3, 2, 4, 3, 6, 5, 6, 6, 10, 7, 13, 10, 15, 13, 20, 15, 26, 21, 28, 26, 36, 31, 44, 42, 49, 50, 61, 57, 75, 73, 84, 85, 103, 97, 123, 121, 137, 140, 166, 159, 194, 194, 216, 225, 256, 253, 295, 304, 330, 346, 389, 387, 446, 456, 498, 516, 579, 576
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 02 2020

Keywords

Examples

			a(6) = 3 because we have [3, 3], [3, 1, 1, 1] and [1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000217, A007294, A027187, A292519, A339364, A339374, A339375, A339376.

Programs

  • Mathematica
    nmax = 65; CoefficientList[Series[(1/2) (Product[1/(1 - x^(k (k + 1)/2)), {k, 1, nmax}] + Product[1/(1 + x^(k (k + 1)/2)), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (Product_{k>=1} 1 / (1 - x^(k*(k + 1)/2)) + Product_{k>=1} 1 / (1 + x^(k*(k + 1)/2))).
a(n) = (A007294(n) + A292519(n)) / 2.

A339418 Number of compositions (ordered partitions) of n into an even number of squares.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 1, 4, 2, 6, 9, 8, 20, 16, 35, 44, 55, 102, 105, 196, 242, 344, 540, 652, 1084, 1380, 2037, 2964, 3912, 6042, 7976, 11776, 16634, 22968, 33963, 46156, 67457, 94510, 133180, 192316, 266514, 385338, 540138, 767008, 1094576, 1534704, 2200821, 3094248
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 03 2020

Keywords

Examples

			a(9) = 6 because we have [4, 1, 1, 1, 1, 1], [1, 4, 1, 1, 1, 1], [1, 1, 4, 1, 1, 1], [1, 1, 1, 4, 1, 1], [1, 1, 1, 1, 4, 1] and [1, 1, 1, 1, 1, 4].

Links

Crossrefs

Cf. A000290, A006456, A034008, A317665, A339364, A339416, A339419.

Programs

  • Maple
    b:= proc(n, t) option remember; local r, f, g;
      if n=0 then t else r, f, g:=$0..2; while f<=n
      do r, f, g:= r+b(n-f, 1-t), f+2*g-1, g+1 od; r fi
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 03 2020
  • Mathematica
    nmax = 47; CoefficientList[Series[4/(3 + 2 EllipticTheta[3, 0, x] - EllipticTheta[3, 0, x]^2), {x, 0, nmax}], x]

Formula

G.f.: 4 / (3 + 2 * theta_3(x) - theta_3(x)^2), where theta_3() is the Jacobi theta function.
a(n) = (A006456(n) + A317665(n)) / 2.
a(n) = Sum_{k=0..n} A006456(k) * A317665(n-k).

A339368 Number of partitions of n into an even number of cubes.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 4, 2, 4, 3, 4, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 4, 6, 4, 6, 5, 6, 5, 7, 5, 7, 6, 7, 6, 8, 6, 8, 7, 9, 8, 9, 9, 9, 9, 10, 9, 11, 10, 11, 10, 12, 10, 12, 11, 13, 12, 14, 13, 14, 13
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 01 2020

Keywords

Examples

			a(16) = 2 because we have [8, 8] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000578, A003108, A027187, A292560, A339364, A339369.

Programs

  • Mathematica
    nmax = 85; CoefficientList[Series[(1/2) (Product[1/(1 - x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}] + Product[1/(1 + x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (Product_{k>=1} 1 / (1 - x^(k^3)) + Product_{k>=1} 1 / (1 + x^(k^3))).
a(n) = (A003108(n) + A292560(n)) / 2.
Showing 1-6 of 6 results.

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