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 31-34 of 34 results.

A301502 Number of compositions (ordered partitions) of n into triangular parts (A000217) such that no two adjacent parts are equal (Carlitz compositions).

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 1, 3, 3, 3, 7, 9, 6, 10, 20, 20, 20, 36, 50, 54, 75, 109, 126, 156, 233, 302, 352, 480, 676, 838, 1053, 1447, 1896, 2374, 3152, 4225, 5368, 6923, 9297, 12133, 15472, 20353, 26959, 34779, 45092, 59551, 77717, 100475, 131714, 172949, 224316, 291987, 383418
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 22 2018

Keywords

Examples

			a(12) = 6 because we have [3, 6, 3], [3, 1, 3, 1, 3, 1], [1, 10, 1], [1, 6, 1, 3, 1], [1, 3, 1, 6, 1] and [1, 3, 1, 3, 1, 3].

Links

Crossrefs

Cf. A000217, A003242, A023361, A301503.

Programs

  • Mathematica
    nmax = 52; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1)/2)/(1 + x^(k (k + 1)/2)), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k>=1} x^(k*(k+1)/2)/(1 + x^(k*(k+1)/2))).

A303908 Expansion of 1/(2 + x - theta_2(sqrt(x))/(2*x^(1/8))), where theta_2() is the Jacobi theta function.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 2, 0, 0, 3, 1, 0, 5, 2, 0, 9, 5, 0, 15, 10, 1, 27, 20, 3, 46, 40, 9, 80, 78, 22, 139, 152, 51, 242, 290, 114, 427, 550, 247, 753, 1034, 525, 1340, 1933, 1092, 2396, 3602, 2237, 4312, 6685, 4519, 7813, 12380, 9027, 14239, 22877, 17866, 26110, 42214, 35072, 48123, 77829, 68379
Offset: 0

Views

Author

Ilya Gutkovskiy, May 02 2018

Keywords

Comments

Number of compositions (ordered partitions) of n into triangular numbers > 1.

Links

Crossrefs

Cf. A000217, A023361, A212804, A280542, A303668, A303906, A303907.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
      add(a(n-j*(j+1)/2), j=2..isqrt(2*n))))
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, May 02 2018
  • Mathematica
    nmax = 62; CoefficientList[Series[1/(2 + x - EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8))), {x, 0, nmax}], x]
nmax = 62; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1)/2), {k, 2, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[SquaresR[1, 8 k + 1] a[n - k], {k, 2, n}]/2; Table[a[n], {n, 0, 62}]

Formula

G.f.: 1/(1 - Sum_{k>=2} x^(k*(k+1)/2)).

A332005 Number of compositions (ordered partitions) of n into distinct centered triangular numbers.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 0, 0, 0, 0, 1, 2, 0, 0, 2, 6, 0, 0, 0, 1, 2, 0, 0, 2, 6, 0, 0, 0, 0, 2, 6, 1, 2, 6, 24, 2, 6, 0, 0, 0, 0, 2, 6, 0, 0, 6, 25, 2, 0, 0, 4, 12, 0, 0, 6, 24, 2, 6, 0, 0, 12, 48, 0, 0, 25, 124, 6, 0, 2, 12, 24, 0, 0, 0, 2, 12, 24, 2, 12, 48, 120, 6, 24, 2, 6, 1, 2, 12
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2020

Keywords

Examples

			a(15) = 6 because we have [10, 4, 1], [10, 1, 4], [4, 10, 1], [4, 1, 10], [1, 10, 4] and [1, 4, 10].

Links

Crossrefs

Cf. A005448, A023361, A280950, A281081, A282502, A331843, A331919, A332006.

A369341 Number of compositions (ordered partitions) of n into triangular numbers not greater than sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745, 4023, 5896, 8641, 12664, 18560, 27201, 39865, 58425, 85626, 125491, 183916, 269542, 395033, 2823330, 4343681, 6682741, 10281375, 15817857, 24335721, 37440426, 57601964
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 20 2024

Keywords

Crossrefs

Cf. A023361, A364526.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 - Sum[x^(k (k + 1)/2), {k, 1, Floor[(Sqrt[1 + 8 Sqrt[n]] - 1)/2]}]), {x, 0, n}], {n, 0, 43}]
Previous Showing 31-34 of 34 results.

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