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.

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A340445 Number of partitions of n into 3 parts that are not all the same.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 2, 4, 5, 6, 8, 10, 11, 14, 16, 18, 21, 24, 26, 30, 33, 36, 40, 44, 47, 52, 56, 60, 65, 70, 74, 80, 85, 90, 96, 102, 107, 114, 120, 126, 133, 140, 146, 154, 161, 168, 176, 184, 191, 200, 208, 216, 225, 234, 242, 252, 261, 270, 280, 290, 299, 310, 320, 330
Offset: 0

Views

Author

Wesley Ivan Hurt, Jan 07 2021

Keywords

Comments

Conjecturally the same as A230059 (apart from the offset). - R. J. Mathar, Jan 14 2021

Examples

			a(6) = 2; [4,1,1], [3,2,1] ( [2,2,2] not counted ),
a(7) = 4; [5,1,1], [4,2,1], [3,3,1], [3,2,2],
a(8) = 5; [6,1,1], [5,2,1], [4,3,1], [4,2,2], [3,3,2],
a(9) = 6; [7,1,1], [6,2,1], [5,3,1], [4,4,1], [5,2,2], [4,3,2] ( [3,3,3] not counted ).

Links

Crossrefs

Cf. A036410, A069905.

Programs

  • Mathematica
    Table[Sum[Sum[(1 - KroneckerDelta[i, k, n - i - k]), {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 0, 80}]

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1 - [k = i = n-i-k]), where [ ] is the (generalized) Iverson bracket.
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1 - [k = i] * [2*i = n-k] * [2*k = n-i]), where [ ] is the Iverson bracket.
From Alois P. Heinz, Jan 07 2021: (Start)
G.f.: x^4*(x^2-x-1)/((x+1)*(x^2+x+1)*(x-1)^3).
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6), n>6. (End)
a(n) = A036410(n-1)-1. - Hugo Pfoertner, Jan 09 2021
a(n) + A079978(n) = A069905(n), n>0. - R. J. Mathar, Jan 18 2021
72*a(n) = -16*A099837(n+3) -9*(-1)^n +6*n^2 -31. - R. J. Mathar, Jun 09 2022

A094909 Let p_k(n) = number of partitions of n into exactly k parts; sequence gives p_3(n-3) + p_2(n-2) + 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 4, 6, 7, 9, 10, 13, 14, 17, 19, 22, 24, 28, 30, 34, 37, 41, 44, 49, 52, 57, 61, 66, 70, 76, 80, 86, 91, 97, 102, 109, 114, 121, 127, 134, 140, 148, 154, 162, 169, 177, 184, 193, 200, 209, 217, 226, 234, 244, 252, 262, 271, 281, 290, 301, 310, 321
Offset: 0

Views

Author

N. J. A. Sloane, Jun 18 2004

Keywords

Links

Crossrefs

p_k(n) = A008284(n,k).

Programs

  • PARI
    Vec((x^7-x^6-x^5-x^4+x^3+x^2-1)/((1+x)*(x^2+x+1)*(x-1)^3) + O(x^80)) \\ Michel Marcus, Jul 19 2015

Formula

G.f.: (x^7-x^6-x^5-x^4+x^3+x^2-1)/((1+x)*(x^2+x+1)*(x-1)^3). - Alois P. Heinz, Jul 19 2015
72*a(n) = 8*A099837(n+3) +27*(-1)^n +29 +6*n^2, (n>1). - R. J. Mathar, Nov 15 2019

Extensions

a(0)=1 prepended by Alois P. Heinz, Jul 19 2015

A107351 Expansion of (1+x^3)/((1-x)^3*(1-x^2)^3*(1-x^3)).

Original entry on oeis.org

1, 3, 9, 21, 45, 87, 159, 273, 450, 712, 1092, 1626, 2364, 3360, 4686, 6420, 8661, 11517, 15121, 19617, 25179, 31995, 40287, 50295, 62298, 76596, 93534, 113482, 136860, 164118, 195762, 232332, 274431, 322701, 377853, 440643, 511903, 592515, 683445, 785715
Offset: 0

Views

Author

N. J. A. Sloane, May 23 2005

Keywords

References

  • Y. Teranishi, Linear Diophantine equations and invariant theory of matrices, in Commutative algebra and combinatorics (Kyoto, 1985), pp. 259-275, Adv. Stud. Pure Math., 11, North-Holland, Amsterdam, 1987.

Links

Formula

G.f.: ( -1+x-x^2 ) / ( (1+x+x^2)*(1+x)^2*(x-1)^7 ). - R. J. Mathar, Dec 18 2014
81*a(n) = 2*A099837(n+3) +(134946*n +22680*n^3 +77688*n^2 +3690*n^4 +324*n^5 +12*n^6 +87625)/1280 +(-1)^n*243 *(9+2*n)/256 . - R. J. Mathar, Dec 17 2017
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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