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 1-3 of 3 results.

A238622 Number of partitions of n such that floor(n/2) or ceiling(n/2) is a part.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 7, 5, 11, 7, 17, 11, 25, 15, 36, 22, 51, 30, 71, 42, 97, 56, 132, 77, 177, 101, 235, 135, 310, 176, 406, 231, 527, 297, 681, 385, 874, 490, 1116, 627, 1418, 792, 1793, 1002, 2256, 1255, 2829, 1575, 3532, 1958, 4393, 2436, 5445, 3010, 6727
Offset: 1

Views

Author

Clark Kimberling, Mar 02 2014

Keywords

Examples

			a(7) counts these partitions:  43, 421, 4111, 331, 322, 3211, 31111.

Crossrefs

Cf. A238623, A238624.

Programs

  • Mathematica
    z=40; g[n_] := g[n] = IntegerPartitions[n];
t1 = Table[Count[g[n], p_ /; Or[MemberQ[p, Floor[n/2]], MemberQ[p, Ceiling[n/2]]]], {n, z}]   (* A238622 [or] *)
t2 = Table[Count[g[n], p_ /; Nor[MemberQ[p, Floor[n/2]], MemberQ[p, Ceiling[n/2]]]], {n, z}]   (* A238623 [nor] *)
t3 = Table[Count[g[n], p_ /; Xnor[MemberQ[p, Floor[n/2]], MemberQ[p, Ceiling[n/2]]]], {n, z}]   (* A238624 [xnor] *)

Formula

a(n) + A238623(n) = A000041(n).

A238623 Number of partitions of n such that neither floor(n/2) nor ceiling(n/2) is a part.

Original entry on oeis.org

0, 1, 1, 3, 3, 8, 8, 17, 19, 35, 39, 66, 76, 120, 140, 209, 246, 355, 419, 585, 695, 946, 1123, 1498, 1781, 2335, 2775, 3583, 4255, 5428, 6436, 8118, 9616, 12013, 14202, 17592, 20763, 25525, 30069, 36711, 43165, 52382, 61468, 74173, 86878, 104303, 121925
Offset: 1

Views

Author

Clark Kimberling, Mar 02 2014

Keywords

Examples

			a(7) counts these 8 partitions:  7, 61, 52, 511, 2221, 22111, 211111, 1111111.

Crossrefs

Cf. A238622, A238624.

Programs

  • Mathematica
    z=40; g[n_] := g[n] = IntegerPartitions[n];
t1 = Table[Count[g[n], p_ /; Or[MemberQ[p, Floor[n/2]], MemberQ[p, Ceiling[n/2]]]], {n, z}]   (* A238622 [or] *)
t2 = Table[Count[g[n], p_ /; Nor[MemberQ[p, Floor[n/2]], MemberQ[p, Ceiling[n/2]]]], {n, z}]   (* A238623 [nor] *)
t3 = Table[Count[g[n], p_ /; Xnor[MemberQ[p, Floor[n/2]], MemberQ[p, Ceiling[n/2]]]], {n, z}]   (* A238624 [xnor] *)

Formula

a(n) + A238622(n) = A000041(n).

A238625 Number of partitions p of n such that 1 + (1/2)*max(p) is a part of p.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 5, 6, 9, 11, 14, 19, 24, 31, 41, 51, 65, 84, 105, 132, 167, 207, 257, 321, 395, 486, 599, 731, 892, 1089, 1319, 1597, 1933, 2327, 2798, 3361, 4021, 4805, 5736, 6825, 8109, 9625, 11393, 13469, 15905, 18738, 22049, 25915, 30401, 35620
Offset: 1

Views

Author

Clark Kimberling, Mar 02 2014

Keywords

Examples

			a(6) counts these partitions:  222, 2211, 21111.

Links

Crossrefs

Cf. A238479, A238624.

Programs

  • Mathematica
    Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 1 + Max[p]/2]], {n, 50}]
p[n_, m_] := If[m > n, 0, If[n == m, 1, p[n, m] = Sum[p[n - m, j], {j, m}]]]; a[1] = 0; a[n_] := 1 + Sum[p[n-k-1, 2*k], {k, n/2}]; Array[a,100] (* Giovanni Resta, Mar 07 2014 *)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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