A090864 Complement of generalized pentagonal numbers (A001318).

3, 4, 6, 8, 9, 10, 11, 13, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84

Offset

1,1

Comments

Also n for which A006906(n) is even, or equivalently n for which A000009(n) is even (since A006906 and A000009 have the same parity).

The number of partitions of a(n) into distinct parts with an even number of parts equals the number of such partitions with an odd number of parts: A067661(a(n)) = A067659(a(n)). See, e.g., the Freitag-Busam reference, p. 410 given in A036499. - Wolfdieter Lang, Jan 19 2016

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

A080995(a(n)) = 0; A000009(a(n)) = A118303(n). - Reinhard Zumkeller, Apr 22 2006

A010815(a(n)) = A067661(a(n)) - A067659(a(n)) = 0, n >= 1. See a comment above. - Wolfdieter Lang, Jan 19 2016

a(n) = n+1 + A085141(n-1) + A111651(n). - Kevin Ryde, Sep 15 2024

Mathematica

Complement[Range[200], Select[Accumulate[Range[0, 200]]/3, IntegerQ]] (* G. C. Greubel, Jun 06 2017 *)

Prog

(Python)
from math import isqrt
def A090864(n):
    def f(x): return n+(m:=isqrt(24*x+1)+1)//6+(m-2)//6
    kmin, kmax = 0, 1
    while f(kmax) > kmax:
        kmax <<= 1
    while kmax-kmin > 1:
        kmid = kmax+kmin>>1
        if f(kmid) <= kmid:
            kmax = kmid
        else:
            kmin = kmid
    return kmax # Chai Wah Wu, Aug 29 2024
(PARI) a(n) = my(q, r); [q, r]=divrem(sqrtint(24*n), 3); n + q + (r >= bitnegimply(1, q)); \\ Kevin Ryde, Sep 15 2024

Crossrefs

Cf. A000009, A001318, A067659, A067661, A080995, A006906, A010815, A118303.

Cf. A085141, A111651.

Sequence in context: A080702 A156167 A376560 * A118300 A263098 A317391

Adjacent sequences: A090861 A090862 A090863 * A090865 A090866 A090867

Keyword

nonn,easy

Author

Jon Perry, Feb 12 2004

Extensions

More terms from Reinhard Zumkeller, Apr 22 2006

Edited by Ray Chandler, Dec 14 2011

Edited by Jon E. Schoenfield, Nov 25 2016

Status

approved