A052276 Nonnegative numbers of the form n^3 (+/-) 3, n >= 0.

3, 4, 5, 11, 24, 30, 61, 67, 122, 128, 213, 219, 340, 346, 509, 515, 726, 732, 997, 1003, 1328, 1334, 1725, 1731, 2194, 2200, 2741, 2747, 3372, 3378, 4093, 4099, 4910, 4916, 5829, 5835, 6856, 6862, 7997, 8003, 9258, 9264, 10645, 10651

Offset: 1

Author

N. J. A. Sloane, Feb 05 2000

Keywords

• nonn
• easy

Comments

It is conjectured that A003325 and A052276 (the current sequence) have infinitely many numbers in common, although only one example (128) is known.

The next such example must be larger than 2*10^12. - M. F. Hasler, Jan 10 2021

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Programs

Maple

3,4,op(map(n -> (n^3-3,n^3+3), [$2..100])); # Robert Israel, Jul 09 2015

PARI

Vec(x*(5*x^7-8*x^6-9*x^5+19*x^4+3*x^3-8*x^2+x+3)/((x-1)^4*(x+1)^3) + O(x^100)) \\ Colin Barker, Jul 09 2015

PARI

apply( {A052276(n)=(n\/2)^3+3*(-1)^n+(n==1)*5}, [1..99]) \\ M. F. Hasler, Jan 10 2021

Python

def A052276(n): return (n+1>>1)**3+(-3 if n&1 else 3) if n>1 else 3 # Chai Wah Wu, Jun 27 2025

Formula

a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7) for n>8. - Colin Barker, Jul 09 2015

G.f.: x*(5*x^7-8*x^6-9*x^5+19*x^4+3*x^3-8*x^2+x+3) / ((x-1)^4*(x+1)^3). - Colin Barker, Jul 09 2015

a(n) = ((2*n+1)*(n^2+n+1) - (-1)^n*(3*n^2+3*n-47))/16 for n >= 2. - Robert Israel, Jul 09 2015

a(n) = ceiling(n/2)^3 + 3*(-1)^n for all n > 1. - M. F. Hasler, Jan 10 2021