A057590 Numbers of the form k*(k^3 +- 1)/2.

0, 1, 7, 9, 39, 42, 126, 130, 310, 315, 645, 651, 1197, 1204, 2044, 2052, 3276, 3285, 4995, 5005, 7315, 7326, 10362, 10374, 14274, 14287, 19201, 19215, 25305, 25320, 32760, 32776, 41752, 41769, 52479, 52497, 65151, 65170, 79990, 80010, 97230, 97251, 117117

Offset: 1

Author

N. J. A. Sloane, Oct 05 2000

Keywords

• nonn
• easy

Comments

This sequence is also the integers k such that x^6 - 8*k*x^4 - 1 is reducible over the integers, and, for k > 0 such that the factors are two irreducible cubics. - James R. Buddenhagen, May 29 2010

Integer solutions of x + y = (x - y)^4. If x = a(n) then y = a(n - (-1)^n). - Thomas Scheuerle, Mar 06 2023

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Programs

Maple

map(k -> (k*(k^3-1)/2, k*(k^3+1)/2), [$1..100]); # Robert Israel, Apr 28 2015

PARI

concat(0, Vec(-x^2*(x^4+6*x^3-2*x^2+6*x+1)/((x-1)^5*(x+1)^4) + O(x^100))) \\ Colin Barker, Apr 28 2015

Formula

G.f.: -x^2*(x^4+6*x^3-2*x^2+6*x+1) / ((x-1)^5*(x+1)^4). - Colin Barker, Apr 28 2015

From Robert Israel, Apr 28 2015: (Start)

a(n) = (2*n^4 + 4*n^3 + 6*n^2 + 4*n - 7)/64 + (-1)^n * (7 - 2*n - 2*n^2)*(1 + 2*n)/64.

a(n+8) - 4*a(n+6) + 6*a(n+4) - 4*a(n+2) + a(n) = 12. (End)