A014112 a(n) = n*(n-1) + (n-2)*(n-3) + ... + 1*0 + 1 for n odd; otherwise, a(n) = n*(n-1) + (n-2)*(n-3) + ... + 2*1.

1, 2, 7, 14, 27, 44, 69, 100, 141, 190, 251, 322, 407, 504, 617, 744, 889, 1050, 1231, 1430, 1651, 1892, 2157, 2444, 2757, 3094, 3459, 3850, 4271, 4720, 5201, 5712, 6257, 6834, 7447, 8094, 8779, 9500, 10261, 11060, 11901, 12782, 13707, 14674, 15687

Offset: 1

Author

Jon Wild, Jul 14 1997

Keywords

• nonn
• easy

Examples

			From _Bruno Berselli_, Mar 12 2018: (Start)
n=1: 1;
n=2: 1*2;
n=3: 1 + 0*1 + 2*3 = 7;
n=4: 1*2 + 3*4 = 14;
n=5: 1 + 0*1 + 2*3 + 4*5 = 27;
n=6: 1*2 + 3*4 + 5*6 = 44;
n=7: 1 + 0*1 + 2*3 + 4*5 + 6*7 = 69, etc.
(End)
		

Links

• Delbert L. Johnson, Table of n, a(n) for n = 1..20000
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Crossrefs

Cf. A064999, A178218 (first differences).

Programs

Magma

[n le 2 select n else Self(n-2)+n*(n-1):n in [1..50]]; // Vincenzo Librandi, Feb 28 2016
(C#) public BigInteger a(BigInteger n) => (n * (n + 2) * (2 * n - 1) + 9) / 12; // Delbert L. Johnson, Mar 19 2023

Mathematica

LinearRecurrence[{3, -2, -2, 3, -1}, {1, 2, 7, 14, 27}, 50] (* Vincenzo Librandi, Feb 28 2016 *)

Formula

a(n) = a(n-2) + n*(n-1) for n > 2, a(1)=1, a(2)=2.

G.f.: x*(1 - x^3 + 3*x^2 - x)/((x + 1)*(x - 1)^4). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009

a(n) + a(n+1) = A064999(n). - R. J. Mathar, Feb 27 2016

a(n) = n*(n + 2)*(2*n - 1)/12 + 3*(1 - (-1)^n)/8. - Bruno Berselli, Mar 12 2018

Extensions

More terms from Erich Friedman