A110344 a(n) = Sum_{k=0..n-1} (n+k) = n(3n-1)/2 if n is even; a(n) = Sum_{k=0..n-1} (n-k) = n(n+1)/2 if n is odd.

1, 5, 6, 22, 15, 51, 28, 92, 45, 145, 66, 210, 91, 287, 120, 376, 153, 477, 190, 590, 231, 715, 276, 852, 325, 1001, 378, 1162, 435, 1335, 496, 1520, 561, 1717, 630, 1926, 703, 2147, 780, 2380, 861, 2625, 946, 2882, 1035, 3151, 1128, 3432, 1225, 3725, 1326

Offset: 1

Author

Amarnath Murthy, Jul 20 2005

Keywords

• nonn
• easy

Examples

			a(3) = 3 + 2 +1 = 6.
a(6) = 6 + 7 + 8 + 9 + 10 + 11 = 51.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Cf. A000217, A000326, A049452.

Programs

Maple

a:=proc(n) if n mod 2=0 then n*(3*n-1)/2 else n*(n+1)/2 fi end: seq(a(n),n=1..60); # Emeric Deutsch

Mathematica

a[n_] := n*(2*n + (n - 1)*(-1)^n)/2; Array[a, 50] (* Amiram Eldar, Sep 11 2022 *)

PARI

Vec(-x*(7*x^3+3*x^2+5*x+1)/((x-1)^3*(x+1)^3) + O(x^100)) \\ Colin Barker, Feb 17 2015

Formula

From Emeric Deutsch, Aug 01 2005: (Start)

a(2n+1) = A000217(2n+1) = (n+1)(2n+1) (triangular numbers with odd index).

a(2n) = A000326(2n) = A049452(n) = n(6n-1) (pentagonal numbers with even index).

(End)

a(n) = n*( 2*n + (n-1)*(-1)^n )/2. - Luce ETIENNE, Jul 08 2014

From Colin Barker, Feb 17 2015: (Start)

a(n) = 3*a(n-2)-3*a(n-4)+a(n-6).

G.f.: -x*(7*x^3+3*x^2+5*x+1) / ((x-1)^3*(x+1)^3). (End)

Sum_{n>=1} 1/a(n) = 4*log(2) + 3*log(3)/2 - sqrt(3)*Pi/2. - Amiram Eldar, Sep 11 2022

Extensions

More terms from Emeric Deutsch, Aug 01 2005