A061317 Split positive integers into extending even groups and sum: 1+2, 3+4+5+6, 7+8+9+10+11+12, 13+14+15+16+17+18+19+20, ...

0, 3, 18, 57, 132, 255, 438, 693, 1032, 1467, 2010, 2673, 3468, 4407, 5502, 6765, 8208, 9843, 11682, 13737, 16020, 18543, 21318, 24357, 27672, 31275, 35178, 39393, 43932, 48807, 54030, 59613, 65568, 71907, 78642, 85785, 93348, 101343, 109782

Offset: 0

Author

Henry Bottomley, Feb 13 2002

Keywords

• nonn
• easy

Comments

5*a(n+1) is the sum of the products of the 10 distinct combinations of three consecutive numbers starting with n (using 1,2,3 the 10 combinations are 111 112 113 122 123 133 222 223 233 333; 1*1*1 + 1*1*2 + 1*1*3 + 1*2*2 + 1*2*3 + 1*3*3 + 2*2*2 + 2*2*3 + 2*3*3 + 3*3*3 = 90 = 5*a(2)). - J. M. Bergot, Mar 28 2014 [expanded by Jon E. Schoenfield, Feb 22 2015]

Examples

			1+2 = 3; 3+4+5+6 = 18; 7+8+9+10+11+12 = 57; 13+14+15+16+17+18+19+20 = 132.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A005898, A006003.

Cf. A000217, A002378, A110450.

Programs

Maple

A061317:=n->2*n^3+n; seq(A061317(n), n=0..100); # Wesley Ivan Hurt, Mar 20 2014

Mathematica

Table[2n^3+n,{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Dec 04 2010 *)
LinearRecurrence[{4,-6,4,-1},{0,3,18,57},40] (* Harvey P. Dale, Aug 23 2015 *)
With[{nn=40},Total/@TakeList[Range[nn+nn^2],2Range[0,nn]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Mar 10 2018 *)

PARI

a(n) = { 2*n^3 + n } \\ Harry J. Smith, Jul 21 2009

Formula

a(n) = 2*n^3 + n.

a(n) = A000217(A002378(n)) - A000217(A002378(n-1)).

a(n) = 3 * A005900(n).

a(n) = A001477(n) * A058331(n).

a(n) = A000578(n) + A034262(n).

G.f.: 3*x*(1+x)^2/(x-1)^4.

a(n) = A110450(n) - A110450(n-1). - J.S. Seneschal, Jul 01 2025