A260513 a(n) = (8*n+13*n^3+3*n^5)/24; also the sum of triangular numbers taken in successive groups of increasing size (see Example).

1, 9, 46, 164, 460, 1091, 2289, 4376, 7779, 13045, 20856, 32044, 47606, 68719, 96755, 133296, 180149, 239361, 313234, 404340, 515536, 649979, 811141, 1002824, 1229175, 1494701, 1804284, 2163196, 2577114, 3052135, 3594791, 4212064, 4911401, 5700729, 6588470

Offset: 1

Author

Harvey P. Dale, Jul 27 2015

Keywords

• nonn
• easy

Examples

			The first ten triangular numbers are 1,3,6,10,15,21,28,36,45,and 55.  Take them in groups, respectively, of 1, 2, 3, and 4 = (1), (3, 6), (10, 15, 21), and (28, 36, 45, 55).  Summing each group separately = 1, 9, 46, 164.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A000217.

Programs

Mathematica

Table[1/24*(8*x+13*x^3+3*x^5),{x,50}]
Module[{nn=40},Total/@TakeList[Accumulate[Range[(nn(nn+1))/2]],Range[nn]]] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1,9,46,164,460,1091},40] (* Harvey P. Dale, Aug 09 2023 *)

PARI

Vec(x*(x^4+3*x^3+7*x^2+3*x+1)/(x-1)^6 + O(x^100)) \\ Colin Barker, Aug 07 2015

Formula

From Colin Barker, Aug 07 2015: (Start)

a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6) for n>6.

G.f.: x*(x^4+3*x^3+7*x^2+3*x+1) / (x-1)^6. (End)

E.g.f.: exp(x)*x*(24 + 84*x + 88*x^2 + 30*x^3 + 3*x^4)/24. - Stefano Spezia, May 14 2024