A129895 a(1)=1. a(n) = a(n-1) + number of triangular numbers among the first (n-1) terms of the sequence.

1, 2, 3, 5, 7, 9, 11, 13, 15, 18, 21, 25, 29, 33, 37, 41, 45, 50, 55, 61, 67, 73, 79, 85, 91, 98, 105, 113, 121, 129, 137, 145, 153, 162, 171, 181, 191, 201, 211, 221, 231, 242, 253, 265, 277, 289, 301, 313, 325, 338, 351, 365, 379, 393, 407, 421, 435, 450, 465, 481

Offset: 1

Author

Leroy Quet, Jun 04 2007

Keywords

• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2, 0, -2, 0, 2, 0, -2, 1).

Crossrefs

Cf. A097602.

Programs

Maple

T := {seq((1/2)*j*(j+1), j = 1 .. 40)}: a[1] := 1; for n from 2 to 60 do a[n] := a[n-1]+nops(`intersect`(T, {seq(a[i], i = 1 .. n-1)})) end do: seq(a[n], n = 1 .. 60); # Emeric Deutsch, Jun 21 2007

Mathematica

nxt[{a_,t_}]:=Module[{x=t},{a+t,If[IntegerQ[(Sqrt[8(a+t)+1]-1)/2], x+1, x]}]; Transpose[NestList[nxt,{1,1},70]][[1]] (* or *) LinearRecurrence[ {2,0,-2,0,2,0,-2,1},{1,2,3,5,7,9,11,13},70] (* Harvey P. Dale, May 16 2014 *)

Formula

For k=1,3: a(8*n+k) = (4*n+k)*(2*n+1). - Reinhard Zumkeller, Dec 20 2007

G.f.: -x*(x^7 - x^6 - x^5 + x^4 + x^3 - x^2 + 1) / ((x-1)^3*(x+1)*(x^4+1)). - Colin Barker, Mar 29 2013

a(n) = 2*a(n-1) - 2*a(n-3) + 2*a(n-5) - 2*a(n-7) + a(n-8); a(1)=1, a(2)=2, a(3)=3, a(4)=5, a(5)=7, a(6)=9, a(7)=11, a(8)=13. - Harvey P. Dale, May 16 2014

Extensions

More terms from Emeric Deutsch, Jun 21 2007