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

1,2

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).

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

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 *)

Crossrefs

Cf. A097602.

Sequence in context: A046654 A280724 A023543 * A256212 A360107 A096149

Adjacent sequences: A129892 A129893 A129894 * A129896 A129897 A129898

Keyword

nonn

Author

Leroy Quet, Jun 04 2007

Extensions

More terms from Emeric Deutsch, Jun 21 2007

Status

approved