A153280 Eigensequence of triangle A153279.

1, 3, 15, 165, 4785, 397155, 97302975, 71128474725, 155700231173025, 1021860617188563075, 20115326249356864131375, 1187830130350772183821825125, 210422919761508941591852499068625, 111827787746815596446398867662527275875

Offset: 0

Author

Gary W. Adamson, Dec 23 2008

Keywords

• nonn

Examples

			Triangle M =
1;
1;
2, 1;
4, 2, 3;
8, 4, 6, 9;
16, 8, 12, 18, 27;
...
M^n rapidly converges to this sequence with sufficiently large n.
a(0) = 1, a(1) = 1*(2+3^0) = 3, a(2) = 3*(2+3^1) = 15, a(3) = 15*(2+3^2) = 165, a(4) = 165*(2+3^3) = 4785, ... - _Philippe Deléham_, Sep 27 2014
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..65

Crossrefs

Cf. A153279, A057735.

Programs

Mathematica

RecurrenceTable[{a[n+1] == a[n]*(2 + 3^n), a[0] == 1}, a, {n, 0, 15}] (* Vaclav Kotesovec, Jan 22 2023 *)
Table[2^n * QPochhammer[-1/2, 3, n], {n, 0, 15}] (* Vaclav Kotesovec, Jan 22 2023 *)
nxt[{n_,a_}]:={n+1,a(2+3^n)}; NestList[nxt,{0,1},20][[;;,2]] (* Harvey P. Dale, Mar 28 2024 *)

Formula

Given triangle A153279, let a new triangle = M shifted down one row, inserting a "1" in (0,0). Triangle equals lim_{n->oo} M^n.

a(n+1) = a(n)*(2+3^n), a(0) = 1. - Philippe Deléham, Sep 27 2014

a(n) ~ c * 3^(n*(n-1)/2), where c = QPochhammer(-2, 1/3) = 6.80914656805984199... - Vaclav Kotesovec, Jan 22 2023

Extensions

More terms from Philippe Deléham, Sep 27 2014