A095127 a(n+3) = 2*a(n+2) + 3*a(n+1) - a(n); with a(1) = 1, a(2) = 4, a(3) = 10.

1, 4, 10, 31, 88, 259, 751, 2191, 6376, 18574, 54085, 157516, 458713, 1335889, 3890401, 11329756, 32994826, 96088519, 279831760, 814934251, 2373275263, 6911521519, 20127934576, 58617158446, 170706599101, 497136738964

Offset: 1

Author

Gary W. Adamson, May 29 2004

Keywords

• nonn
• easy

Comments

A sequence generated from the characteristic polynomial of A095125 and A095126.

a(n)/a(n-1) tends to a 2.9122291784..., a root of the polynomial x^3 - 2x^2 - 3x + 1; e.g. a(16)/a(15) = 11329756/3890401 = 2.912233...

Examples

			a(7) = 751 = 2*a(6) + 3*a(5) - a(4) = 2*259 + 3*88 - 31.
a(4) = 31 = center term in M^4 * [1 1 1] = [10 31 88].
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,3,-1).

Crossrefs

Cf. A095125, A095126, A095128.

Programs

Mathematica

a[1] = 1; a[2] = 4; a[3] = 10; a[n_] := a[n] = 2a[n - 1] + 3a[n - 2] - a[n - 3]; Table[ a[n], {n, 25}] (* Robert G. Wilson v, Jun 01 2004 *)
nxt[{a_,b_,c_}]:={b,c,2c+3b-a}; NestList[nxt,{1,4,10},30][[All,1]] (* or *) LinearRecurrence[{2,3,-1},{1,4,10},30] (* Harvey P. Dale, Feb 08 2022 *)

PARI

Vec(x*(1 + 2*x - x^2) / (1 - 2*x - 3*x^2 + x^3) + O(x^30)) \\ Colin Barker, Aug 31 2019

Formula

M = a matrix having the same eigenvalues as the roots of the characteristic polynomial of A095125 and A095126: (x^3 - 2x^2 - 3x + 1). Then M^n * [1 1 1] = [p q r] where q = a(n) and p, r, are offset members of the same sequence.

G.f.: x*(1 + 2*x - x^2) / (1 - 2*x - 3*x^2 + x^3). - Colin Barker, Aug 31 2019

Extensions

Edited, corrected and extended by Robert G. Wilson v, Jun 01 2004