A135364 First column of a triangle - see Comments lines.

1, 2, 3, 7, 17, 40, 93, 216, 502, 1167, 2713, 6307, 14662, 34085, 79238, 184206, 428227, 995507, 2314273, 5380032, 12507057, 29075380, 67592058, 157132471, 365288677, 849193147, 1974134558, 4589306057, 10668842202

Offset

0,2

Comments

...1;

...2,...1;

...3,...3,...1;

...7,...5,...4,...1;

..17,..10,...7,...5,...1;

..40,..24,..13,...9,...6,...1;

..93,..57,..31,..16,..11,...7,...1;

From the second, the sum of a row gives the first term of the following one. Diagonal differences are the first term upon. First column is a(n).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Richard Choulet, Transformation à la Curtz. Curtz like Transformation, March 2008.

Index entries for linear recurrences with constant coefficients, signature (3,-2,1).

Formula

From Richard Choulet, Jan 06 2008: (Start)

a(n+1) = a(n) + a(n-1) + (n-1)*a(1) + (n-2)*a(2) + ... + 2*a(n-2) for n>=3.

O.g.f.: 1 + x*(2 - 3*x + 2*x^2) / (1 - 3*x + 2*x^2 - x^3).

a(n+3) = 3*a(n+2) - 2*a(n+1) + a(n). (End)

a(n) = A034943(n) + A034943(n+1). - R. J. Mathar, Apr 09 2008

a(0) = 1, a(n) = term (1,3) in the 1 X 3 matrix [7,3,2].[3,1,0; -2,0,1; 1,0,0]^(n-1) (n>0). - Alois P. Heinz, Jul 24 2008

a(n) = 2*A095263(n-1) -3*A095263(n-2) +2*A095263(n-3) with a(0) = 1. - G. C. Greubel, Apr 19 2021

Maple

a:= n-> `if`(n=0, 1, (<<7|3|2>> .<<3|1|0>, <-2|0|1>, <1|0|0>>^(n-1))[1, 3]):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 24 2008

Mathematica

LinearRecurrence[{3, -2, 1}, {1, 2, 3, 7, 17}, 51] (* G. C. Greubel, Oct 11 2016; Apr 19 2021 *)

Prog

(Magma) I:=[3, 7, 17]; [1, 2] cat [n le 3 select I[n] else 3*Self(n-1) -2*Self(n-2) +Self(n-3): n in [1..51]]; // G. C. Greubel, Apr 19 2021
(Sage)
@CachedFunction
def A095263(n): return sum( binomial(n+j+2, 3*j+2) for j in (0..n//2) )
def A135364(n): return 1 if n==0 else 2*A095263(n-1) -3*A095263(n-2) +2*A095263(n-3)
[A135364(n) for n in (0..50)] # G. C. Greubel, Apr 19 2021

Crossrefs

Cf. A034943, A097550, A136302, A136303, A136304, A136305, A137229, A137234, A137249.

Sequence in context: A191033 A105554 A145230 * A051291 A178178 A333499

Adjacent sequences: A135361 A135362 A135363 * A135365 A135366 A135367

Keyword

nonn

Author

Paul Curtz, Dec 09 2007

Extensions

More terms from Richard Choulet, Jan 06 2008

Status

approved