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

Author

Paul Curtz, Dec 09 2007

Keywords

• nonn

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).

Crossrefs

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

Programs

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

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 *)

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

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

Extensions

More terms from Richard Choulet, Jan 06 2008