A180676 The Gi3 sums of the Pell-Jacobsthal triangle A013609.

1, 1, 1, 1, 17, 337, 6641, 130865, 2578785, 50816737, 1001378849, 19732860833, 388849631729, 7662550168241, 150995835638929, 2975477077435217, 58633827885912001, 1155420016046016193, 22768348266078953793

Offset: 0

Author

Johannes W. Meijer, Sep 21 2010

Keywords

• easy
• nonn

Comments

The a(n+3) represent the Gi3 sums of the Pell-Jacobsthal triangle A013609. See A180662 for information about these giraffe and other chess sums.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..750
• Index entries for linear recurrences with constant coefficients, signature (20,-6,4,-1).

Crossrefs

Cf. A052942 (Gi1), A008999 (Gi2), this sequence (Gi3), A180677 (Gi4).

Cf. A013609, A180662.

Programs

GAP

a:=[1,1,1,1];; for n in [5..30] do a[n]:=20*a[n-1]-6*a[n-2] + 4*a[n-3]-a[n-4]; od; a; # G. C. Greubel, Jun 11 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-19*x-13*x^2-17*x^3)/(1-20*x+6*x^2-4*x^3+x^4) )); // G. C. Greubel, Jun 11 2019

Maple

nmax:=18: a(0):=1: a(1):=1: a(2):=1: a(3):=1: for n from 4 to nmax do a(n) := 20*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) od: seq(a(n),n=0..nmax);

Mathematica

LinearRecurrence[{20,-6,4,-1},{1,1,1,1},20] (* Harvey P. Dale, Jun 07 2015 *)

PARI

my(x='x+O('x^30)); Vec((1-19*x-13*x^2-17*x^3)/(1-20*x+6*x^2-4*x^3 +x^4)) \\ G. C. Greubel, Jun 11 2019

Sage

((1-19*x-13*x^2-17*x^3)/(1-20*x+6*x^2-4*x^3+x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 11 2019

Formula

a(n) = 20*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) with a(0)=1, a(1)=1, a(2)=1 and a(3)=1.

a(n+3) = Sum_{k=0..n} A013609(n+3*k,4*k).

G.f.: (1-19*x-13*x^2-17*x^3)/(1-20*x+6*x^2-4*x^3+x^4).