A180678 The Ze2 sums of the Pell-Jacobsthal triangle A013609.

1, 2, 5, 16, 57, 206, 737, 2612, 9213, 32442, 114205, 402072, 1415713, 4985126, 17554489, 61816252, 217679141, 766531986, 2699251381, 9505089568, 33471028105, 117864194430, 415044573969, 1461529529924, 5146600421325

Offset: 0

Author

Johannes W. Meijer, Sep 21 2010

Keywords

• easy
• nonn

Comments

The a(n) represent the Ze2 sums of the Pell-Jacobsthal triangle A013609. See A180662 for information about these zebra and other chess sums.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-11,8).

Crossrefs

Cf. A140413 (Ze1), A180678 (Ze2), A097117 (Ze3), A055588 (Ze4).

Programs

GAP

a:=[1,2,5];; for n in [4..30] do a[n]:=6*a[n-1]-11*a[n-2]+8*a[n-3]; od; a; # G. C. Greubel, Jun 06 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-4*x+4*x^2)/(1-6*x+11*x^2-8*x^3) )); // G. C. Greubel, Jun 06 2019

Maple

nmax:=24: a(0):=1: a(1):=2: a(2):=5: for n from 3 to nmax do a(n) := 6*a(n-1)-11*a(n-2)+8*a(n-3) od: seq(a(n),n=0..nmax);

Mathematica

LinearRecurrence[{6,-11,8}, {1,2,5}, 30] (* or *) CoefficientList[ Series[(1-4*x+4*x^2)/(1-6*x+11*x^2-8*x^3), {x,0,30}], x] (* G. C. Greubel, Jun 06 2019 *)

PARI

my(x='x+O('x^30)); Vec((1-4*x+4*x^2)/(1-6*x+11*x^2-8*x^3)) \\ G. C. Greubel, Jun 06 2019

Sage

((1-4*x+4*x^2)/(1-6*x+11*x^2-8*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 06 2019

Formula

a(n) = 6*a(n-1) - 11*a(n-2) + 8*a(n-3) with a(0)=1, a(1)=2 and a(2)= 5.

a(n) = Sum_{k=0..floor(n/2)} A013609(n+k,n-2*k).

G.f.: (1-4*x+4*x^2)/(1-6*x+11*x^2-8*x^3).