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A105635 a(n) = (2*Pell(n+2) - (1+(-1)^n))/4. 4
0, 1, 2, 6, 14, 35, 84, 204, 492, 1189, 2870, 6930, 16730, 40391, 97512, 235416, 568344, 1372105, 3312554, 7997214, 19306982, 46611179, 112529340, 271669860, 655869060, 1583407981, 3822685022, 9228778026, 22280241074, 53789260175 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Transform of Pell(n) under the Riordan array (1/(1-x^2), x).

Starting (1, 2, 6, 14, 35, ...) equals row sums of triangle A157901. - Gary W. Adamson, Mar 08 2009

Starting with 1 = row sums of a triangle with the Pell series shifted down twice for columns > 1. - Gary W. Adamson, Mar 03 2010

LINKS

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

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

FORMULA

G.f.: x/((1-x^2)*(1-2*x-x^2)).

a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4).

a(n) = Sum_{k=0..floor((n-1)/2)} Pell(n-2k).

a(n) = Sum_{k=0..n} Pell(k)*(1-(-1)^(n+k-1))/2.

a(n) = term (4,1) in the 4 X 4 matrix [1,1,0,0; 3,0,1,0; 1,0,0,0; 1,0,0,1]^n. - Alois P. Heinz, Jul 24 2008

a(n) = ( A033539(n+3) - A09776(n+3) )/2. - Gary Detlefs Dec 19 2010

MAPLE

with(combinat): seq(iquo(fibonacci(n+1, 2), 2), n=0..30); # Zerinvary Lajos, Apr 20 2008

# second Maple program:

a:= n-> (Matrix([[1, 1, 0, 0], [3, 0, 1, 0], [1, 0, 0, 0], [1, 0, 0, 1]])^(n))[4, 1]; seq(a(n), n=0..50); # Alois P. Heinz, Jul 24 2008

MATHEMATICA

Table[(Fibonacci[n+1, 2] - Fibonacci[n+1, 0])/2, {n, 0, 30}] (* G. C. Greubel, Oct 27 2019 *)

PROG

(PARI) my(x='x+O('x^30)); concat([0], Vec(x/((1-x^2)*(1-2*x-x^2)))) \\ G. C. Greubel, Oct 27 2019

(MAGMA) R<x>:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x/((1-x^2)*(1-2*x-x^2)) )); // G. C. Greubel, Oct 27 2019

(Sage)

def A105635_list(prec):

    P.<x> = PowerSeriesRing(ZZ, prec)

    return P(x/((1-x^2)*(1-2*x-x^2))).list()

A105635_list(30) # G. C. Greubel, Oct 27 2019

(GAP) a:=[0, 1, 2, 6];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]-2*a[n-3] -a[n-4]; od; a; # G. C. Greubel, Oct 27 2019

CROSSREFS

Cf. A000129.

Cf. A157901. - Gary W. Adamson, Mar 08 2009

Sequence in context: A307068 A269506 A292816 * A178320 A297187 A231509

Adjacent sequences:  A105632 A105633 A105634 * A105636 A105637 A105638

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Apr 16 2005

STATUS

approved

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Last modified July 28 10:29 EDT 2021. Contains 346326 sequences. (Running on oeis4.)