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A105635
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a(n) = (2*Pell(n+1) - (1+(-1)^n))/4.
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5
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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, 129858761424, 313506783024
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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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).
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FORMULA
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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) - A097076(n+3))/2. - Gary Detlefs Dec 19 2010
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MAPLE
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with(combinat): seq(iquo(fibonacci(n+1, 2), 2), n=0..30); # Zerinvary Lajos, Apr 20 2008
# second Maple program:
a:= n-> (<<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
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MATHEMATICA
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Table[(Fibonacci[n+1, 2] - Fibonacci[n+1, 0])/2, {n, 0, 30}] (* G. C. Greubel, Oct 27 2019 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Barry, Apr 16 2005
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STATUS
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approved
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