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A105635
a(n) = (2*Pell(n+1) - (1+(-1)^n))/4.
6
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
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
Also the matching and vertex cover numbers of the n-Pell graph. - Eric W. Weisstein, Aug 01 2023
LINKS
Eric Weisstein's World of Mathematics, Matching Number
Eric Weisstein's World of Mathematics, Pell Graph
Eric Weisstein's World of Mathematics, Vertex Cover Number
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) - A097076(n+3))/2. - Gary Detlefs, Dec 19 2010
a(n) = floor(Pell(n)/2). - Eric W. Weisstein, Aug 01 2023
MAPLE
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
MATHEMATICA
Table[(Fibonacci[n + 1, 2] - Fibonacci[n + 1, 0])/2, {n, 0, 30}] (* G. C. Greubel, Oct 27 2019 *)
Floor[Fibonacci[Range[20], 2]/2] (* Eric W. Weisstein, Aug 01 2023 *)
Table[(2 Fibonacci[n + 1, 2] - (-1)^n - 1)/4, {n, 0, 10}] (* Eric W. Weisstein, Aug 01 2023 *)
CoefficientList[Series[x/(1 - 2 x - 2 x^2 + 2 x^3 + x^4), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 01 2023 *)
LinearRecurrence[{2, 2, -2, -1}, {0, 1, 2, 6, 14}, 20] (* Eric W. Weisstein, Aug 01 2023 *)
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
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 16 2005
STATUS
approved