A105635 a(n) = (2*Pell(n+1) - (1+(-1)^n))/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, 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

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

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

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) - 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

Adjacent sequences: A105632 A105633 A105634 * A105636 A105637 A105638

Keyword

easy,nonn

Author

Paul Barry, Apr 16 2005

Status

approved