login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A105635 a(n) = (2*Pell(n+1) - (1+(-1)^n))/4. 5
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 (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) - A097076(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-> (<<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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 9 17:46 EST 2022. Contains 358703 sequences. (Running on oeis4.)