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A100131 Sum C(n-2k,2k)2^(n-4k), k=0..floor(n/4). 5
1, 2, 4, 8, 17, 38, 88, 208, 497, 1194, 2876, 6936, 16737, 40398, 97520, 235424, 568353, 1372114, 3312564, 7997224, 19306993, 46611190, 112529352, 271669872, 655869073, 1583407994, 3822685036, 9228778040, 22280241089, 53789260190 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Binomial transform of 1,1,1,1,2,2,4,4,8,8,... (g.f. (1-x)(1+x)^2/(1-2x^2)).

Row sums of number triangle A108350. - Paul Barry, May 31 2005

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1).

FORMULA

G.f.: (1-2x)/((1-2x)^2-x^4)=(1-2x)/((1-x)^2(1-2x-x^2)); a(n)=4a(n-1)-4a(n-2)+a(n-4); a(n)=((sqrt(2)+1)^(n+1)+(sqrt(2)-1)^(n+1)(-1)^n)/(4sqrt(2))+(n+1)/2; a(n)=sum{k=0..n, (1-k)A000129(n-k+1)}.

a(n)=sUm{k=0..n, sum{j=0..n-k, C(k, j)*C(n-j, k)*mod(j+1, 2)}}; - Paul Barry, May 31 2005

(1/2) [Pell(n) + n + 1 ], with Pell(n) = A000129(n). - Ralf Stephan, May 15 2007

MAPLE

with(combinat):seq((n+fibonacci(n, 2))/2, n=1..30); - Zerinvary Lajos, Jun 02 2008

MATHEMATICA

CoefficientList[Series[(1-2x)/((1-2x)^2-x^4), {x, 0, 40}], x]  (* Harvey P. Dale, Mar 22 2011 *)

LinearRecurrence[{4, -4, 0, 1}, {1, 2, 4, 8}, 40] (* Vincenzo Librandi, Jun 25 2012 *)

PROG

(MAGMA) I:=[1, 2, 4, 8]; [n le 4 select I[n] else 4*Self(n-1)-4*Self(n-2)+Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 25 2012

CROSSREFS

Cf. A098576, A100132, A100133.

Sequence in context: A193050 A107597 A082499 * A119685 A025276 A006461

Adjacent sequences:  A100128 A100129 A100130 * A100132 A100133 A100134

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Nov 06 2004

STATUS

approved

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Last modified October 21 17:10 EDT 2018. Contains 316427 sequences. (Running on oeis4.)