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A081057 E.g.f.: Sum_{n>=0} a(n)*x^n/n! = {Sum_{n>=0} F(n+1)*x^n/n!}^2, where F(n) is the n-th Fibonacci number. 1
1, 2, 6, 18, 58, 186, 602, 1946, 6298, 20378, 65946, 213402, 690586, 2234778, 7231898, 23402906, 75733402, 245078426, 793090458, 2566494618, 8305351066, 26876680602, 86974765466, 281456253338, 910811568538, 2947448150426 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) ~ c*(sqrt(5)+1)^n, where c = (sqrt(5)+3)/10.

The inverse binomial transform is 1,1,3,5,... (1 followed by A056487). Partial sum of 1,1,4,12,..., i.e., 1 plus n-th partial sum of A087206. [R. J. Mathar, Oct 04 2010]

From R. J. Mathar, Oct 12 2010: (Start)

Apparently the row n=4 of an array which counts walks with k steps on an n X n board, starting at a corner, each step to one of the <= 4 adjacent squares:

1,2,4,8,16,32,64,128,256,512,1024,2048,4096,

1,2,6,16,48,128,384,1024,3072,8192,24576,65536,196608,

1,2,6,18,58,186,602,1946,6298,20378,65946,213402,690586,

1,2,6,18,60,198,684,2322,8100,27702,96876,331938,1161540,

1,2,6,18,60,200,698,2432,8658,30762,110374,395428,1422916,

1,2,6,18,60,200,700,2448,8800,31552,115104,418176,1537536,

1,2,6,18,60,200,700,2450,8818,31730,116182,425172,1573416,

1,2,6,18,60,200,700,2450,8820,31750,116400,426600,1583400,

1,2,6,18,60,200,700,2450,8820,31752,116422,426862,1585246,

1,2,6,18,60,200,700,2450,8820,31752,116424,426886,1585556,

1,2,6,18,60,200,700,2450,8820,31752,116424,426888,1585582,

(End)

Decomposing rook walks of length=n on a 4 X 4 board into combinations of independent vertical and horizontal walks in 4-wide corridors leads to an exponential convolution of the Fibonacci numbers, cf. A052899. [David Scambler, Oct 17 2010]

LINKS

Table of n, a(n) for n=0..25.

Index entries for linear recurrences with constant coefficients, signature (3,2,-4).

FORMULA

G.f.: (1-x-2x^2)/(1-3x-2x^2+4x^3). - Michael Somos, Mar 04 2003

a(n) - 2*a(n-1) = A014334(n), n > 0. - Vladeta Jovovic, Mar 05 2003

From Vladeta Jovovic, Mar 05 2003: (Start)

a(n) = 2/5 + (3/10 - 1/10*5^(1/2))*(1 - 5^(1/2))^n + (3/10 + 1/10*5^(1/2))*(1 + 5^(1/2))^n.

Recurrence: a(n) = 3*a(n-1) + 2*a(n-2) - 4*a(n-3).

G.f.: (1+x)*(1-2*x)/(1-2*x-4*x^2)/(1-x). (End)

a(n) = Sum_{k=0..n} ( F(k+1) * F(n-k+1) * C(n,k) ), where F(k) = Fibonacci(k). - David Scambler, Oct 17 2010

CROSSREFS

a(n) = A052899(n-1) + A052899(n). a(n) - 2*a(n-1) = A014334(n).

Sequence in context: A324166 A307755 A304200 * A000137 A151282 A193777

Adjacent sequences:  A081054 A081055 A081056 * A081058 A081059 A081060

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Mar 03 2003

EXTENSIONS

Corrected and extended by Vladeta Jovovic and Michael Somos, Mar 05 2003

STATUS

approved

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Last modified February 27 21:03 EST 2020. Contains 332309 sequences. (Running on oeis4.)