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A094292 Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n, s(0) = 2, s(n) = 4. 2
0, 0, 1, 3, 9, 25, 68, 182, 483, 1275, 3355, 8811, 23112, 60580, 158717, 415715, 1088661, 2850645, 7463884, 19541994, 51163695, 133951675, 350695511, 918141623, 2403740304, 6293097000, 16475579353, 43133687427, 112925557953 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

In general a(n,m,j,k)=2/m*Sum(r,1,m-1,Sin(j*r*Pi/m)Sin(k*r*Pi/m)(1+2Cos(Pi*r/m))^n) is the number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < m and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n, s(0) = j, s(n) = k.

a(n+1) is an inverse Catalan transform of F(3n)/F(3). The g.f. may be obtained from that of A001076 under the mapping g(x)-> g(x(1-x)). - Paul Barry, Nov 17 2004

A transform of Fib(2n) : Fib(2n) may be recovered as sum{k=0..2n, sum{j=0..k, C(0,2n-k)C(k,j)(-1)^(k-j)*A094292(j)}}. - Paul Barry, Jun 10 2005

LINKS

Table of n, a(n) for n=0..28.

É. Czabarka, R. Flórez, L. Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), #15.1.6.

Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).

FORMULA

a(n) = (2/5)*Sum(k, 1, 4, Sin(2Pi*k/5)Sin(4Pi*k/5)(1+2Cos(Pi*k/5))^n).

a(n) = 4*a(n-1)-3*a(n-2)-2*a(n-3)+a(n-4) G.f.: (x^2-x^3)/(1-4x+3x^2+2x^3-x^4) - Herbert Kociemba, Jun 16 2004

a(n) = (Fibonacci(2*n)-Fibonacci(n))/2. - Vladeta Jovovic, Jul 17 2004

a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k, k)(-1)^k*F(3n-3k)/F(3). - Paul Barry, Nov 17 2004

a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)Fib(2k). - Paul Barry, Jun 10 2005

a(n) = Sum_{k=0..n-1} Fibonacci(n+k-1)/2. [Gary Detlefs, Feb 22 2011]

a(n) = Fibonacci(n)*(Lucas(n) - 1)/2. - Vladimir Reshetnikov, Sep 27 2016

MATHEMATICA

Table[Sum[Fibonacci[n - 1 + i]/2, {i, 0, n - 1}], {n, 0, 27}]  (* Zerinvary Lajos, Jul 12 2009 *)

Table[Fibonacci[n] (LucasL[n] - 1)/2, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 27 2016 *)

PROG

(Mupad)(numlib::fibonacci(2*n)-numlib::fibonacci(n))/2 $ n = 0..35; // Zerinvary Lajos, May 09 2008

CROSSREFS

Cf. A000032, A000045, A049681.

Sequence in context: A069403 A291021 A226710 * A291019 A236570 A201533

Adjacent sequences:  A094289 A094290 A094291 * A094293 A094294 A094295

KEYWORD

easy,nonn

AUTHOR

Herbert Kociemba, Jun 02 2004

EXTENSIONS

a(0) = a(1) = 0 added and offset changed to 0 by Vladimir Reshetnikov, Oct 04 2016

STATUS

approved

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Last modified November 18 14:15 EST 2017. Contains 294893 sequences.