A056014 a(n) = (Fibonacci(2n-1) - Fibonacci(n+1))/2.
0, 0, 0, 1, 4, 13, 38, 106, 288, 771, 2046, 5401, 14212, 37324, 97904, 256621, 672336, 1760997, 4611642, 12075526, 31617520, 82781215, 216732890, 567428401, 1485570024, 3889310328, 10182407328, 26657986681, 69791674108
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Éva Czabarka, Rigoberto Flórez, and Leandro Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), Article 15.1.6.
- Henrik Eriksson and Markus Jonsson, Level sizes of the Bulgarian solitaire game tree, Fib. Q., 35:3 (2017), 243-251.
- Hideyuki Othsuka, Problem B-1345, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 62, No. 1 (2024), p. 85.
- Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).
Crossrefs
Programs
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Magma
I:=[0, 0, 0, 1]; [n le 4 select I[n] else 4*Self(n-1)-3*Self(n-2)-2*Self(n-3)+Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 23 2012
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Mathematica
Table[(Fibonacci[2n-1]-Fibonacci[n+1])/2,{n,0,40}] (* Harvey P. Dale, Mar 24 2011 *) LinearRecurrence[{4,-3,-2,1},{0,0, 0,1},40] (* Vincenzo Librandi, Jun 23 2012 *) -
PARI
a(n)=(fibonacci(2*n-1)-fibonacci(n+1))/2
Formula
a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3) + a(n-4), a(0)=a(1)=a(2)=0, a(3)=1.
Convolution of Fibonacci numbers F(n) with F(2n). - Benoit Cloitre, Jun 07 2004
G.f.: x^3/((1 - x - x^2)*(1 - 3*x + x^2)). - Herbert Kociemba, Jun 16 2004
Binomial transform of x^3/(1-3x^2+x^4), or (essentially) F(2n) with interpolated zeros. a(n)=sum{k=0..n, binomial(n, k)((3/2-sqrt(5)/2)^(k/2)((sqrt(5)/20+1/4)(-1)^k-sqrt(5)/20-1/4)+ (sqrt(5)/2+3/2)^(k/2)((sqrt(5)/20-1/4)(-1)^k-sqrt(5)/20+1/4))}. - Paul Barry, Jul 26 2004
Convolution of the powers of 2 (A000079) with the number of positive rational knots with 2n+1 crossings (A051450), with three leading zeros. - Graeme McRae, Jun 28 2006
a(n) = Sum_{k=1..n-1} binomial(n-1, k) * A094966(k-1) (Othsuka, 2024). - Amiram Eldar, Feb 29 2024
Comments