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A372015
Product of Fibonacci and self-convolution of Fibonacci numbers: a(n) = A000045(n+1)*A001629(n+1).
2
0, 1, 4, 15, 50, 160, 494, 1491, 4420, 12925, 37380, 107136, 304764, 861445, 2421700, 6775755, 18879734, 52413856, 145038890, 400183575, 1101277060, 3023462521, 8282790024, 22646131200, 61805595000, 168399404425, 458128878724, 1244567262471, 3376576740410, 9149594423200
OFFSET
0,3
COMMENTS
Conjecture: a(n) is the total number of pairs of adjacent parts that are the same color in all n-color compositions of n+1. - John Tyler Rascoe, Jul 30 2024
FORMULA
a(n) = F(n+1)*((n+2)*F(n) + (n)*F(n+2))/5 where F(n) = A000045(n) is the Fibonacci numbers.
G.f.: x*(1-x)/((1+x)*(1-3*x+x^2)^2).
MAPLE
a := proc(n) option remember; if n < 3 then return n^2 fi;
-((2 - 2*n^2 + n)*a(n - 1) + (1 - 2*n^2 + 3*n)*a(n - 2) + n^2*a(n - 3))/(n - 1)^2 end: seq(a(n), n = 0..29); # Peter Luschny, Apr 16 2024
MATHEMATICA
CoefficientList[Series[x(1-x)/((1+x)*(1-3*x+x^2)^2), {x, 0, 29}], x] (* Stefano Spezia, Apr 16 2024 *)
PROG
(PARI)
A_x(N)= {my(x='x+O('x^N)); concat([0], Vec(x*(1-x)/((1+x)*(1-3*x+x^2)^2)))}
A_x(40) \\ John Tyler Rascoe, Jul 29 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vladimir Kruchinin, Apr 15 2024
STATUS
approved