A292498 G.f.: Sum_{n=-oo..+oo} Fibonacci(n+1) * x^n * (1-x^n)^n.

1, 1, 2, 3, 3, 8, 10, 21, 18, 67, 54, 144, 196, 377, 470, 1077, 1321, 2584, 3905, 6765, 10014, 18173, 27084, 46368, 73001, 121448, 191530, 319827, 505038, 832040, 1335766, 2178309, 3497250, 5710862, 9183554, 14931192, 24093521, 39088169, 63117470, 102363639, 165353391, 267914296, 433177813, 701408733, 1134249308, 1836422951, 2970148632

Offset

0,3

Comments

Compare g.f. to: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n = 0.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1030

Formula

G.f. P(x) + Q(x) where

P(x) = Sum_{n>=0} Fibonacci(n+1) * x^n * (1-x^n)^n,

Q(x) = Sum_{n>=1} Fibonacci(n-1) * x^(n^2-n) / (1-x^n)^n.

a(n) ~ phi^(n+1) / sqrt(5), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 10 2017

Example

G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 3*x^4 + 8*x^5 + 10*x^6 + 21*x^7 + 18*x^8 + 67*x^9 + 54*x^10 + 144*x^11 + 196*x^12 + 377*x^13 + 470*x^14 + 1077*x^15 + 1321*x^16 + 2584*x^17 + 3905*x^18 + 6765*x^19 + 10014*x^20 +...
where A(x) = Sum_{n=-oo..+oo} Fibonacci(n+1) * x^n * (1-x^n)^n.
RELATED SERIES.
G.f. A(x) = P(x) + Q(x) where
P(x) = 1 + 1*x*(1-x) + 2*x^2*(1-x^2)^2 + 3*x^3*(1-x^3)^3 + 5*x^4*(1-x^4)^4 + 8*x^5*(1-x^5)^5 + 13*x^6*(1-x^6)^6 + 21*x^7*(1-x^7)^7 + 34*x^8*(1-x^8)^8 + 55*x^9*(1-x^9)^9 + 89*x^10*(1-x^10)^10 +...+ Fibonacci(n+1)*x^n*(1-x^n)^n +...
Q(x) = 0*x^0/(1-x) + 1*x^2/(1-x^2)^2 + 1*x^6/(1-x^3)^3 + 2*x^12/(1-x^4)^4 + 3*x^20/(1-x^5)^5 + 5*x^30/(1-x^6)^6 + 8*x^42/(1-x^7)^7 + 13*x^56/(1-x^8)^8 + 21*x^72/(1-x^9)^9 +...+ Fibonacci(n-1)*x^(n^2-n)/(1-x^n)^n +...
Explicitly,
P(x) = 1 + x + x^2 + 3*x^3 + x^4 + 8*x^5 + 6*x^6 + 21*x^7 + 14*x^8 + 64*x^9 + 49*x^10 + 144*x^11 + 182*x^12 + 377*x^13 + 463*x^14 + 1067*x^15 + 1305*x^16 + 2584*x^17 + 3881*x^18 + 6765*x^19 + 9981*x^20 + 18152*x^21 +...
Q(x) = x^2 + 2*x^4 + 4*x^6 + 4*x^8 + 3*x^9 + 5*x^10 + 14*x^12 + 7*x^14 + 10*x^15 + 16*x^16 + 24*x^18 + 33*x^20 + 21*x^21 + 11*x^22 + 80*x^24 + 15*x^25 +...
The reciprocal of the g.f. begins:
1/A(x) = 1 - x - x^2 + 2*x^4 - 4*x^5 + x^6 - 2*x^7 + 19*x^8 - 50*x^9 + 58*x^10 - 42*x^11 + 75*x^12 - 267*x^13 + 566*x^14 - 827*x^15 + 1225*x^16 - 2431*x^17 + 4972*x^18 - 8438*x^19 + 13089*x^20 - 22333*x^21 + 43831*x^22 +...

Prog

(PARI) {a(n) = my(A, P, Q, Ox=x*O(x^n));
P = sum(m=0, n, fibonacci(m+1) * x^m * (1-x^m +Ox)^m);
Q = sum(m=1, sqrtint(2*n+9), fibonacci(m-1) * x^(m^2-m) / (1-x^m +Ox)^m);
A = P + Q; polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))

Crossrefs

Sequence in context: A108381 A238761 A261469 * A108692 A157126 A357655

Adjacent sequences: A292495 A292496 A292497 * A292499 A292500 A292501

Keyword

nonn

Author

Paul D. Hanna, Oct 09 2017

Status

approved