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A245282 G.f.: Sum_{n>=1} Fibonacci(n+1) * x^n / (1 - x^n). 4
1, 3, 4, 8, 9, 19, 22, 42, 59, 100, 145, 257, 378, 634, 999, 1639, 2585, 4255, 6766, 11051, 17736, 28804, 46369, 75316, 121402, 196798, 317870, 514868, 832041, 1347372, 2178310, 3526217, 5703035, 9230052, 14930382, 24162310, 39088170, 63252754, 102334536, 165591226, 267914297 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..1000

FORMULA

G.f.: Sum_{n>=1} x^n * (1 + x^n) / (1 - x^n - x^(2*n)).

G.f.: Sum_{n>=1} x^n * Sum_{d|n} (1 + x^d)^(n/d).

a(n) ~ 1/sqrt(5) * ((1+sqrt(5))/2)^(n+1). - Vaclav Kotesovec, Aug 22 2014

EXAMPLE

G.f.: A(x) = x + 3*x^2 + 4*x^3 + 8*x^4 + 9*x^5 + 19*x^6 + 22*x^7 +...

where by definition

A(x) = 1*x/(1-x) + 2*x^2/(1-x^2) + 3*x^3/(1-x^3) + 5*x^4/(1-x^4) + 8*x^5/(1-x^5) + 13*x^6/(1-x^6) + 21*x^7/(1-x^7) + 34*x^8/(1-x^8) + 55*x^9/(1-x^9) + 89*x^10/(1-x^10) + 144*x^11/(1-x^11) +...+ Fibonacci(n+1)*x^n/(1-x^n) +...

The g.f. is also given by the series identity:

A(x) = x*(1+x)/(1-x-x^2) + x^2*(1+x^2)/(1-x^2-x^4) + x^3*(1+x^3)/(1-x^3-x^6) + x^4*(1+x^4)/(1-x^4-x^8) + x^5*(1+x^5)/(1-x^5-x^10) + x^6*(1+x^6)/(1-x^6-x^12) + x^7*(1+x^7)/(1-x^7-x^14) +...+ x^n*(1+x^n)/(1-x^n-x^(2*n)) +...

And also we have the series:

A(x) = x*(1 + x) + x^2*((1+x)^2 + (1+x^2)) + x^3*((1+x)^3 + (1+x^3))

+ x^4*((1+x)^4 + (1+x^2)^2 + (1+x^4)) + x^5*((1+x)^5 + (1+x^5))

+ x^6*((1+x)^6 + (1+x^2)^3 + (1+x^3)^2 + (1+x^6))

+ x^7*((1+x)^7 + (1+x^7))

+ x^8*((1+x)^8 + (1+x^2)^4 + (1+x^4)^2 + (1+x^8))

+ x^9*((1+x)^9 + (1+x^3)^3 + (1+x^9))

+ x^10*((1+x)^10 + (1+x^2)^5 + (1+x^5)^2 + (1+x^10))

+ x^11*((1+x)^11 + (1+x^11))

+ x^12*((1+x)^12 + (1+x^2)^6 + (1+x^3)^4 + (1+x^4)^3 + (1+x^6)^2 + (1+x^12))

+...+ x^n * Sum_{d|n} (1 + x^d)^(n/d) +...

or, more explicitly,

A(x) = x*(1 + x) + x^2*(2 + 2*x + 2*x^2) + x^3*(2 + 3*x + 3*x^2 + 2*x^3)

+ x^4*(3 + 4*x + 8*x^2 + 4*x^3 + 3*x^4)

+ x^5*(2 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + 2*x^5)

+ x^6*(4 + 6*x + 18*x^2 + 22*x^3 + 18*x^4 + 6*x^5 + 4*x^6)

+ x^7*(2 + 7*x + 21*x^2 + 35*x^3 + 35*x^4 + 21*x^5 + 7*x^6 + 2*x^7)

+ x^8*(4 + 8*x + 32*x^2 + 56*x^3 + 78*x^4 + 56*x^5 + 32*x^6 + 8*x^7 + 4*x^8)

+ x^9*(3 + 9*x + 36*x^2 + 87*x^3 + 126*x^4 + 126*x^5 + 87*x^6 + 36*x^7 + 9*x^8 + 3*x^9)

+ x^10*(4 + 10*x + 50*x^2 + 120*x^3 + 220*x^4 + 254*x^5 + 220*x^6 + 120*x^7 + 50*x^8 + 10*x^9 + 4*x^10)

+ x^11*(2 + 11*x + 55*x^2 + 165*x^3 + 330*x^4 + 462*x^5 + 462*x^6 + 330*x^7 + 165*x^8 + 55*x^9 + 11*x^10 + 2*x^11)

+ x^12*(6 + 12*x + 72*x^2 + 224*x^3 + 513*x^4 + 792*x^5 + 952*x^6 + 792*x^7 + 513*x^8 + 224*x^9 + 72*x^10 + 12*x^11 + 6*x^12) + ...

PROG

(PARI) {a(n)=polcoeff(sum(m=1, n, fibonacci(m+1)*x^m/(1-x^m +x*O(x^n))), n)}

for(n=1, 50, print1(a(n), ", "))

(PARI) {a(n)=polcoeff(sum(m=1, n, x^m*(1+x^m)/(1-x^m-x^(2*m) +x*O(x^n)) ), n)}

for(n=1, 50, print1(a(n), ", "))

(PARI) {a(n)=local(A=x+x^2); A=sum(m=1, n, x^m*sumdiv(m, d, (1 + x^(m/d) +x*O(x^n))^d) ); polcoeff(A, n)}

for(n=1, 50, print1(a(n), ", "))

CROSSREFS

Cf. A034729.

Sequence in context: A096185 A137640 A165362 * A320355 A113057 A253606

Adjacent sequences:  A245279 A245280 A245281 * A245283 A245284 A245285

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 21 2014

STATUS

approved

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Last modified January 22 05:19 EST 2022. Contains 350481 sequences. (Running on oeis4.)