A245282 G.f.: Sum_{n>=1} Fibonacci(n+1) * x^n / (1 - x^n).
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
Offset: 1
Keywords
Examples
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) + ...
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..1000
Programs
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Maple
with(numtheory): seq(add(combinat[fibonacci](d+1), d in divisors(n)), n=1..60); # Ridouane Oudra, Apr 10 2025
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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),", "))
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
a(n) = Sum_{d|n} Fibonacci(d+1). - Ridouane Oudra, Apr 10 2025