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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A203413 G.f.: exp( Sum_{n>=1} A203414(n)*x^n/n ) where A203414(n) = n*Pell(n)*Sum_{d|n} 1/(d*Pell(d)).

Original entry on oeis.org

1, 1, 3, 8, 25, 64, 200, 512, 1528, 4048, 11654, 30585, 88601, 231295, 651713, 1733011, 4814031, 12685230, 35225415, 92628772, 254268558, 672643614, 1826716115, 4814931851, 13086575526, 34391797265, 92637759753, 244294085952, 654813738224, 1720509596070, 4606408076053
Offset: 0

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Author

Paul D. Hanna, Jan 01 2012

Keywords

Comments

Note: x/(1-2*x-x^2) = exp(Sum_{n>=1} A002203(n)*x^n/n) is the g.f. of the Pell numbers and A002203 is the companion Pell numbers.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 25*x^4 + 64*x^5 + 200*x^6 + 512*x^7 +...
where
log(A(x)) = x/(1-2*x-x^2) + (x^2/2)/(1-6*x^2+x^4) + (x^3/3)/(1-14*x^3-x^6) + (x^4/4)/(1-34*x^4+x^8) +...+ (x^n/n)/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) +...
Equivalently, log(A(x)) = Sum_{n>=1} G_n(x^n) * x^n/n
where G_n(x) = exp( Sum_{k>=1} A002203(n*k)*x^k/k ), which begin:
G_1(x) = x*(1 + 2*x + 5*x^2 + 12*x^3 + 29*x^4 +...+ Pell(n+1)*x^n +...
G_2(x) = 1 + 6*x^2 + 35*x^4 + 204*x^6 +...+ Pell(2*n+2)/2*x^(2*n) +...
G_3(x) = 1 + 14*x^3 + 197*x^6 + 2772*x^9 +...+ Pell(3*n+3)/5*x^(3*n) +...
G_4(x) = 1 + 34*x^4 + 1155*x^8 + 39236*x^12 +...+ Pell(4*n+4)/12*x^(4*n) +...
G_5(x) = 1 + 82*x^5 + 6725*x^10 + 551532*x^15 +...+ Pell(5*n+5)/29*x^(5*n) +...
G_6(x) = 1 + 198*x^6 + 39203*x^12 + 7761996*x^18 +...+ Pell(6*n+6)/70*x^(6*n) +...
For n>=1, G_n(x) = 1/(1 - A002203(n)*x + (-1)^n*x^2),
where the companion Pell numbers (offset 1) begin:
A002203 = [2,6,14,34,82,198,478,1154,2786,6726 16238,...].
The logarithmic derivative of this sequence begins:
A203414 = [1,5,16,61,146,554,1184,4149,9457,29890,63152,...].

Links

Crossrefs

Cf. A203413, A203319, A203321; A000129 (Pell), A002203 (companion Pell).

Programs

  • PARI
    /* Subroutines used in PARI programs below: */
{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{A002203(n)=Pell(n-1)+Pell(n+1)}
  • PARI
    {a(n)=local(A=1);A=exp(sum(m=1,n+1,x^m*Pell(m)*sumdiv(m, d, 1/(d*Pell(d))) +x*O(x^n)));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1);A=exp(sum(m=1,n+1,(x^m/m)/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1);A=exp(sum(m=1,n+1,(x^m/m)*exp(sum(k=1,floor((n+1)/m),A002203(m*k)*x^(m*k)/k)+x*O(x^n))));polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+2*x+x*O(x^n),G=1/(1-2*x-x^2+x*O(x^n)));A=exp(sum(m=1,n+1,(x^m/m)*round(prod(k=0,m-1,subst(G,x,exp(2*Pi*I*k/m)*x+x*O(x^n))))));polcoeff(A, n)}

Formula

G.f.: exp( Sum_{n>=1} (x^n/n) / (1 - A002203(n)*x^n + (-1)^n*x^(2*n)) ).
G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} A002203(n*k)*x^(n*k)/k ) ).
G.f.: exp( Sum_{n>=1} G_n(x^n) * x^n/n ) such that G_n(x^n) = Product_{k=0..n-1} G(u^k*x) where G(x) = 1/(1-2*x-x^2) and u is an n-th root of unity.

A203319 a(n) = n*Fibonacci(n) * Sum_{d|n} 1/(d*Fibonacci(d)).

Original entry on oeis.org

1, 3, 7, 19, 26, 81, 92, 267, 358, 848, 980, 3061, 3030, 7976, 11042, 25099, 27150, 78642, 79440, 219884, 270704, 584862, 659112, 1977909, 1950651, 4735370, 6204499, 14189096, 14912642, 43168586, 41734340, 110786987, 135815060, 290854380, 339428752, 953889058, 893839230
Offset: 1

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Author

Paul D. Hanna, Jan 01 2012

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 26*x^5/5 + 81*x^6/6 +...
where
L(x) = x*(1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 +...+ F(n+1)*x^n +...) +
x^2/2*(1 + 3*x^2 + 8*x^4 + 21*x^6 + 55*x^8 +...+ F(2*n+2)*x^(2*n) +...) +
x^3/3*(1 + 4*x^3 + 17*x^6 + 72*x^9 +...+ F(3*n+3)/2*x^(3*n) +...) +
x^4/4*(1 + 7*x^4 + 48*x^8 + 329*x^12 +...+ F(4*n+4)/3*x^(4*n) +...) +
x^5/5*(1 + 11*x^5 + 122*x^10 + 1353*x^15 +...+ F(5*n+5)/5*x^(5*n) +...) +
x^6/6*(1 + 18*x^6 + 323*x^12 + 5796*x^18 +...+ F(6*n+6)/8*x^(6*n) +...) +...
here F(n) = Fibonacci(n) = A000045(n).
Equivalently,
L(x) = x/(1-x-x^2) + (x^2/2)/(1-3*x^2+x^4) + (x^3/3)/(1-4*x^3-x^6) + (x^4/4)/(1-7*x^4+x^8) +...+ (x^n/n)/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
here Lucas(n) = A000032(n).
Exponentiation of the l.g.f. equals the g.f. of A203318:
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 36*x^6 + 64*x^7 +...+ A203318(n)*x^n +...

Links

Crossrefs

Cf. A203318, A203321; A203414 (Pell variant).
Cf. A000032 (Lucas), A000045 (Fibonacci), A001906, A001076, A004187, A049666, A049660, A049667.

Programs

  • Mathematica
    a[n_] := n Fibonacci[n] DivisorSum[n, 1/(# Fibonacci[#]) &]; Array[a, 40] (* Jean-François Alcover, Dec 23 2015 *)
  • PARI
    {a(n)=if(n<1,0, n*fibonacci(n)*sumdiv(n,d,1/(d*fibonacci(d))) )}
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=n*polcoeff(sum(m=1,n+1,(x^m/m)/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(L=x); L=sum(m=1, n, x^m/m*exp(sum(k=1, floor((n+1)/m), Lucas(m*k)*x^(m*k)/k)+x*O(x^n))); n*polcoeff(L,n)}
  • PARI
    {a(n)=local(A=1+x+x*O(x^n),F=1/(1-x-x^2+x*O(x^n))); A=exp(sum(m=1, n+1, x^m/m*round(prod(k=0, m-1, subst(F, x, exp(2*Pi*I*k/m)*x+x*O(x^n)))))); n*polcoeff(log(A), n)}

Formula

Equals the logarithmic derivative of A203318.
L.g.f.: L(x) = Sum_{n>=1} a(n)*x^n/n satisfies:
(1) L(x) = Sum_{n>=1} x^n/n * Sum_{k>=0} F(n*k+n)/F(n) * x^(n*k) where F(n) = Fibonacci(n).
(2) L(x) = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} Lucas(n*k)*x^(n*k)/k ) where Lucas(n) = A000032(n).
(3) L(x) = Sum_{n>=1} x^n/n * 1/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) where Lucas(n) = A000032(n).
(4) L(x) = Sum_{n>=1} x^n/n * G_n(x^n) where G_n(x^n) = Product_{k=0..n-1} G(u^k*x) where G(x) = 1/(1-x-x^2) and u is an n-th root of unity.
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