cp's OEIS Frontend

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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A203850 G.f.: Product_{n>=1} (1 - Lucas(n)*x^n + (-x^2)^n) / (1 + Lucas(n)*x^n + (-x^2)^n) where Lucas(n) = A000204(n).

Original entry on oeis.org

1, -2, -4, 0, 14, 16, 0, 0, 4, -152, -188, 0, 0, -44, 0, 0, 4414, 5456, -4, 0, 1288, 0, 0, 0, 0, -335406, -414728, 0, 0, -97904, 0, 0, 4, 0, -8828, 0, 66770564, 82532956, 0, 0, 19483388, -304, 0, 0, 0, 1756816, 0, 0, 0, -34787592002, -42999828492, 0, 60508, -10150882544, 0, 0, 0, 0, -915304508, 0, 0, 796
Offset: 0

Views

Author

Paul D. Hanna, Jan 07 2012

Keywords

Comments

Compare to: Product_{n>=1} (1-q^n)/(1+q^n) = 1 + 2*Sum_{n>=1} (-1)^n*q^(n^2), the Jacobi theta_4 function, which has the g.f: exp( Sum_{n>=1} -(sigma(2*n)-sigma(n)) * q^n/n ).

Examples

			G.f.: A(x) = 1 - 2*x - 4*x^2 + 14*x^4 + 16*x^5 + 4*x^8 - 152*x^9 - 188*x^10 +...
-log(A(x)) = 2*x + 4*3*x^2/2 + 8*4*x^3/3 + 8*7*x^4/4 + 12*11*x^5/5 + 16*18*x^6/6 +...+ (sigma(2*n)-sigma(n))*Lucas(n)*x^n/n +...
Compare to the logarithm of Jacobi theta4 H(x) = 1 + 2*Sum_{n>=1} (-1)^n*x^(n^2):
-log(H(x)) = 2*x + 4*x^2/2 + 8*x^3/3 + 8*x^4/4 + 12*x^5/5 + 16*x^6/6 + 16*x^7/7 +...+ (sigma(2*n)-sigma(n))*x^n/n +...
The g.f. equals the product:
A(x) = (1-x-x^2)/(1+x-x^2) * (1-3*x^2+x^4)/(1+3*x^2+x^4) * (1-4*x^3-x^6)/(1+4*x^3-x^6) * (1-7*x^4+x^8)/(1+7*x^4+x^8) * (1-11*x^5-x^10)/(1+11*x^5-x^10) *...* (1 - Lucas(n)*x^n + (-x^2)^n)/(1 + Lucas(n)*x^n + (-x^2)^n) *...
Positions of zeros form A022544:
[3,6,7,11,12,14,15,19,21,22,23,24,27,28,30,31,33,35,38,39,42,43,44,...]
which are numbers that are not the sum of 2 squares.

Links

Crossrefs

Cf. A203860, A203861, A022544, A000204 (Lucas), A203318, A225524.

Programs

  • PARI
    /* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
  • PARI
    {a(n)=polcoeff(prod(m=1, n, 1 - Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))/prod(m=1, n, 1 + Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n)), n)}
  • PARI
    {a(n)=polcoeff(prod(m=1, n\2+1, (1 - Lucas(2*m-1)*x^(2*m-1) - x^(4*m-2))^2*(1 - Lucas(2*m)*x^(2*m) + x^(4*m) +x*O(x^n))), n)}
  • PARI
    {a(n)=polcoeff(exp(sum(k=1, n,-(sigma(2*k)-sigma(k))*Lucas(k)*x^k/k)+x*O(x^n)), n)}

Formula

a(n) = 0 iff n is not the sum of 2 squares (A022544).
G.f.: Product_{n>=1} (1 - Lucas(2*n-1)*x^(2*n-1) - x^(4*n-2))^2 * (1 - Lucas(2*n)*x^(2*n) + x^(4*n)).
G.f.: exp( Sum_{n>=1} -(sigma(2*n)-sigma(n)) * Lucas(n) * x^n/n ) where Lucas(n) = A000204(n).

A225525 a(n) = (sigma(2*n) - sigma(n))*Lucas(n) where Lucas(n) = A000204(n) and sigma(n) = A000203(n) is the sum of divisors of n.

Original entry on oeis.org

2, 12, 32, 56, 132, 288, 464, 752, 1976, 2952, 4776, 10304, 14588, 26976, 65472, 70624, 128556, 300456, 373960, 726096, 1566464, 1900944, 3075792, 6635648, 10401182, 15200808, 35136320, 45481408, 68991060, 178607808, 192662336, 311734208, 756594816, 918147096, 1980790944, 3472069328
Offset: 1

Views

Author

Paul D. Hanna, May 09 2013

Keywords

Comments

Compare l.g.f. to log(theta_4(x)) = Sum_{n>=1} (sigma(2*n)-sigma(n))*x^n/n, where Jacobi theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n*x^(n^2).

Examples

			L.g.f.: L(x) = 2*x + 4*3*x^2/2 + 8*4*x^3/3 + 8*7*x^4/4 + 12*11*x^5/5 + 16*18*x^6/6 +...
where
exp(-L(x)) = 1 - 2*x - 4*x^2 + 14*x^4 + 16*x^5 + 4*x^8 - 152*x^9 - 188*x^10 +...+ A203850(n)*x^n +...
Also,
exp(L(x)) = 1 + 2*x + 8*x^2 + 24*x^3 + 66*x^4 + 184*x^5 + 488*x^6 + 1248*x^7 +...+ A225524(n)*x^n +...
Exponentiation yields the product:
exp(L(x)) = (1+x-x^2)/(1-x-x^2) * (1+3*x^2+x^4)/(1-3*x^2+x^4) * (1+4*x^3-x^6)/(1-4*x^3-x^6) * (1+7*x^4+x^8)/(1-7*x^4+x^8) * (1+11*x^5-x^10)/(1-11*x^5-x^10) *...* (1 + Lucas(n)*x^n + (-x^2)^n)/(1 - Lucas(n)*x^n + (-x^2)^n) *...

Crossrefs

Cf. A203850, A225524.

Programs

  • Mathematica
    Table[(DivisorSigma[1,2n]-DivisorSigma[1,n])LucasL[n],{n,40}] (* Harvey P. Dale, Sep 10 2016 *)
  • PARI
    {a(n)=(sigma(2*n) - sigma(n))*(fibonacci(n-1)+fibonacci(n+1))}
for(n=1,40,print1(a(n),", "))
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=n*polcoeff(-log(prod(m=1, n\2+1, (1 - Lucas(2*m-1)*x^(2*m-1) - x^(4*m-2))^2*(1 - Lucas(2*m)*x^(2*m) + x^(4*m) +x*O(x^n)))), n)}
for(n=1,40,print1(a(n),", "))

Formula

Logarithmic derivative of A225524 and A203850 (up to sign).
L.g.f.: Sum_{n>=1} log( (1 + Lucas(n)*x^n + (-x^2)^n) / (1 - Lucas(n)*x^n + (-x^2)^n) ) = Sum_{n>=1} a(n)*x^n/n.
a(n) == 0 (mod 2); a(n) == 2 (mod 4) iff n is congruent to 1 or 5 mod 6 (A007310).
Showing 1-2 of 2 results.

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