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.

Showing 1-5 of 5 results.

A203847 a(n) = tau(n)*Fibonacci(n), where tau(n) = A000005(n), the number of divisors of n.

Original entry on oeis.org

1, 2, 4, 9, 10, 32, 26, 84, 102, 220, 178, 864, 466, 1508, 2440, 4935, 3194, 15504, 8362, 40590, 43784, 70844, 57314, 370944, 225075, 485572, 785672, 1906866, 1028458, 6656320, 2692538, 13069854, 14098312, 22811548, 36909860, 134373168, 48315634, 156352676, 252983944
Offset: 1

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Author

Paul D. Hanna, Jan 11 2012

Keywords

Comments

Compare g.f. to the Lambert series identity: Sum_{n>=1} x^n/(1-x^n) = Sum_{n>=1} tau(n)*x^n.
Related identities:
(1) Sum_{n>=1} n^k*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_{k}(n)*Fibonacci(n)*x^n for k>=0.
(2) Sum_{n>=1} phi(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} n*Fibonacci(n)*x^n.
(3) Sum_{n>=1} moebius(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = x.
(4) Sum_{n>=1} lambda(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} Fibonacci(n^2)*x^(n^2).

Examples

			G.f.: A(x) = x + 2*x^2 + 4*x^3 + 9*x^4 + 10*x^5 + 32*x^6 + 26*x^7 +...
where A(x) = x/(1-x-x^2) + x^2/(1-3*x^2+x^4) + 2*x^3/(1-4*x^3-x^6) + 3*x^4/(1-7*x^4+x^8) + 5*x^5/(1-11*x^5-x^10) + 8*x^6/(1-18*x^6+x^12) +...+ Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...

Links

Crossrefs

Cf. A203848, A203849, A203838, A204060, A000005 (tau), A000204 (Lucas), A000045.
Cf. A205507, A205882, A205963, A205964, A205965, A205966.
Cf. A205967, A205968, A205969, A205970, A205971.
Cf. A205972, A205973, A205974, A205975, A205976.
Cf. A204291, A203801, A203850, A203860, A203861.

Programs

  • Mathematica
    Table[DivisorSigma[0, n]*Fibonacci[n], {n, 50}] (* G. C. Greubel, Jul 17 2018 *)
  • PARI
    {a(n)=sigma(n,0)*fibonacci(n)}
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1,n,fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
  • PARI
    a(n) = numdiv(n)*fibonacci(n); \\ Michel Marcus, Jul 18 2018

Formula

G.f.: Sum_{n>=1} Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} tau(n)*Fibonacci(n)*x^n, where Lucas(n) = A000204(n).

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).

A203861 G.f.: Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^3 where Lucas(n) = A000204(n).

Original entry on oeis.org

1, -3, -9, 20, 45, 0, -151, -231, 0, 140, 1107, 2052, 49, -1305, 0, -15004, -28260, 0, 17710, 0, 81, 324040, 589953, 0, -375570, -1089, 0, -124124, -10659705, -19764180, -121, 12605358, 0, 0, 4158315, 0, 567552368, 1052295189, -780030, -669901660, 0, 0, -221399431, -85965, 0
Offset: 0

Views

Author

Paul D. Hanna, Jan 07 2012

Keywords

Comments

a(A020757(n)) = 0 where A020757 lists numbers that are not the sum of two triangular numbers.

Examples

			G.f.: A(x) = 1 - 3*x - 9*x^2 + 20*x^3 + 45*x^4 - 151*x^6 - 231*x^7 +...
-log(A(x))/3 = x + 3*3*x^2/2 + 4*4*x^3/3 + 7*7*x^4/4 + 6*11*x^5/5 + 12*18*x^6/6 +...+ sigma(n)*A000204(n)*x^n/n +...
The g.f. equals the product:
A(x) = (1-x-x^2)^3 * (1-3*x^2+x^4)^3 * (1-4*x^3-x^6)^3 * (1-7*x^4+x^8)^3 * (1-11*x^5-x^10)^3 * (1-18*x^6+x^12)^3 *...* (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^3 *...
Positions of zeros form A020757:
[5,8,14,17,19,23,26,32,33,35,40,41,44,47,50,52,53,54,59,62,63,...]
which are numbers that are not the sum of two triangular numbers.

Links

Crossrefs

Cf. A203860, A203850, A156234, A020757.

Programs

  • PARI
    /* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
  • PARI
    {a(n)=polcoeff(exp(sum(k=1, n, -3*sigma(k)*Lucas(k)*x^k/k)+x*O(x^n)), n)}
  • PARI
    {a(n)=polcoeff(prod(m=1, n, 1 - Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))^3, n)}

Formula

G.f.: exp( Sum_{n>=1} -3 * sigma(n) * A000204(n) * x^n/n ).

A204382 G.f.: Product_{n>=1} (1 - A002203(n)*x^n + (-1)^n*x^(2*n)) where A002203(n) is the companion Pell numbers.

Original entry on oeis.org

1, -2, -7, -2, 1, 82, 34, 464, 198, -82, -1, 0, -39208, -16238, 6725, -551614, -228486, 95120, 0, 82, 6726, 0, 263673800, 109216786, -45239073, 0, 8957108166, 3706940654, -1536796802, -551614, -1, -109216786, 0, -18738638, -6726, -24954506565518, -10336495061766
Offset: 0

Views

Author

Paul D. Hanna, Jan 14 2012

Keywords

Comments

a(A093519(n)) = 0 where A093519 lists numbers that are not equal to the sum of two generalized pentagonal numbers.

Examples

			G.f.: A(x) = 1 - 2*x - 7*x^2 - 2*x^3 + x^4 + 82*x^5 + 34*x^6 + 464*x^7 +...
-log(A(x)) = 1*2*x + 3*6*x^2/2 + 4*14*x^3/3 + 7*34*x^4/4 + 6*82*x^5/5 + 12*198*x^6/6 +...+ sigma(n)*A002203(n)*x^n/n +...
The g.f. equals the product:
A(x) = (1-2*x-x^2) * (1-6*x^2+x^4) * (1-14*x^3-x^6) * (1-34*x^4+x^8) * (1-82*x^5-x^10) * (1-198*x^6+x^12) *...* (1 - A002203(n)*x^n + (-1)^n*x^(2*n)) *...
Positions of zeros form A093519:
[11,18,21,25,32,39,43,46,49,54,60,65,67,68,74,76,81,87,88,90,...].
which are numbers that are not the sum of two generalized pentagonal numbers.

Links

Crossrefs

Cf. A203860, A204383, A204384, A203534, A093519, A002203.

Programs

  • PARI
    /* Subroutine used in PARI programs below: */
{A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}
  • PARI
    {a(n)=polcoeff(exp(sum(k=1, n, -sigma(k)*A002203(k)*x^k/k)+x*O(x^n)), n)}
  • PARI
    {a(n)=polcoeff(prod(m=1, n, 1 - A002203(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n)), n)}

Formula

G.f.: exp( Sum_{n>=1} -sigma(n) * A002203(n) * x^n/n ).

A203801 G.f.: Product_{n>=1} (1 + Lucas(n)*x^n + (-1)^n*x^(2*n)) where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 1, 2, 7, 9, 27, 53, 109, 206, 463, 907, 1756, 3591, 6849, 13706, 27132, 51477, 99168, 195160, 366269, 707173, 1355524, 2558372, 4836092, 9186600, 17245564, 32428375, 61057276, 113946770, 212495896, 397836811, 737325660, 1368659832, 2544085015, 4694930535
Offset: 0

Views

Author

Paul D. Hanna, Jan 11 2012

Keywords

Comments

Analog to Euler's identity: Product_{n>=1} (1+x^n) = Product_{n>=1} 1/(1-x^(2*n-1)), which is the g.f. for the number of partitions into distinct parts.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 9*x^4 + 27*x^5 + 53*x^6 +...
where A(x) = (1+x-x^2) * (1+3*x^2+x^4) * (1+4*x^3-x^6) * (1+7*x^4+x^8) * (1+11*x^5-x^10) * (1+18*x^6+x^12) *...* (1 + Lucas(n)*x^n + (-1)^n*x^(2*n)) *...
and 1/A(x) = (1-x-x^2) * (1-4*x^3-x^6) * (1-11*x^5-x^10) * (1-29*x^7-x^14) * (1-76*x^9-x^18) * (1-199*x^11-x^22) *...* (1 - Lucas(2*n-1)*x^(2*n-1) + (-1)^n*x^(4*n-2)) *...
Also, the logarithm of the g.f. equals the series:
log(A(x)) = x + 1*3*x^2/2 + 4*4*x^3/3 + 1*7*x^4/4 + 6*11*x^5/5 + 4*18*x^6/6 + 8*29*x^7/7 + 1*47*x^8/8 +...+ A000593(n)*Lucas(n)*x^n/n +...

Links

Crossrefs

Cf. A156234, A203860, A203850, A000204, A000593.

Programs

  • Mathematica
    max = 40; s = Product[1 + LucasL[n]*x^n + (-1)^n*x^(2*n), {n, 1, max}] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 14 2015 *)
  • PARI
    /* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
  • PARI
    {a(n)=polcoeff(prod(k=1,n,1+Lucas(k)*x^k+(-1)^k*x^(2*k) +x*O(x^n)),n)}
  • PARI
    {a(n)=polcoeff(1/prod(k=1,n,1-Lucas(2*k-1)*x^(2*k-1)-x^(4*k-2) +x*O(x^n)),n)}
  • PARI
    /* Exponential form using sum of odd divisors of n: */
{A000593(n)=if(n<1, 0, sumdiv(n, d, (-1)^(d+1)*n/d))}
{a(n)=polcoeff(exp(sum(k=1, n, A000593(k)*Lucas(k)*x^k/k)+x*O(x^n)), n)}

Formula

G.f.: Product_{n>=1} 1/(1 - Lucas(2*n-1)*x^(2*n-1) + (-1)^n*x^(4*n-2)).
G.f.: exp( Sum_{n>=1} A000593(n) * Lucas(n) * x^n/n ) where A000593(n) = sum of odd divisors of n.
a(n) = (1/n)*Sum_{k=1..n} A000593(k)*Lucas(k)*a(n-k) for n>0, with a(0) = 1.
Showing 1-5 of 5 results.

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