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 1-7 of 7 results.

A077916 Expansion of (1-x)^(-1)/(1 + 2*x - 2*x^2 - x^3).

Original entry on oeis.org

1, -1, 5, -10, 30, -74, 199, -515, 1355, -3540, 9276, -24276, 63565, -166405, 435665, -1140574, 2986074, -7817630, 20466835, -53582855, 140281751, -367262376, 961505400, -2517253800, 6590256025, -17253514249, 45170286749, -118257345970, 309601751190, -810547907570
Offset: 0

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Author

N. J. A. Sloane, Nov 17 2002

Keywords

Links

Crossrefs

Cf. A002571.
Bisections are A103433 and A103434.
Cf. A064831, A000045. A203803, A207969, A207970.

Programs

  • Mathematica
    a[0] = 1; a[1] = -1; a[2] = 5; a[3] = -10; a[n_] := a[n] = -a[n-1] + 4 a[n-2] - a[n-3] - a[n-4]; Table[a[n], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 28 2015 *)
CoefficientList[Series[(1 - x)^(-1)/(1 + 2*x - 2*x^2 - x^3), {x, 0, 50}], x] (* G. C. Greubel, Dec 25 2017 *)
Table[If[OddQ[n], (Fibonacci[2n+2]+n+1)/5, -(Fibonacci[2n+2]-n-1)/5], {n,1,20}] (* Rigoberto Florez, May 09 2019 *)
  • PARI
    Vec((1-x)^(-1)/(1+2*x-2*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012
  • PARI
    Vec(1/((1-x)^2*(1+3*x+x^2)) + O(x^100)) \\ Altug Alkan, Oct 28 2015

Formula

a(n-1) = Sum_{i=1..n} (-1)^(i+1)*Fibonacci(i)*Fibonacci(i+1), n >= 1. - Alexander Adamchuk, Jun 16 2006
From R. J. Mathar, Mar 14 2011: (Start)
G.f.: 1/((1-x)^2*(1+3*x+x^2)).
a(n) = ((-1)^n*A001906(n+2)+n+2)/5. (End)
O.g.f.: exp( Sum_{n >= 1} Lucas(n)^2*(-x)^n/n ) = 1 - x + 5*x^2 - 10*x^3 + .... Cf. A203803. See also A207969 and A207970. - Peter Bala, Apr 03 2014
From Vladimir Reshetnikov, Oct 28 2015: (Start)
Recurrence (5-term): a(0) = 1, a(1) = -1, a(2) = 5, a(3) = -10, a(n) = -a(n-1) + 4*a(n-2) - a(n-3) - a(n-4).
Recurrence (4-term): a(0) = 1, a(1) = -1, a(2) = 5, n*a(n) = (1-2*n)*a(n-1) + (3*n+3)*a(n-2) + (n+1)*a(n-3).
(End)
a(n) = (F(2n+2)+n+1)/5 if n is odd and a(n)= -(F(2n+2)-n-1)/5 if n is even, where F(n) = Fibonacci numbers (A000045). - Rigoberto Florez, May 09 2019

A207970 G.f.: exp( Sum_{n>=1} 5*Fibonacci(n)^6 * x^n/n ).

Original entry on oeis.org

1, 5, 15, 140, 1505, 21875, 319620, 4936985, 77358485, 1236083870, 19982821875, 326511608255, 5379199407890, 89249496596015, 1489580814490755, 24988546214618750, 421055477328447620, 7122346563647277860, 120891417096833214485, 2058225554792946621495
Offset: 0

Views

Author

Paul D. Hanna, Feb 22 2012

Keywords

Comments

Conjecture: exp( Sum_{n>=1} 5*Fibonacci(n)^(2*k) * x^n/n ) is an integer series for integers k >= 0.
Note that exp( Sum_{n>=1} 5*Fibonacci(n)^(2*k+1) * x^n/n ) is not an integer series for integers k.
Note that exp( Sum_{n>=1} Fibonacci(n)^(2*k) * x^n/n ) is not an integer series for integers k.

Examples

			G.f.: A(x) = 1 + 5*x + 15*x^2 + 140*x^3 + 1505*x^4 + 21875*x^5 + 319620*x^6 + ...
such that
log(A(x))/5 = x + x^2/2 + 2^6*x^3/3 + 3^6*x^4/4 + 5^6*x^5/5 + 8^6*x^6/6 + 13^6*x^7/7 + ... + Fibonacci(n)^6*x^n/n + ...

Crossrefs

Cf. A054888, A207969, A207971, A207972, A207834, A207834. A077916, A203806.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(k=1,n,5*fibonacci(k)^6*x^k/k)+x*O(x^n)),n)}
for(n=0,31,print1(a(n),", "))

Formula

The o.g.f. A(x) = 1 + 5*x + 15*x^2 + 140*x^3 + ... is an algebraic function: A(x)^25 = ( (1 + 2*x + x^2)^10*(1 + 7*x + x^2)^6 )/( (1 - 3*x + x^2)^15*(1 - 18*x + x^2) ). Cf. A203806. - Peter Bala, Apr 03 2014
a(n) ~ 2^(17/25) * 5^(13/50) * phi^(6*n) / (Gamma(1/25) * 3^(3/5) * n^(24/25)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 18 2020

A207972 Expansion of g.f.: exp( Sum_{n>=1} 5*Fibonacci(n^2) * x^n/n ).

Original entry on oeis.org

1, 5, 20, 115, 1665, 82650, 12847310, 5620114060, 6659421195205, 21082748688390045, 177217804775828062850, 3941798437750184226876305, 231505293200405380457355524620, 35848160499603817968830380832049915, 14619744406297572472084577939841875791890
Offset: 0

Views

Author

Paul D. Hanna, Feb 22 2012

Keywords

Comments

Moss and Ward prove that this is an integral sequence. - Peter Bala, Nov 28 2022
Let A(x) be the g.f. for this sequence. Note that the expansion of A(x)^(1/5) = exp( Sum_{n>=1} Fibonacci(n^2) * x^n/n ) does not have integer coefficients.

Examples

			G.f.: A(x) = 1 + 5*x + 20*x^2 + 115*x^3 + 1665*x^4 + 82650*x^5 + ...
such that
log(A(x))/5 = x + 3*x^2/2 + 34*x^3/3 + 987*x^4/4 + 75025*x^5/5 + 14930352*x^6/6 + 7778742049*x^7/7 + ... + Fibonacci(n^2)*x^n/n + ...

Links

Crossrefs

Cf. A054888, A207969, A207970, A207971, A054783, A207834, A211892.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(k=1,n,5*fibonacci(k^2)*x^k/k)+x*O(x^n)),n)}
for(n=0,16,print1(a(n),", "))

A207971 G.f.: exp( Sum_{n>=1} 5*Fibonacci(n)^(2*n) * x^n/n ).

Original entry on oeis.org

1, 5, 15, 140, 8795, 9808325, 57315191130, 2812698182891585, 894119494320160426760, 2048089587570930007354766745, 32079229816919862900907520464756250, 3500720882833094608324749707338857577696205, 2633228648869966875007549667526201212159637714889015
Offset: 0

Views

Author

Paul D. Hanna, Feb 22 2012

Keywords

Comments

Give g.f. A(x), note that A(x)^(1/5) is not an integer series.

Examples

			G.f.: A(x) = 1 + 5*x + 15*x^2 + 140*x^3 + 8795*x^4 + 9808325*x^5 +...
such that
log(A(x))/5 = x + x^2/2 + 2^6*x^3/3 + 3^8*x^4/4 + 5^10*x^5/5 + 8^12*x^6/6 + 13^14*x^7/7 +...+ Fibonacci(n)^(2*n)*x^n/n +...

Crossrefs

Cf. A054888, A207969, A207970, A207972, A207834, A207834.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(k=1,n,5*fibonacci(k)^(2*k)*x^k/k)+x*O(x^n)),n)}
for(n=0,31,print1(a(n),", "))

A211894 G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n)^2 * x^n/n ), where Jacobsthal(n) = A001045(n).

Original entry on oeis.org

1, 3, 6, 18, 57, 195, 684, 2460, 8970, 33102, 123204, 461868, 1741410, 6597750, 25099584, 95822928, 366943881, 1408947675, 5422742910, 20915079258, 80820382425, 312839889219, 1212812010804, 4708415402772, 18302630040504, 71230126892088, 277514015733168
Offset: 0

Views

Author

Paul D. Hanna, Apr 25 2012

Keywords

Comments

Given g.f. A(x), note that A(x)^(1/3) is not an integer series.

Examples

			G.f.: A(x) = 1 + 3*x + 6*x^2 + 18*x^3 + 57*x^4 + 195*x^5 + 684*x^6 +...
such that
log(A(x))/3 = x + x^2/2 + 3^2*x^3/3 + 5^2*x^4/4 + 11^2*x^5/5 + 21^2*x^6/6 + 43^2*x^7/7 +...+ Jacobsthal(n)^2*x^n/n +...
Jacobsthal numbers begin:
A001045 = [1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,...].

Crossrefs

Cf. A211893, A211895, A211896, A054888, A207969, A001045 (Jacobsthal).

Programs

  • Mathematica
    CoefficientList[Series[(1+2*x)^(2/3) / ((1-x)*(1-4*x))^(1/3), {x, 0, 30}], x] (* Vaclav Kotesovec, Oct 18 2020 *)
  • PARI
    {Jacobsthal(n)=polcoeff(x/(1-x-2*x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, 3*Jacobsthal(k)^2*x^k/k)+x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff(((1+2*x)^2/((1-x)*(1-4*x) +x*O(x^n)))^(1/3),n)}

Formula

G.f.: (1+2*x)^(2/3) / ((1-x)*(1-4*x))^(1/3).
G.f.: exp( Sum_{n>=1} (2^n - (-1)^n)^2 / 3 * x^n/n ).
a(n) ~ 3^(1/3) * 2^(2*n) / (n^(2/3) * Gamma(1/3)). - Vaclav Kotesovec, Oct 18 2020

A211896 G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n)^4 * x^n/n ), where Jacobsthal(n) = A001045(n).

Original entry on oeis.org

1, 3, 6, 90, 723, 10689, 130428, 1862580, 25594611, 368313993, 5289203262, 77279744418, 1134460916361, 16798605635235, 249994099311288, 3740771822960664, 56208829313956998, 847934859174601650, 12834366187138678836, 194855374723972622988, 2966358133685609559042
Offset: 0

Views

Author

Paul D. Hanna, Apr 25 2012

Keywords

Comments

Given g.f. A(x), note that A(x)^(1/3) is not an integer series.

Examples

			G.f.: A(x) = 1 + 3*x + 6*x^2 + 90*x^3 + 723*x^4 + 10689*x^5 + 130428*x^6 +...
such that
log(A(x))/3 = x + x^2/2 + 3^4*x^3/3 + 5^4*x^4/4 + 11^4*x^5/5 + 21^4*x^6/6 + 43^4*x^7/7 +...+ Jacobsthal(n)^4*x^n/n +...
Jacobsthal numbers begin:
A001045 = [1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,...].

Crossrefs

Cf. A211893, A211894, A211895, A207969, A001045 (Jacobsthal).

Programs

  • PARI
    {Jacobsthal(n)=polcoeff(x/(1-x-2*x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, 3*Jacobsthal(k)^4*x^k/k)+x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff(((1+2*x)^4*(1+8*x)^4/((1-x)*(1-4*x)^6*(1-16*x))+x*O(x^n))^(1/27),n)}

Formula

G.f.: ( (1+2*x)^4*(1+8*x)^4 / ((1-x)*(1-4*x)^6*(1-16*x)) )^(1/27).
G.f.: exp( Sum_{n>=1} (2^n - (-1)^n)^4 / 27 * x^n/n ).
a(n) ~ 3^(5/27) * 2^(4*n) / (5^(1/27) * Gamma(1/27) * n^(26/27)). - Vaclav Kotesovec, Oct 18 2020

A208055 G.f.: exp( Sum_{n>=1} 2*Pell(n)^4 * x^n/n ), where Pell(n) = A000129(n).

Original entry on oeis.org

1, 2, 18, 450, 11362, 311426, 8857426, 259072706, 7730804098, 234255654466, 7184570715602, 222512186923010, 6947171244623714, 218374183252085826, 6903938704875627410, 219355658720815861378, 6999679608428089841154, 224210965624588803552642
Offset: 0

Views

Author

Paul D. Hanna, Feb 22 2012

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 450*x^3 + 11362*x^4 + 311426*x^5 +...
such that, by definition,
log(A(x))/2 = x + 2^4*x^2/2 + 5^4*x^3/3 + 12^4*x^4/4 + 29^4*x^5/5 + 70^4*x^6/6 + 169^4*x^7/7 + 408^4*x^8/8 +...+ Pell(n)^4*x^n/n +...

Crossrefs

Cf. A000129, A208034, A208056, A204061, A204062, A207969.

Programs

  • PARI
    {Pell(n)=polcoeff(x/(1-2*x-x^2 +x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(m=1,n,2*Pell(m)^4*x^m/m) +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

Formula

The o.g.f. A(x) = 1 + 2*x + 18*x^2 + 450*x^3 + ... is an algebraic function: A(x)^32 = (1 + 6*x + x^2)^4/( (1 - 34*x + x^2)*(1 - 2*x + x^2)^3 ). Cf. A207969. - Peter Bala, Apr 03 2014
From Vaclav Kotesovec, Oct 31 2024: (Start)
G.f.: (1 + x*(6 + x))^(1/8) / ((1 - x)^(3/16)*(1 + (-17 + 12*sqrt(2))*x)^(1/32) * (1 - (17 + 12*sqrt(2))*x)^(1/32)).
a(n) ~ 5^(1/8) * (1 + sqrt(2))^(4*n) / (2^(13/64) * 3^(1/32) * Gamma(1/32) * n^(31/32)). (End)
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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