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 1-8 of 8 results.

A203803 G.f.: exp( Sum_{n>=1} A000204(n)^3 * x^n/n ) where A000204 is the Lucas numbers.

Original entry on oeis.org

1, 1, 14, 35, 205, 744, 3414, 13926, 60060, 252330, 1072902, 4537272, 19234463, 81452015, 345084970, 1461714517, 6192083147, 26229794928, 111111714300, 470675847900, 1993816532280, 8445939457380, 35777578796220, 151556246864400, 642002579853325, 2719566542567917
Offset: 0

Views

Author

Paul D. Hanna, Jan 06 2012

Keywords

Comments

More generally, exp(Sum_{k>=1} A000204(k)^(2*n+1) * x^k/k) = Product_{k=0..n} 1/(1 - (-1)^(n-k)*A000204(2*k+1)*x - x^2)^binomial(2*n+1,n-k).

Examples

			G.f.: A(x) = 1 + x + 14*x^2 + 35*x^3 + 205*x^4 + 744*x^5 + 3414*x^6 +...
where
log(A(x)) = x + 3^3*x^2/2 + 4^3*x^3/3 + 7^3*x^4/4 + 11^3*x^5/5 + 18^3*x^6/6 + 29^3*x^7/7 + 47^3*x^8/8 +...+ Lucas(n)^3*x^n/n +...

Links

Crossrefs

Cf. A002571, A203804, A203805, A203806, A203807, A203808, A203809.
Cf. A203853, A203800.

Programs

  • Mathematica
    CoefficientList[Series[1/((1 + x - x^2)^3*(1 - 4*x - x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 24 2017 *)
  • PARI
    /* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
  • PARI
    {a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^3*x^k/k)+x*O(x^n)), n)}
  • PARI
    {a(n,m=1)=polcoeff(prod(k=0,m, 1/(1 - (-1)^(m-k)*Lucas(2*k+1)*x - x^2+x*O(x^n))^binomial(2*m+1,m-k)),n)}

Formula

G.f.: 1/( (1+x-x^2)^3 * (1-4*x-x^2) ).
G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203853(n) where A203853(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^2.

A203854 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^3, where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 13, 21, 79, 266, 957, 3484, 12935, 48768, 185951, 716418, 2781675, 10878520, 42789478, 169181010, 671866245, 2678678730, 10716651456, 43007270292, 173072549610, 698235680844, 2823329210391, 11439823946306, 46440709210035, 188856966693230, 769241291729020
Offset: 1

Views

Author

Paul D. Hanna, Jan 07 2012

Keywords

Examples

			G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^13 * (1-4*x^3-x^6)^21 * (1-7*x^4+x^8)^79 * (1-11*x^5-x^10)^266 * (1-18*x^6+x^12)^957 *...* (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) *...)
where F(x) = exp( Sum_{n>=1} Lucas(n)^4 * x^n/n ) = g.f. of A203804:
F(x) = 1 + x + 41*x^2 + 126*x^3 + 1526*x^4 + 7854*x^5 + 63629*x^6 +...
where
log(F(x)) = x + 3^4*x^2/2 + 4^4*x^3/3 + 7^4*x^4/4 + 11^4*x^5/5 + 18^4*x^6/6 + 29^4*x^7/7 + 47^4*x^8/8 +...+ Lucas(n)^4*x^n/n +...

Links

Crossrefs

Cf. A203804, A203853, A203855, A203856, A203857, A203858, A203859, A203800.

Programs

  • Mathematica
    a[n_] := 1/n DivisorSum[n, MoebiusMu[n/#] LucasL[#]^3&]; Array[a, 30] (* Jean-François Alcover, Dec 04 2015 *)
  • PARI
    {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^3)/n)}
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^4*x^m/m)+x*O(x^n)));if(n==1,1,polcoeff(F*prod(k=1,n-1,(1 - Lucas(k)*x^k + (-1)^k*x^(2*k) +x*O(x^n))^a(k)),n)/Lucas(n))}

Formula

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} Lucas(n)^4 * x^n/n), which is the g.f. of A203804.
a(n) ~ phi^(3*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 02 2017

A203855 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^4, where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 40, 85, 580, 2928, 17440, 101040, 609660, 3706880, 22887192, 142567200, 895855380, 5667708960, 36072949560, 230763023408, 1482822818820, 9565561745040, 61920953016320, 402074969960400, 2618069854211784, 17090016552803440, 111812320834030800
Offset: 1

Views

Author

Paul D. Hanna, Jan 07 2012

Keywords

Examples

			G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^40 * (1-4*x^3-x^6)^85 *
(1-7*x^4+x^8)^580 * (1-11*x^5-x^10)^2928 * (1-18*x^6+x^12)^17440 *...* (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) *...)
where F(x) = exp( Sum_{n>=1} Lucas(n)^5 * x^n/n ) = g.f. of A203805:
F(x) = 1 + x + 122*x^2 + 463*x^3 + 11985*x^4 + 85456*x^5 +...
where
log(F(x)) = x + 3^5*x^2/2 + 4^5*x^3/3 + 7^5*x^4/4 + 11^5*x^5/5 + 18^5*x^6/6 + 29^5*x^7/7 + 47^5*x^8/8 +...+ Lucas(n)^5*x^n/n +...

Links

Crossrefs

Cf. A203805, A203853, A203854, A203856, A203857, A203858, A203859, A203800.

Programs

  • Mathematica
    a[n_] := 1/n DivisorSum[n, MoebiusMu[n/#] LucasL[#]^4 &]; Array[a, 30] (* Jean-François Alcover, Dec 04 2015 *)
  • PARI
    {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^4)/n)}
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^5*x^m/m)+x*O(x^n)));if(n==1,1,polcoeff(F*prod(k=1,n-1,(1 - Lucas(k)*x^k + (-1)^k*x^(2*k) +x*O(x^n))^a(k)),n)/Lucas(n))}

Formula

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} Lucas(n)^5 * x^n/n), which is the g.f. of A203805.
a(n) ~ phi^(4*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 02 2017

A203856 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^5, where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 121, 341, 4141, 32210, 314717, 2930164, 28666025, 281724928, 2815289555, 28370872818, 288468152625, 2952876368200, 30409537607218, 314760765272250, 3272590619892675, 34158620991538050, 357779277130203136, 3758998894159780092, 39603542856374168550
Offset: 1

Views

Author

Paul D. Hanna, Jan 07 2012

Keywords

Examples

			G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^121 * (1-4*x^3-x^6)^341 * (1-7*x^4+x^8)^4141 * (1-11*x^5-x^10)^32210 * (1-18*x^6+x^12)^314717 * ... * (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) * ...)
where F(x) = exp( Sum_{n>=1} Lucas(n)^6 * x^n/n ) = g.f. of A203806:
F(x) = 1 + x + 365*x^2 + 1730*x^3 + 97390*x^4 + 948562*x^5 + ...
where
log(F(x)) = x + 3^6*x^2/2 + 4^6*x^3/3 + 7^6*x^4/4 + 11^6*x^5/5 + 18^6*x^6/6 + 29^6*x^7/7 + 47^6*x^8/8 + ... + Lucas(n)^6*x^n/n + ...

Links

Crossrefs

Cf. A203806, A203853, A203854, A203855, A203857, A203858, A203859, A203800.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, MoebiusMu[n/#]*LucasL[#]^5&]/n; Array[a, 20] (* Jean-François Alcover, Dec 07 2015 *)
  • PARI
    {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^5)/n)}
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^6*x^m/m)+x*O(x^n)));if(n==1,1,polcoeff(F*prod(k=1,n-1,(1 - Lucas(k)*x^k + (-1)^k*x^(2*k) +x*O(x^n))^a(k)),n)/Lucas(n))}

Formula

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} Lucas(n)^6 * x^n/n), which is the g.f. of A203806.
a(n) ~ phi^(5*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 02 2017

A203857 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^6, where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 364, 1365, 29230, 354312, 5667900, 84974760, 1347387210, 21411102720, 346282421940, 5645803690800, 92886793449030, 1538448587832240, 25635241395476100, 429333683845968552, 7222607529064709670, 121980435560782376760, 2067248664062116147200
Offset: 1

Views

Author

Paul D. Hanna, Jan 07 2012

Keywords

Examples

			G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^364 * (1-4*x^3-x^6)^1365 * (1-7*x^4+x^8)^29230 * (1-11*x^5-x^10)^354312 * (1-18*x^6+x^12)^5667900 * ... * (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) * ...)
where F(x) = exp( Sum_{n>=1} Lucas(n)^7 * x^n/n ) = g.f. of A203807:
F(x) = 1 + x + 1094*x^2 + 6555*x^3 + 809765*x^4 + 10676072*x^5 + ...
where log(F(x)) = x + 3^7*x^2/2 + 4^7*x^3/3 + 7^7*x^4/4 + 11^7*x^5/5 + 18^7*x^6/6 + 29^7*x^7/7 + 47^7*x^8/8 + ... + Lucas(n)^7*x^n/n + ...

Links

Crossrefs

Cf. A203807, A203853, A203854, A203855, A203856, A203858, A203859, A203800.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, MoebiusMu[n/#]*LucasL[#]^6&]/n; Array[a, 18] (* Jean-François Alcover, Dec 07 2015 *)
  • PARI
    {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^6)/n)}
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^7*x^m/m)+x*O(x^n)));if(n==1,1,polcoeff(F*prod(k=1,n-1,(1 - Lucas(k)*x^k + (-1)^k*x^(2*k) +x*O(x^n))^a(k)),n)/Lucas(n))}

Formula

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} Lucas(n)^7 * x^n/n), which is the g.f. of A203807.
a(n) ~ phi^(6*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 02 2017

A203858 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^7, where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 1093, 5461, 205339, 3897434, 102033577, 2464268044, 63327787115, 1627243839488, 42592757748839, 1123514934469218, 29909548355299575, 801531714260597080, 21610508530971123358, 585611144766061271010, 15940294818101008311105, 435592135387553867410170
Offset: 1

Views

Author

Paul D. Hanna, Jan 07 2012

Keywords

Examples

			G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^1093 * (1-4*x^3-x^6)^5461 * (1-7*x^4+x^8)^205339 * (1-11*x^5-x^10)^3897434 * (1-18*x^6+x^12)^102033577 * ... * (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) * ...)
where F(x) = exp( Sum_{n>=1} Lucas(n)^8 * x^n/n ) = g.f. of A203808:
F(x) = 1 + x + 3281*x^2 + 25126*x^3 + 6845526*x^4 + 121368902*x^5 + ...
where
log(F(x)) = x + 3^8*x^2/2 + 4^8*x^3/3 + 7^8*x^4/4 + 11^8*x^5/5 + 18^8*x^6/6 + 29^8*x^7/7 + 47^8*x^8/8 + ... + Lucas(n)^8*x^n/n + ...

Links

Crossrefs

Cf. A203808, A203853, A203854, A203855, A203856, A203857, A203859, A203800.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, MoebiusMu[n/#]*LucasL[#]^7 &]/n; Array[a, 50] (* G. C. Greubel, Mar 05 2018 *)
  • PARI
    {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^7)/n)}
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^8*x^m/m)+x*O(x^n)));if(n==1,1,polcoeff(F*prod(k=1,n-1,(1 - Lucas(k)*x^k + (-1)^k*x^(2*k) +x*O(x^n))^a(k)),n)/Lucas(n))}

Formula

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} Lucas(n)^8 * x^n/n), which is the g.f. of A203808.
a(n) ~ phi^(7*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 02 2017

A203859 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^8, where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 3280, 21845, 1439560, 42871776, 1836648080, 71463773280, 2976410112120, 123670531932160, 5238909421389744, 223579471959374400, 9630874585937597160, 417598023129771078720, 18217658692611614215920, 798773601460909332885856, 35180230663617319463871240
Offset: 1

Views

Author

Paul D. Hanna, Jan 07 2012

Keywords

Examples

			G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^3280 * (1-4*x^3-x^6)^21845 * (1-7*x^4+x^8)^1439560 * (1-11*x^5-x^10)^42871776 *...* (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) *...)
where F(x) = exp( Sum_{n>=1} Lucas(n)^9 * x^n/n ) = g.f. of A203809:
F(x) = 1 + x + 9842*x^2 + 97223*x^3 + 58608265*x^4 + 1390114224*x^5 +...
where
log(F(x)) = x + 3^9*x^2/2 + 4^9*x^3/3 + 7^9*x^4/4 + 11^9*x^5/5 + 18^9*x^6/6 + 29^9*x^7/7 + 47^9*x^8/8 +...+ Lucas(n)^9*x^n/n +...

Links

Crossrefs

Cf. A203809, A203853, A203854, A203855, A203856, A203857, A203858, A203800.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, MoebiusMu[n/#]*LucasL[#]^8&]/n; Array[a, 30] (* G. C. Greubel, Dec 19 2017 *)
  • PARI
    {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^8)/n)}
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^9*x^m/m)+x*O(x^n)));if(n==1,1,polcoeff(F*prod(k=1,n-1,(1 - Lucas(k)*x^k + (-1)^k*x^(2*k) +x*O(x^n))^a(k)),n)/Lucas(n))}

Formula

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} Lucas(n)^9 * x^n/n), which is the g.f. of A203809.
a(n) ~ phi^(8*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 02 2017

A212443 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * A002203(d)^2, where A002203 is the companion Pell numbers.

Original entry on oeis.org

4, 16, 64, 280, 1344, 6496, 32640, 166320, 862400, 4523232, 23970240, 128063040, 689008320, 3728973120, 20285199872, 110841302880, 608029121280, 3346972244000, 18480871268160, 102328688556864, 568014587806720, 3160148362953120, 17617881702072960
Offset: 1

Views

Author

Paul D. Hanna, May 17 2012

Keywords

Crossrefs

Cf. A008683, A212442, A203853, A002203.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, MoebiusMu[n/#] * LucasL[#, 2]^2 &] / n; Array[a, 25] (* Amiram Eldar, Aug 22 2023 *)
  • PARI
    {A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*A002203(d)^2)/n)}
for(n=1,30,print1(a(n),","))

Formula

G.f.: 1/Product_{n>=1} (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} A002203(n)^3 * x^n/n), which equals the g.f. of A212442.
Showing 1-8 of 8 results.

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