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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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
Previous Showing 11-14 of 14 results.

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