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 1-2 of 2 results.

A152189 Product_{k=1..floor((n-1)/2)} (1 + 4*cos(k*Pi/n)^2)*(1 + 4*sin(k*Pi/n)^2).

Original entry on oeis.org

1, 1, 1, 8, 9, 55, 64, 377, 441, 2584, 3025, 17711, 20736, 121393, 142129, 832040, 974169, 5702887, 6677056, 39088169, 45765225, 267914296, 313679521, 1836311903, 2149991424, 12586269025, 14736260449, 86267571272, 101003831721, 591286729879, 692290561600
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Nov 28 2008

Keywords

Comments

It appears that limit(sqrt(a(n+2)/a(n)), n->Infinity) = 1+(sqrt(5)+1)/2.

Crossrefs

Cf. A152191.

Programs

  • Mathematica
    f[n_] = Product[(1 + 4*Cos[k*Pi/n]^2)*(1 + 4*Sin[k*Pi/n]^2), {k, 1, Floor[(n - 1)/2]}]; Table[N[f[n]], {n, 0, 30}]; Round[%] (* corrected by Colin Barker, Apr 11 2014 *)
  • PARI
    a(n) = round(prod(k=1, floor((n-1)/2), (1+4*cos(k*Pi/n)^2)*(1+4*sin(k*Pi/n)^2))) \\ Colin Barker, Apr 11 2014

Formula

Empirical g.f.: (x^6+x^5-9*x^4+7*x^2-x-1) / ((x-1)*(x+1)*(x^2-3*x+1)*(x^2+3*x+1)). - Colin Barker, Apr 11 2014

Extensions

Two initial terms added, and several terms corrected by Colin Barker, Apr 11 2014

A152192 a(n) = Product_{k=1..floor((n-1)/2)} (1 + 4*cos(2*Pi*k/n)^2).

Original entry on oeis.org

1, 1, 1, 2, 1, 5, 4, 13, 9, 34, 25, 89, 64, 233, 169, 610, 441, 1597, 1156, 4181, 3025, 10946, 7921, 28657, 20736, 75025, 54289, 196418, 142129, 514229, 372100, 1346269, 974169, 3524578, 2550409, 9227465, 6677056, 24157817, 17480761, 63245986, 45765225
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Nov 28 2008

Keywords

Links

Crossrefs

Cf. A152189, A152191.

Programs

  • Mathematica
    a[n_] := Product[(1 + 4*Cos[2*Pi*k/n]^2), {k, 1, Floor[(n - 1)/2]}]; a = Table[N[a[n]], {n, 0, 30}]
Join[{1}, Table[If[EvenQ[n], Fibonacci[(n)/2]^2, Fibonacci[n]], {n, 1, 30}]] (* Greg Dresden, Oct 16 2021 *)
  • PARI
    a(n) = round(prod(k=1, floor((n-1)/2), (1+4*cos(2*Pi*k/n)^2))) \\ Colin Barker, Apr 11 2014

Formula

Lim_{n->infinity} sqrt(a(n+2)/a(n)) = (sqrt(5) + 1)/2.
G.f.: (1+x^6-x^5-3*x^4-x^2+x)/((x^2+1)*(x^2+x-1)*(x^2-x-1)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009
For n > 0, a(n) = Fibonacci(n) for n odd, and Fibonacci(n/2)^2 for n even. - Greg Dresden, Oct 16 2021

Extensions

More terms and edited by Colin Barker, Michel Marcus, and Joerg Arndt, Apr 11 2014
Showing 1-2 of 2 results.

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