A341782 -id:A341782 - cp's OEIS frontend

A341535 a(n) = sqrt(Product_{1<=j,k<=n} (4sin((2j-1)Pi/(2n))^2 + 4sin((2k-1)*Pi/n)^2)).

1, 2, 36, 224, 38416, 2540032, 4115479104, 3044533460992, 48656376372265216, 387018647188487143424, 62441634466575620320306176, 5221063878050546380074377019392, 8590392749565593082105293619707908096, 7476351474500749779460880888573410601336832

Offset: 0

Seiichi Manyama, Feb 13 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..62

Main diagonal of A341533.

Cf. A341478, A341479, A341782.

Table[Sqrt[Product[4*Sin[(2*j - 1)*Pi/(2*n)]^2 + 4*Sin[(2*k - 1)*Pi/n]^2, {k, 1, n}, {j, 1, n}]], {n, 0, 20}] // Round (* Vaclav Kotesovec, Feb 14 2021 *)

default(realprecision, 120);
a(n) = round(sqrt(prod(j=1, n, prod(k=1, n, 4*sin((2*j-1)*Pi/(2*n))^2+4*sin((2*k-1)*Pi/n)^2))));

a(n) ~ 2^(1/4)*(1 + sqrt(2)*(1 + (-1)^n)/2) * exp(2*G*n^2/Pi), where G is Catalan's constant A006752. - Vaclav Kotesovec, Feb 14 2021

If n is odd, a(n) = 2*A341478(n). - Seiichi Manyama, Feb 19 2021

A341738 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = sqrt( Product_{a=1..n} Product_{b=1..k-1} (4sin((2a-1)Pi/(2n))^2 + 4sin(2b*Pi/k)^2) ).

Original entry on oeis.org

1, 2, 1, 7, 2, 1, 16, 25, 2, 1, 41, 72, 112, 2, 1, 98, 361, 400, 529, 2, 1, 239, 1250, 4961, 2312, 2527, 2, 1, 576, 5041, 25088, 77841, 13456, 12100, 2, 1, 1393, 18432, 200999, 559682, 1270016, 78408, 57967, 2, 1

Offset: 1

Seiichi Manyama, Feb 18 2021

			Square array begins:
  1, 2,     7,    16,       41,        98, ...
  1, 2,    25,    72,      361,      1250, ...
  1, 2,   112,   400,     4961,     25088, ...
  1, 2,   529,  2312,    77841,    559682, ...
  1, 2,  2527, 13456,  1270016,  12771458, ...
  1, 2, 12100, 78408, 20967241, 292820000, ...

Main diagonal gives A341782.

Cf. A341533, A341739, A341741.

default(realprecision, 120);
T(n, k) = round(sqrt(prod(a=1, n, prod(b=1, k-1, 4*sin((2*a-1)*Pi/(2*n))^2+4*sin(2*b*Pi/k)^2))));

If k is odd, T(n,k) = A341533(n,k)/2.

cp's OEIS Frontend

A341535 a(n) = sqrt(Product_{1<=j,k<=n} (4sin((2j-1)Pi/(2n))^2 + 4sin((2k-1)*Pi/n)^2)).

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A341738 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = sqrt( Product_{a=1..n} Product_{b=1..k-1} (4sin((2a-1)Pi/(2n))^2 + 4sin(2b*Pi/k)^2) ).

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Programs

PARI

Formula

cp's OEIS Frontend

A341535 a(n) = sqrt(Product_{1<=j,k<=n} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin((2*k-1)*Pi/n)^2)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A341738 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = sqrt( Product_{a=1..n} Product_{b=1..k-1} (4*sin((2*a-1)*Pi/(2*n))^2 + 4*sin(2*b*Pi/k)^2) ).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A341535 a(n) = sqrt(Product_{1<=j,k<=n} (4sin((2j-1)Pi/(2n))^2 + 4sin((2k-1)*Pi/n)^2)).

A341738 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = sqrt( Product_{a=1..n} Product_{b=1..k-1} (4sin((2a-1)Pi/(2n))^2 + 4sin(2b*Pi/k)^2) ).