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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A349070 a(n) = T(3*n, n), where T(n, x) is the Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 1, 1351, 3880899, 28355806081, 429364731169925, 11731978174095849671, 525735133485219615486151, 36049049892983023583045990401, 3588952618789973294871796462342089, 497937643558017209960022199517744044999, 93156956377055671178035977181412016527566091
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 07 2021

Keywords

Comments

In general, for k>=1, T(k*n, n) ~ 2^(k*n - 1) * n^(k*n).

Links

Crossrefs

Cf. A053120, A115066, A173129, A349072, A349074.

Programs

  • Mathematica
    Table[ChebyshevT[3*n, n], {n, 0, 13}]
  • PARI
    a(n) = polchebyshev(3*n, 1, n); \\ Michel Marcus, Nov 07 2021

Formula

a(n) = cosh(3*n*arccosh(n)).
a(n) = ((n + sqrt(n^2-1))^(3*n) + (n - sqrt(n^2-1))^(3*n))/2.
a(n) ~ 2^(3*n-1) * n^(3*n).

A173171 a(n) = - sin^2((2n-1)*arccos(sqrt n)) = sin^2((2n-1)*arcsin(sqrt n)) - 1.

Original entry on oeis.org

-1, 0, 49, 23762, 25421763, 48225038404, 142786923879605, 608447515452613206, 3527836867501829594887, 26710782540478226038759688, 255922222218837615280903143609, 3026917140685147530327256796600410
Offset: 0

Views

Author

Artur Jasinski, Feb 11 2010

Keywords

Links

Crossrefs

Cf. A132592, A146311, A146312, A146313, A173115, A173116 A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A173148, A173151, A173170.

Programs

  • Mathematica
    Table[Round[-N[ Sin[(2 n - 1) ArcCos[Sqrt[n]]]^2, 100]], {n, 0, 20}] (* Artur Jasinski, Feb 11 2010; Typo fixed by Vincenzo Librandi, Jun 29 2014 *)

A173175 a(n) = sinh^2( 2n*arcsinh(sqrt n)).

Original entry on oeis.org

0, 8, 2400, 1825200, 2687489280, 6503780163000, 23436548406180000, 117725514040791821024, 786292024016459316676608, 6739465778247681589030301160, 72110357818535214970387726284000, 942092946853627620313318842336862608, 14758709413836719039368938494112056160000
Offset: 0

Views

Author

Artur Jasinski, Feb 11 2010

Keywords

Links

Crossrefs

Cf. A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A173148, A173151, A173170, A173171, A322699.

Programs

  • Maple
    A173175 := proc(n) sinh(2*n*arcsinh(sqrt(n))) ; %^2 ; expand(%); simplify(%) ; end proc: # R. J. Mathar, Feb 26 2011
  • Mathematica
    Table[Round[N[Sinh[(2 n) ArcSinh[Sqrt[n]]]^2, 100]], {n, 0, 20}]
  • PARI
    {a(n) = (polchebyshev(2*n, 1, 2*n+1)-1)/2} \\ Seiichi Manyama, Jan 02 2019
  • PARI
    {a(n) = 1/2*(-1+sum(k=0, 2*n, binomial(4*n, 2*k)*(n+1)^(2*n-k)*n^k))} \\ Seiichi Manyama, Jan 02 2019

Formula

From Seiichi Manyama, Jan 02 2019: (Start)
a(n) = A322699(n,2*n).
a(n) = (T_{2*n}(2*n+1) - 1)/2 where T_{n}(x) is a Chebyshev polynomial of the first kind.
a(n) = 1/2 * (-1 + Sum_{k=0..2*n} binomial(4*n,2*k)*(n+1)^(2*n-k)*n^k). (End)
a(n) ~ exp(1) * 2^(4*n - 2) * n^(2*n). - Vaclav Kotesovec, Jan 02 2019

Extensions

a(11)-a(12) from Seiichi Manyama, Jan 02 2019

A173194 a(n) = -sin^2 (2*n*arccos n) = - sin^2 (2*n*arcsin n).

Original entry on oeis.org

0, 0, 9408, 384199200, 54471499791360, 20405558846592060000, 16793517249722147195701440, 26730228454204365035835498694848, 75019085697452515216001640927169855488, 346154755746154620929434271983392498083891520
Offset: 0

Views

Author

Artur Jasinski, Feb 12 2010

Keywords

Links

Crossrefs

Cf. A132592, A146311, A146312, A146313, A173115, A173116 A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A173148, A173151, A173170, A173171, A173174, A173175, A173176.

Programs

  • Maple
    A173194 := proc(n) ((n+sqrt(n^2-1))^(2*n)-(n-sqrt(n^2-1))^(2*n))^2 ; expand(%/4) ; simplify(%) ; end proc: # R. J. Mathar, Feb 26 2011
  • Mathematica
    Round[Table[ -N[Sin[2 n ArcSin[n]], 100]^2, {n, 0, 15}]] (* Artur Jasinski *)
Table[FullSimplify[(-1/2 (x - Sqrt[ -1 + x^2])^(2 x) + 1/2 (x + Sqrt[ -1 + x^2])^(2 x))^2], {x, 0, 7}] (* Artur Jasinski, Feb 17 2010 *)
Table[(n^2-1)*ChebyshevU[2*n-1, n]^2, {n, 0, 20}] (* Vaclav Kotesovec, Jan 05 2019 *)
  • PARI
    {a(n) = (n^2-1)*n^2*(sum(k=0, n-1, binomial(2*n, 2*k+1)*(n^2-1)^(n-1-k)*n^(2*k)))^2} \\ Seiichi Manyama, Jan 05 2019
  • PARI
    {a(n) = (n^2-1)*polchebyshev(2*n-1, 2, n)^2} \\ Seiichi Manyama, Jan 05 2019

Formula

4*a(n) = ( (n+sqrt(n^2-1))^(2*n) - (n-sqrt(n^2-1))^(2*n) )^2. - Artur Jasinski, Feb 17 2010
From Seiichi Manyama, Jan 05 2019: (Start)
a(n) = (n^2-1) * n^2 * (Sum_{k=0..n-1} binomial(2*n,2*k+1)*(n^2-1)^(n-1-k)*n^(2*k))^2.
For n > 0, a(n) = (n^2-1) * U_{2*n-1}(n)^2 where U_{n}(x) is a Chebyshev polynomial of the second kind. (End)
a(n) ~ 2^(4*n - 2) * n^(4*n). - Vaclav Kotesovec, Jan 05 2019

Extensions

a(9) from Seiichi Manyama, Jan 05 2019

A173150 a(n) = sinh^2 (2n*arccosh(sqrt n)).

Original entry on oeis.org

0, 0, 288, 235224, 354079488, 865363202000, 3134808545188320, 15796198853361763368, 105717380511014096025600, 907380314352243226001152800, 9718304978537581699085289156000
Offset: 0

Views

Author

Artur Jasinski, Feb 11 2010

Keywords

Comments

Also a(n) = -sin(2n*arccos(sqrt(n)))^2 = -sin(2n*arcsin(sqrt(n)))^2.

Links

Crossrefs

Cf. A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A173148.

Programs

  • Maple
    A173150 := proc(n) sinh(2*n*arccosh(sqrt(n))) ; %^2 ; expand(%) ; simplify(%) ;end proc: # R. J. Mathar, Feb 26 2011
  • Mathematica
    Table[Round[-Sin[2 n ArcCos[Sqrt[n]]]^2], {n, 0, 20}] (* Artur Jasinski, Feb 11 2010 *)

Extensions

Minor edits by Vaclav Kotesovec, Apr 05 2016
Previous Showing 11-15 of 15 results.

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