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 1-5 of 5 results.

A173129 a(n) = cosh(2 * n * arccosh(n)).

Original entry on oeis.org

1, 1, 97, 19601, 7380481, 4517251249, 4097989415521, 5170128475599457, 8661355881006882817, 18605234632923999244961, 49862414878754347585980001, 163104845048002042971670685041, 639582975902942936737758325440001
Offset: 0

Views

Author

Artur Jasinski, Feb 10 2010

Keywords

Links

Crossrefs

Cf. A001079, A037270, A053120 (Chebyshev polynomial), A058331, A115066, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173148.
Cf. A349070, A349071, A349073.

Programs

  • Maple
    seq(orthopoly[T](2*n,n), n=0..50); # Robert Israel, Dec 27 2018
  • Mathematica
    Table[Round[Cosh[2 n ArcCosh[n]]], {n, 0, 20}] (* Artur Jasinski, Feb 10 2010 *)
Round[Table[1/2 (x - Sqrt[ -1 + x^2])^(2 x) + 1/2 (x + Sqrt[ -1 + x^2])^(2 x), {x, 0, 10}]] (* Artur Jasinski, Feb 14 2010 *)
Table[ChebyshevT[2*n, n], {n, 0, 15}] (* Vaclav Kotesovec, Nov 07 2021 *)
  • PARI
    {a(n) = sum(k=0, n, binomial(2*n, 2*k)*(n^2-1)^(n-k)*n^(2*k))} \\ Seiichi Manyama, Dec 27 2018
  • PARI
    {a(n) = polchebyshev(2*n, 1, n)} \\ Seiichi Manyama, Dec 28 2018
  • PARI
    {a(n) = polchebyshev(n, 1, 2*n^2-1)} \\ Seiichi Manyama, Dec 29 2018

Formula

a(n) = (1/2)*((n+sqrt(n^2-1))^(2*n) + (n-sqrt(n^2-1))^(2*n)). - Artur Jasinski, Feb 14 2010, corrected by Vaclav Kotesovec, Apr 05 2016
a(n) = Sum_{k=0..n} binomial(2*n,2*k)*(n^2-1)^(n-k)*n^(2*k). - Seiichi Manyama, Dec 27 2018
a(n) = T_{2n}(n) where T_{2n} is a Chebyshev polynomial of the first kind. - Robert Israel, Dec 27 2018
a(n) = T_{n}(2*n^2-1) where T_{n}(x) is a Chebyshev polynomial of the first kind. - Seiichi Manyama, Dec 29 2018

A323118 a(n) = U_{n}(n) where U_{n}(x) is a Chebyshev polynomial of the second kind.

Original entry on oeis.org

1, 2, 15, 204, 3905, 96030, 2883167, 102213944, 4178507265, 193501094490, 10011386405999, 572335117886532, 35827847605137601, 2437406399741075126, 179059769134174484415, 14127079203550978667760, 1191321539697176278429697, 106935795565608726499866930
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2019

Keywords

Links

Crossrefs

Main diagonal of A323182.
Cf. A115066, A318192, A323117, A349073, A349074, A349075, A349076.

Programs

  • Mathematica
    Table[ChebyshevU[n, n], {n, 0, 20}] (* Vaclav Kotesovec, Jan 05 2019 *)
  • PARI
    a(n) = polchebyshev(n, 2, n);
  • PARI
    a(n) = sum(k=0, n\2, (n^2-1)^k*n^(n-2*k)*binomial(n+1, 2*k+1));
  • PARI
    a(n) = sum(k=0, n, (2*n-2)^k*binomial(n+1+k, 2*k+1)); \\ Seiichi Manyama, Mar 03 2021

Formula

a(n) = Sum_{k=0..floor(n/2)} (n^2-1)^k*n^(n-2*k) * binomial(n+1,2*k+1).
a(n) ~ 2^n * n^n. - Vaclav Kotesovec, Jan 05 2019
a(n) = Sum_{k=0..n} (2*n-2)^(n-k) * binomial(2*n+1-k,k) = Sum_{k=0..n} (2*n-2)^k * binomial(n+1+k,2*k+1). - Seiichi Manyama, Mar 03 2021

A349071 a(n) = T(n, 2*n), where T(n, x) is the Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 2, 31, 846, 32257, 1580050, 94558751, 6686381534, 545471324161, 50428155189474, 5210183616019999, 594949288292777902, 74404881332329766401, 10114032809617941274226, 1484781814660796486716447, 234114571438498509048719550, 39459584112457284328544403457
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 07 2021

Keywords

Links

Crossrefs

Cf. A053120, A115066, A173129, A349072, A349073.

Programs

  • Mathematica
    Table[ChebyshevT[n, 2*n], {n, 0, 20}]
  • PARI
    a(n) = polchebyshev(n, 1, 2*n); \\ Michel Marcus, Nov 07 2021
  • Python
    from sympy import chebyshevt
def A349071(n): return chebyshevt(n,n<<1) # Chai Wah Wu, Nov 08 2023

Formula

a(n) = cosh(n*arccosh(2*n)).
a(n) = ((2*n + sqrt(4*n^2-1))^n + (2*n - sqrt(4*n^2-1))^n)/2.
a(n) ~ 2^(2*n-1) * n^n.

A349074 a(n) = U(3*n, n), where U(n, x) is the Chebyshev polynomial of the second kind.

Original entry on oeis.org

1, 4, 2911, 7997214, 57641556673, 867583274883920, 23630375698358890319, 1056918444955456528983706, 72383076947075470731692782081, 7200266529428094485775774835670652, 998383804974887102441600687728515247999, 186701261436825568741051032736345268517903734
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 07 2021

Keywords

Comments

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

Links

Crossrefs

Cf. A323118, A349070, A349073, A349076.

Programs

  • Mathematica
    Table[ChebyshevU[3*n, n], {n, 0, 13}]
  • PARI
    a(n) = polchebyshev(3*n, 2, n); \\ Michel Marcus, Nov 07 2021
  • Python
    from sympy import chebyshevu
def A349074(n): return chebyshevu(3*n,n) # Chai Wah Wu, Nov 08 2023

Formula

For n>1, a(n) = ((n + sqrt(n^2-1))^(3*n+1) - (n - sqrt(n^2-1))^(3*n+1)) / (2*sqrt(n^2-1)).
a(n) ~ 2^(3*n) * n^(3*n).

A349075 a(n) = U(n, 2*n), where U(n, x) is the Chebyshev polynomial of the second kind.

Original entry on oeis.org

1, 4, 63, 1704, 64769, 3168060, 189447551, 13389885712, 1092011153409, 100934312212404, 10426892198423999, 1190514147664125240, 148874434455514989313, 20235554722675691942764, 2970511463324707397138175, 468359315014627272862943520, 78938449723310515780367269889
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 07 2021

Keywords

Links

Crossrefs

Cf. A323118, A349073, A349076.

Programs

  • Mathematica
    Table[ChebyshevU[n, 2*n], {n, 0, 20}]
  • PARI
    a(n) = polchebyshev(n, 2, 2*n); \\ Michel Marcus, Nov 07 2021

Formula

a(n) = ((2*n + sqrt(4*n^2-1))^(n+1) - (2*n - sqrt(4*n^2-1))^(n+1)) / (2*sqrt(4*n^2-1)).
a(n) ~ 4^n * n^n.
Showing 1-5 of 5 results.

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