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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A343261 a(n) = 2 * T(n,(n+2)/2) where T(n,x) is a Chebyshev polynomial of the first kind.

Original entry on oeis.org

2, 3, 14, 110, 1154, 15127, 238142, 4379769, 92198402, 2186871698, 57721023502, 1678243366813, 53301709843202, 1836220544383695, 68200709735854334, 2716906424134261502, 115561578124838522882, 5227260815326346060059, 250566480717349417632398
Offset: 0

Views

Author

Seiichi Manyama, Apr 09 2021

Keywords

Links

Crossrefs

Main diagonal of A299741.
Cf. A115066, A342167, A342206, A343259, A343260.

Programs

  • Mathematica
    Table[2*ChebyshevT[n, (n+2)/2], {n, 0, 18}] (* Amiram Eldar, Apr 09 2021 *)
  • PARI
    a(n) = 2*polchebyshev(n, 1, (n+2)/2);
  • PARI
    a(n) = round(2*cos(n*acos((n+2)/2)));
  • PARI
    a(n) = if(n==0, 2, 2*n*sum(k=0, n, n^k*binomial(n+k, 2*k)/(n+k)));

Formula

a(n) = 2 * cos(n*arccos((n+2)/2)).
a(n) = 2 * n * Sum_{k=0..n} n^k * binomial(n+k,2*k)/(n+k) for n > 0.
a(n) ~ exp(2) * n^n. - Vaclav Kotesovec, Apr 09 2021

A342206 a(n) = T(n,n+2) where T(n,x) is a Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 3, 31, 485, 10081, 262087, 8193151, 299537289, 12545596801, 592479412811, 31154649926687, 1805486216133613, 114342125644787041, 7857107443850071695, 582268591681887560191, 46292552162781456490001, 3930448770533424343942657
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2021

Keywords

Links

Crossrefs

Cf. A107995, A115066, A323117, A342205.

Programs

  • Mathematica
    Table[ChebyshevT[n, n + 2], {n, 0, 16}] (* Amiram Eldar, Mar 05 2021 *)
  • PARI
    a(n) = polchebyshev(n, 1, n+2);
  • PARI
    a(n) = round(cos(n*acos(n+2)));
  • PARI
    a(n) = if(n==0, 1, n*sum(k=0, n, (2*n+2)^k*binomial(n+k, 2*k)/(n+k)));

Formula

a(n) = cos(n*arccos(n+2)).
a(n) = n * Sum_{k=0..n} (2*n+2)^k * binomial(n+k,2*k)/(n+k) for n > 0.
a(n) ~ exp(2) * 2^(n-1) * n^n. - Vaclav Kotesovec, Mar 12 2024

A343260 a(n) = 2 * T(n,(n+1)/2) where T(n,x) is a Chebyshev polynomial of the first kind.

Original entry on oeis.org

2, 2, 7, 52, 527, 6726, 103682, 1874888, 38925119, 912670090, 23855111399, 687808321212, 21687295069442, 742397047217294, 27420344506901023, 1086932029484351248, 46027034321342899967, 2073668380220713167378, 99042070146811639444802
Offset: 0

Views

Author

Seiichi Manyama, Apr 09 2021

Keywords

Links

Crossrefs

Cf. A097691, A115066, A342205, A343259, A343261.

Programs

  • Mathematica
    Table[2*ChebyshevT[n, (n+1)/2], {n, 0, 18}] (* Amiram Eldar, Apr 09 2021 *)
  • PARI
    a(n) = 2*polchebyshev(n, 1, (n+1)/2);
  • PARI
    a(n) = round(2*cos(n*acos((n+1)/2)));
  • PARI
    a(n) = if(n==0, 2, 2*n*sum(k=0, n, (n-1)^k*binomial(n+k, 2*k)/(n+k)));

Formula

a(n) = 2 * cos(n*arccos((n+1)/2)).
a(n) = 2 * n * Sum_{k=0..n} (n-1)^k * binomial(n+k,2*k)/(n+k) for n > 0.
a(n) ~ exp(1) * n^n. - Vaclav Kotesovec, Apr 09 2021
Conjecture: a(p^r) == 1 (mod p^(2*r)) for positive integer r and all primes p >= 5. - Peter Bala, Mar 11 2024
Previous Showing 11-13 of 13 results.

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