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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A316708 Bisection of the odd-indexed Pell numbers A001653: part 1.

Original entry on oeis.org

1, 29, 985, 33461, 1136689, 38613965, 1311738121, 44560482149, 1513744654945, 51422757785981, 1746860020068409, 59341817924539925, 2015874949414289041, 68480406462161287469, 2326317944764069484905, 79026329715516201199301, 2684568892382786771291329, 91196316011299234022705885, 3097990175491791170000708761, 105240469650709600546001391989
Offset: 0

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Author

Wolfdieter Lang, Jul 11 2018

Keywords

Comments

The other part of the bisection is given in A316709.
This sequence gives every other positive proper solutions of the Pell equation b^2 - 2*a^2 = -1 with a1 = a(n) = Pell(4*n+1) and b1 = b1(n) = A002315(2*n), for n >= 0. The other solutions are a2 = A316709(n) = Pell(4*n+3) and b2 = A002315(2*n+1), for n >= 0.

Links

Crossrefs

Cf. A000129, A001653, A029547, A316709.

Programs

  • PARI
    x='x+O('x^99); Vec((1-5*x)/(1-34*x+x^2)) \\ Altug Alkan, Jul 11 2018

Formula

a(n) = Pell(4*n+1) = A000129(4*n+1) = A001653(2*n+1), n >= 0.
a(n) = 34*a(n-1) - a(n-2), with a(-1) = 5 and a(0) = 1.
a(n) = S(n, 34) - 5*S(n-1, 34), where the Chebyshev polynomial S(n, 34) = A029547(n), n >= 0, with S(-1, x) = 0.
G.f.: (1 - 5*x)/(1 - 34*x + x^2).
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