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

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

Offset

0,2

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

Table of n, a(n) for n=0..19.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (34,-1).

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).

Prog

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

Crossrefs

Cf. A000129, A001653, A029547, A316709.

Sequence in context: A167738 A107964 A096699 * A218807 A218193 A218369

Adjacent sequences: A316705 A316706 A316707 * A316709 A316710 A316711

Keyword

nonn,easy

Author

Wolfdieter Lang, Jul 11 2018

Status

approved