A266506 a(n) = 2*a(n-4) + a(n-8) for n >= 8.

2, -1, 2, 1, 1, 3, 3, 5, 4, 5, 8, 11, 9, 13, 19, 27, 22, 31, 46, 65, 53, 75, 111, 157, 128, 181, 268, 379, 309, 437, 647, 915, 746, 1055, 1562, 2209, 1801, 2547, 3771, 5333, 4348, 6149, 9104, 12875, 10497, 14845, 21979, 31083, 25342, 35839, 53062, 75041, 61181, 86523

Offset

0,1

Comments

Previous name was: a(2n) = a(2n - 4) + a(2n - 3) and a(2n + 1) = 2*a(2n - 4) + a(2n - 3), with a(0) = 2, a(1) = -1, a(2) = 2, a(3) = 1. Alternatively, interleave denominators (A266504) and numerators (A266505) of convergents to sqrt(2).

a(2n) gives all x in N | 2*x^2 - 7(-1)^x = y^2. a(2n+1) gives associated y values.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,1).

Formula

From Chai Wah Wu, Sep 17 2016: (Start)

a(n) = 2*a(n-4) + a(n-8) for n > 7.

G.f.: (-3*x^7 + x^6 - 5*x^5 + 3*x^4 - x^3 - 2*x^2 + x - 2)/(x^8 + 2*x^4 - 1).

(End)

Mathematica

CoefficientList[Series[(-3*x^7 + x^6 - 5*x^5 + 3*x^4 - x^3 - 2*x^2 + x - 2)/(x^8 + 2*x^4 - 1), {x, 0, 50}], x] (* G. C. Greubel, Jul 27 2018 *)

Prog

(PARI) x='x+O('x^50); Vec((-3*x^7+x^6-5*x^5+3*x^4-x^3-2*x^2+x-2)/(x^8 + 2*x^4-1)) \\ G. C. Greubel, Jul 27 2018
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((-3*x^7+x^6-5*x^5+3*x^4-x^3-2*x^2+x-2)/(x^8+2*x^4-1))); // G. C. Greubel, Jul 27 2018

Crossrefs

Cf. A266504, A266505.

Sequence in context: A137278 A205810 A139368 * A134303 A078997 A024680

Adjacent sequences: A266503 A266504 A266505 * A266507 A266508 A266509

Keyword

sign,easy,less

Author

Raphie Frank, Dec 30 2015

Extensions

Edited, new name using given formula, Joerg Arndt, Jan 31 2024

Status

approved