A155046 List of pairs: first pair is (1,1); then follow (x,y) with (x+2y, x+y).

1, 1, 3, 2, 7, 5, 17, 12, 41, 29, 99, 70, 239, 169, 577, 408, 1393, 985, 3363, 2378, 8119, 5741, 19601, 13860, 47321, 33461, 114243, 80782, 275807, 195025, 665857, 470832, 1607521, 1136689, 3880899, 2744210, 9369319, 6625109, 22619537, 15994428

Offset: 1

Author

Vincenzo Librandi, Jan 19 2009

Keywords

• nonn
• easy
• tabf

Comments

Sequence of x: A078057(n); sequence of y: A000129(n). - R. J. Mathar, Feb 19 2009

List of pairs (a, b) such that (a, b*sqrt(2)) = (1 + sqrt(2))^n. In the commutative ring Z[sqrt(2)], the set { +/- (1 + sqrt(2))^n} is a multiplicative group. - Michel Lagneau, Nov 27 2015

The fractions a(2*n-1)/a(2*n) are successive convergents of the simple continued fraction of sqrt(2). - Alexander Fraebel, Sep 03 2020

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,1).

Crossrefs

Cf. A002965. - Jaume Oliver Lafont, Feb 08 2009

Cf. A000129, A001333, A078057.

Programs

Haskell

import Data.List (transpose)
a155046 n = a155046_list !! n
a155046_list = concat $ transpose [tail a001333_list, tail a000129_list]
-- Reinhard Zumkeller, Jan 01 2014

Mathematica

LinearRecurrence[{0,2,0,1}, {1,1,3,2}, 40] (* Vincenzo Librandi, Mar 14 2012 *)
NestList[{#[[1]]+2#[[2]],Total[#]}&,{1,1},20]//Flatten (* Harvey P. Dale, Nov 21 2020 *)

PARI

Vec(x*(1+x+x^2)/(1-2*x^2-x^4) + O(x^50)) \\ Michel Marcus, Nov 28 2015

Formula

a(n) = 2*a(n-2) + a(n-4) for n > 4. - R. J. Mathar, Feb 19 2009

a(2k+1)^2 - 2*a(2k+2)^2 = +-1. - Vincenzo Librandi, Mar 14 2012

G.f.: x*(1+x+x^2)/(1-2*x^2-x^4). - Bruno Berselli, Mar 14 2012

Extensions

First term in two pairs corrected by R. J. Mathar, Feb 19 2009