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A155046
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List of pairs: first pair is (1,1); then follow (x,y) with (x+2y, x+y).
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2
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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
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,1).
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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
(PARI) Vec(x*(1+x+x^2)/(1-2*x^2-x^4) + O(x^50)) \\ Michel Marcus, Nov 28 2015
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CROSSREFS
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Cf. A002965. - Jaume Oliver Lafont, Feb 08 2009
Cf. A000129, A001333, A078057.
Sequence in context: A318783 A253564 A232751 * A236388 A033318 A093780
Adjacent sequences: A155043 A155044 A155045 * A155047 A155048 A155049
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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Vincenzo Librandi, Jan 19 2009
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EXTENSIONS
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First term in two pairs corrected by R. J. Mathar, Feb 19 2009
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STATUS
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approved
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