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%I #46 Nov 21 2020 17:39:24
%S 1,1,3,2,7,5,17,12,41,29,99,70,239,169,577,408,1393,985,3363,2378,
%T 8119,5741,19601,13860,47321,33461,114243,80782,275807,195025,665857,
%U 470832,1607521,1136689,3880899,2744210,9369319,6625109,22619537,15994428
%N List of pairs: first pair is (1,1); then follow (x,y) with (x+2y, x+y).
%C Sequence of x: A078057(n); sequence of y: A000129(n). - _R. J. Mathar_, Feb 19 2009
%C 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
%C 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
%H Vincenzo Librandi, <a href="/A155046/b155046.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,1).
%F a(n) = 2*a(n-2) + a(n-4) for n > 4. - _R. J. Mathar_, Feb 19 2009
%F a(2k+1)^2 - 2*a(2k+2)^2 = +-1. - _Vincenzo Librandi_, Mar 14 2012
%F G.f.: x*(1+x+x^2)/(1-2*x^2-x^4). - _Bruno Berselli_, Mar 14 2012
%t LinearRecurrence[{0,2,0,1}, {1,1,3,2}, 40] (* _Vincenzo Librandi_, Mar 14 2012 *)
%t NestList[{#[[1]]+2#[[2]],Total[#]}&,{1,1},20]//Flatten (* _Harvey P. Dale_, Nov 21 2020 *)
%o (Haskell)
%o import Data.List (transpose)
%o a155046 n = a155046_list !! n
%o a155046_list = concat $ transpose [tail a001333_list, tail a000129_list]
%o -- _Reinhard Zumkeller_, Jan 01 2014
%o (PARI) Vec(x*(1+x+x^2)/(1-2*x^2-x^4) + O(x^50)) \\ _Michel Marcus_, Nov 28 2015
%Y Cf. A002965. - _Jaume Oliver Lafont_, Feb 08 2009
%Y Cf. A000129, A001333, A078057.
%K nonn,easy,tabf
%O 1,3
%A _Vincenzo Librandi_, Jan 19 2009
%E First term in two pairs corrected by _R. J. Mathar_, Feb 19 2009
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