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A078343 a(0) = -1, a(1) = 2; a(n) = 2*a(n-1) + a(n-2).

%I A078343 #75 May 26 2024 08:20:37
%S A078343 -1,2,3,8,19,46,111,268,647,1562,3771,9104,21979,53062,128103,309268,
%T A078343 746639,1802546,4351731,10506008,25363747,61233502,147830751,
%U A078343 356895004,861620759,2080136522,5021893803,12123924128,29269742059,70663408246,170596558551,411856525348
%N A078343 a(0) = -1, a(1) = 2; a(n) = 2*a(n-1) + a(n-2).
%C A078343 Inverse binomial transform of -1, 1, 6, 22, 76, 260, ... (see A111566). Binomial transform of -1, 3, -2, 6, -4, 12, -8, 24, -16, ... (see A162255). - _R. J. Mathar_, Oct 02 2012
%D A078343 H. S. M. Coxeter, 1998, Numerical distances among the circles in a loxodromic sequence, Nieuw Arch. Wisk, 16, pp. 1-9.
%H A078343 Reinhard Zumkeller, <a href="/A078343/b078343.txt">Table of n, a(n) for n = 0..1000</a>
%H A078343 H. S. M. Coxeter, <a href="https://doi.org/10.1007/BF03024413">Numerical distances among the spheres in a loxodromic sequence</a>, Math. Intell. 19(4) 1997 pp. 41-47. See page 41. See pp. 46-47.
%H A078343 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H A078343 José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, <a href="http://arxiv.org/abs/1212.1368">A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake</a>, arXiv:1212.1368 [cs.DM], 2012.
%H A078343 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,1).
%F A078343 For the unsigned version: a(1)=1; a(2)=2; a(n) = Sum_{k=2..n-1} (a(k) + a(k-1)).
%F A078343 a(n) is asymptotic to (1/4)*(-2+3*sqrt(2))*(1+sqrt(2))^n.
%F A078343 a(n) = A048746(n-3) + 2, for n > 2. - _Ralf Stephan_, Oct 17 2003
%F A078343 a(n) = 2*A000129(n) - A000129(n-1) if n > 0; abs(a(n)) = Sum_{k=0..floor(n/2)} (C(n-k-1, k) - C(n-k-1, k-1))2^(n-2k). - _Paul Barry_, Dec 23 2004
%F A078343 O.g.f.: (1-4*x)/(-1 + 2*x + x^2). - _R. J. Mathar_, Feb 15 2008
%F A078343 a(n) = 4*Pell(n) - Pell(n+1), where Pell = A000129. - _Vladimir Reshetnikov_, Sep 27 2016
%F A078343 a(n) = -(-1)^n * A048654(-n) = ( (-2+3*sqrt(2))*(1+sqrt(2))^n + (-2-3*sqrt(2))*(1-sqrt(2))^n )/4 for all n in Z. - _Michael Somos_, Jun 30 2022
%F A078343 2*a(n+1)^2 = A048655(n)^2 + (-1)^n*7. - _Philippe Deléham_, Mar 07 2023
%F A078343 E.g.f.: 3*exp(x)*sinh(sqrt(2)*x)/sqrt(2) - exp(x)*cosh(sqrt(2)*x). - _Stefano Spezia_, May 26 2024
%e A078343 G.f. = -1 + 2*x + 3*x^2 + 8*x^3 + 19*x^4 + 46*x^5 + 111*x^6 + ... - _Michael Somos_, Jun 30 2022
%p A078343 f:=proc(n) option remember; if n=0 then RETURN(-1); fi; if n=1 then RETURN(2); fi; 2*f(n-1)+f(n-2); end;
%t A078343 Table[4 Fibonacci[n, 2] - Fibonacci[n + 1, 2], {n, 0, 30}] (* _Vladimir Reshetnikov_, Sep 27 2016 *)
%t A078343 LinearRecurrence[{2,1},{-1,2},40] (* _Harvey P. Dale_, Apr 15 2019 *)
%o A078343 (Haskell)
%o A078343 a078343 n = a078343_list !! n
%o A078343 a078343_list = -1 : 2 : zipWith (+)
%o A078343                         (map (* 2) $ tail a078343_list) a078343_list
%o A078343 -- _Reinhard Zumkeller_, Jan 04 2013
%o A078343 (PARI) a(n)=([0,1;1,2]^n*[-1;2])[1,1] \\ _Charles R Greathouse IV_, Jun 11 2015
%o A078343 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-4*x)/(-1+2*x+x^2))); // _G. C. Greubel_, Jul 26 2018
%Y A078343 Cf. A000129, A048654, A048655, A048746, A111566, A162255, A221172, A221173, A221174, A221175.
%K A078343 sign,easy
%O A078343 0,2
%A A078343 _Benoit Cloitre_, Nov 22 2002
%E A078343 Entry revised by _N. J. A. Sloane_, Apr 29 2004