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 A063782 a(0) = 1, a(1) = 3; for n > 1, a(n) = 2*a(n-1) + 4*a(n-2). 9
 1, 3, 10, 32, 104, 336, 1088, 3520, 11392, 36864, 119296, 386048, 1249280, 4042752, 13082624, 42336256, 137003008, 443351040, 1434714112, 4642832384, 15024521216, 48620371968, 157338828800, 509159145472, 1647673606144 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Ratio of successive terms approaches sqrt(5) + 1. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Harry J. Smith) Index entries for linear recurrences with constant coefficients, signature (2,4). FORMULA For n >= 1, a(n) = 2^(n-1)*Fibonacci(n+3). - Vladeta Jovovic, Oct 25 2003 G.f.: (1 + x)/(1 - 2*x - 4*x^2). - R. J. Mathar, Feb 06 2010 Equals INVERT transform of A006138 and INVERTi transform of A179606. - Gary W. Adamson, Aug 14 2010 a(n) = (1/2)*(1+sqrt(5))^n + (1/5)*(1+sqrt(5))^n*sqrt(5) - (1/5)*sqrt(5)*(1-sqrt(5))^n + (1/2)*(1-sqrt(5))^n. - Alexander R. Povolotsky, Aug 15 2010 It follows that a(n) is the nearest integer to (and is increasingly close to) (1/2 + 1/sqrt(5))*(1+sqrt(5))^n. - N. J. A. Sloane, Aug 10 2012 a(n) = A063727(n) + A063727(n-1). a(n) = M^n(1, 1), with the matrix M= [[3, 1], [1, -1]]. Proof by Cayley-Hamilton, using S(n, -I) = (-I)^n*F(n+1), and S = A049310 and F = A000045. Motivated by A319053. - Wolfdieter Lang, Oct 08 2018 EXAMPLE As the INVERT transform of A006138, (1, 2, 5, 11, 26, 59,...); a(4) = 104 = (26, 11, 5, 2, 1) dot (1, 1, 3, 10, 32) = (26 + 11 + 15 + 20 + 32). MAPLE a := proc(n) option remember: if n=0 then RETURN(1) fi: if n=1 then RETURN(2) fi: 2*a(n-1) + 4*a(n-2); end: for n from 1 to 50 do printf(`%d, `, a(n)+a(n-1)) od: f:=n-> simplify(expand((1/2)*(1+sqrt(5))^n + (1/5)*(1+sqrt(5))^n*sqrt(5) - (1/5)*sqrt(5)*(1-sqrt(5))^n + (1/2)*(1 -sqrt(5))^n )); # N. J. A. Sloane, Aug 10 2012 MATHEMATICA a[n_]:=(MatrixPower[{{1, 5}, {1, 1}}, n].{{2}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *) LinearRecurrence[{2, 4}, {1, 3}, 100] (* G. C. Greubel, Feb 18 2017 *) PROG (PARI) { for (n=0, 200, if (n>1, a=2*a1 + 4*a2; a2=a1; a1=a, if (n, a=a1=2, a=a2=1)); if (n, write("b063782.txt", n, " ", a + a2)) ) } \\ Harry J. Smith, Aug 31 2009 CROSSREFS Cf. A006138. Row sums of A215244. Cf. A000045, A049310, A319053. Sequence in context: A273351 A033505 A297067 * A071718 A261058 A306295 Adjacent sequences:  A063779 A063780 A063781 * A063783 A063784 A063785 KEYWORD nonn,easy AUTHOR Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Aug 17 2001 EXTENSIONS More terms from James A. Sellers, Sep 25 2001 Edited (new offset, new initial term, etc.) by N. J. A. Sloane, Aug 19 2010 STATUS approved

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Last modified December 11 07:44 EST 2019. Contains 329914 sequences. (Running on oeis4.)