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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

%I

%S 1,3,10,32,104,336,1088,3520,11392,36864,119296,386048,1249280,

%T 4042752,13082624,42336256,137003008,443351040,1434714112,4642832384,

%U 15024521216,48620371968,157338828800,509159145472,1647673606144

%N a(0) = 1, a(1) = 3; for n > 1, a(n) = 2*a(n-1) + 4*a(n-2).

%C Ratio of successive terms approaches sqrt(5) + 1.

%H G. C. Greubel, <a href="/A063782/b063782.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Harry J. Smith)

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,4).

%F For n >= 1, a(n) = 2^(n-1)*Fibonacci(n+3). - _Vladeta Jovovic_, Oct 25 2003

%F G.f.: (1 + x)/(1 - 2*x - 4*x^2). - _R. J. Mathar_, Feb 06 2010

%F Equals INVERT transform of A006138 and INVERTi transform of A179606. - _Gary W. Adamson_, Aug 14 2010

%F 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

%F 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

%F a(n) = A063727(n) + A063727(n-1).

%F 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

%e 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).

%p 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:

%p 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

%t 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 *)

%t LinearRecurrence[{2, 4}, {1, 3}, 100] (* _G. C. Greubel_, Feb 18 2017 *)

%o (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

%Y Cf. A006138. Row sums of A215244.

%Y Cf. A000045, A049310, A319053.

%K nonn,easy

%O 0,2

%A Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Aug 17 2001

%E More terms from _James A. Sellers_, Sep 25 2001

%E Edited (new offset, new initial term, etc.) by _N. J. A. Sloane_, Aug 19 2010

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Last modified October 23 23:51 EDT 2019. Contains 328379 sequences. (Running on oeis4.)