A063782 a(0) = 1, a(1) = 3; for n > 1, a(n) = 2*a(n-1) + 4*a(n-2).

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

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