A049680 a(n) = (L(n) + L(2*n))/2, where L = A000032 (the Lucas sequence).

2, 2, 5, 11, 27, 67, 170, 436, 1127, 2927, 7625, 19901, 52002, 135982, 355745, 930931, 2436527, 6377807, 16695530, 43706576, 114420627, 299549527, 784218605, 2053091161, 5375030402, 14071960442, 36840786845, 96450296411

Offset

0,1

Links

Indranil Ghosh, Table of n, a(n) for n = 0..2388

Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).

Formula

Binomial transform of trace(A^n)/4, where A is the adjacency matrix of path graph P_4 (A005248 with interpolated zeros). - Paul Barry, Apr 24 2004

From George F. Johnson, Feb 04 2013: (Start)

G.f.: (1-x)*(2-4*x-x^2)/ ( (1-x-x^2)*(1-3*x+x^2) ).

a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3) + a(n-4) for n>3. (End)

Example

a(8) = (L(8) + L(2 * 8)) / 2 = (47 + 2207) / 2 = 2254 / 2 = 1127. - Indranil Ghosh, Feb 06 2017

Mathematica

LinearRecurrence[{4, -3, -2, 1}, {2, 2, 5, 11}, 30] (* Harvey P. Dale, Nov 22 2015 *)
Table[(LuasL[n] + LucasL[2*n])/2, {n, 0, 30}] (* G. C. Greubel, Dec 02 2017 *)

Prog

(PARI) x='x+O('x^30); Vec((1-x)*(2-4*x-x^2)/ ( (1-x-x^2)*(1-3*x+x^2) )) \\ G. C. Greubel, Dec 02 2017
(Magma) [(Lucas(n) + Lucas(2*n)/2: n in [0..30]]; // G. C. Greubel, Dec 02 2017

Crossrefs

Sequence in context: A112527 A216642 A227999 * A153983 A262714 A058021

Adjacent sequences: A049677 A049678 A049679 * A049681 A049682 A049683

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved