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, 252509934627, 661079236027, 1730727334250

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Examples

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

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).

Programs

Magma

[(Lucas(n) + Lucas(2*n))/2: n in [0..30]]; // G. C. Greubel, Dec 02 2017

Mathematica

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

PARI

my(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

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)