A143648 a(n) = ((4 + sqrt 6)^n + (4 - sqrt 6)^n)/2.

1, 4, 22, 136, 868, 5584, 35992, 232096, 1496848, 9653824, 62262112, 401558656, 2589848128, 16703198464, 107727106432, 694784866816, 4481007870208, 28900214293504, 186391635645952, 1202130942232576, 7753131181401088

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Oct 27 2008

Keywords

• nonn
• easy

Comments

Binomial transform of A084120. Sequences defined by a(n) = ((A + sqrt(B))^n + (A - sqrt(B))^n)/2 have recurrences a(n) = 2*A*a(n-1) + (B - A^2)*a(n-2) and generating functions g.f.: (1-Ax)/(1-2Ax+(A^2-B)x^2). - R. J. Mathar, Nov 01 2008

Examples

			a(3) = 136.
a(4) = ((4 + sqrt(6))^4 + (4 - sqrt(6))^4)/2 = 4^4 + 6*sqrt(6)^2*4^2 + sqrt(6)^4 = 4^4 + 6*6*4^2 + 6^2 = 868. - _Klaus Brockhaus_, Nov 01 2008
		

Links

• Index entries for linear recurrences with constant coefficients, signature (8, -10).

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-6); S:=[ ((4+r6)^n+(4-r6)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 01 2008

Mathematica

Table[MatrixPower[{{4,2},{3,4}},n][[1]][[1]],{n,0,44}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)

Formula

From R. J. Mathar, Klaus Brockhaus and Philippe Deléham, Nov 01 2008: (Start)

a(n) = 8*a(n-1) - 10*a(n-2).

G.f.: (1-4x)/(1-8x+10x^2). (End)

a(n) = (Sum_{k=0..n} A098158(n,k)*4^(2*k)*6^(n-k))/4^n. - Philippe Deléham, Nov 06 2008

Extensions

More terms from Klaus Brockhaus and R. J. Mathar, Nov 01 2008