A188571 Coefficients of the term by sqrt(2) in (1 + sqrt(2) + sqrt(3))^n sequence, denoted as C2(n).

0, 1, 2, 14, 48, 224, 880, 3760, 15360, 64192, 265088, 1101440, 4561920, 18925568, 78447616, 325313536, 1348730880, 5592420352, 23187169280, 96141172736, 398624489472, 1652807303168, 6852965761024, 28414229807104, 117812861337600, 488483370827776

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

Conjecture: a(n) = 4*a(n-1)+4*a(n-2)-16*a(n-3)+8*a(n-4). G.f.: -x*(2*x^2-2*x+1) / (8*x^4-16*x^3+4*x^2+4*x-1). [Colin Barker, Jan 08 2013]

Example

C2(3) is equal to 14, because (1+sqrt(2)+sqrt(3))^3 = 16 + 14*sqrt(2) + 12*sqrt(3) + 6*sqrt(6).

Mathematica

C2[n_] := Sum[Sum[2^(Floor[(n - 1)/2] - k - j) 3^j Multinomial[2 Floor[(n - 1)/2] + 1 - 2 j - 2 k, 2 j, 2 k + 1 - n + 2 Floor[n/2]], {j, 0, Floor[(n - 1)/2] - k + 1}], {k, 0, Floor[(n - 1)/2]}]; Table[C2[n], {n, 0, 25}]
a[n_] := Coefficient[ Expand[(1 + Sqrt[2] + Sqrt[3])^n], Sqrt[2]] /. Sqrt[3] -> 0; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 08 2013 *)

Crossrefs

Cf. A188570.

Sequence in context: A281760 A197885 A200193 * A083102 A270666 A330544

Adjacent sequences: A188568 A188569 A188570 * A188572 A188573 A188574

Keyword

nonn

Author

Mateusz Szymański, Dec 28 2012

Status

approved