A188572 Coefficients of the term by sqrt(3) in (1 + sqrt(2) + sqrt(3))^n sequence, denoted as C3(n).

0, 1, 2, 12, 40, 184, 720, 3072, 12544, 52416, 216448, 899328, 3724800, 15452672, 64052224, 265617408, 1101234176, 4566192128, 18932244480, 78498938880, 325475532800, 1349511512064, 5595423113216, 23200121487360, 96193798471680, 398845002121216

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)

Formula

Conjectures from R. J. Mathar, Jan 09 2013: (Start)

a(n) = +4*a(n-1) +4*a(n-2) -16*a(n-3) +8*a(n-4).

G.f.: x*(-1+2*x)/( -1+4*x+4*x^2-16*x^3+8*x^4 ). (End)

Example

C3(3) is equal to 12, because (1+sqrt(2)+sqrt(3))^3 = 16 + 14*sqrt(2) + 12*sqrt(3) + 6*sqrt(6).

Mathematica

C3[n_] := Sum[Sum[3^(Floor[(n - 1)/2] - k - j) 2^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[C3[n], {n, 0, 25}]
a[n_] := Coefficient[ Expand[(1 + Sqrt[2] + Sqrt[3])^n], Sqrt[3]] /. Sqrt[2] -> 0; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 08 2013 *)

Crossrefs

Cf. A188570, A188571, A188573.

Sequence in context: A240122 A110953 A003683 * A098519 A127725 A371357

Adjacent sequences: A188569 A188570 A188571 * A188573 A188574 A188575

Keyword

nonn

Author

Mateusz Szymański, Dec 28 2012

Status

approved