A188573 Coefficient of the sqrt(6) term in (1 + sqrt(2) + sqrt(3))^n, denoted as C6(n).

0, 0, 2, 6, 32, 120, 528, 2128, 8960, 36864, 153472, 635008, 2635776, 10922496, 45300736, 187800576, 778731520, 3228696576, 13387309056, 55506722816, 230146834432, 954246856704, 3956565671936, 16404954546176, 68019305840640, 282025965649920

Offset

0,3

Links

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

Formula

From G. C. Greubel, Apr 10 2018: (Start)

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

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

Example

C6(3) is equal to 6, because (1+sqrt(2)+sqrt(3))^3 = 16 + 14 sqrt(2) + 12 sqrt(3) + 6 sqrt(6).

Mathematica

C6[n_] := Sum[Sum[2^(Floor[n/2] - j - 1 - k) 3^j Multinomial[2 k + n - 2 Floor[n/2], 2 j + 1, 2 Floor[n/2] - 2 k - 1 - 2 j], {j, 0, Floor[n/2] - k - 1}], {k, 0, Floor[n/2] - 1}]; Table[C6[n], {n, 0, 25}]
a[n_] := Coefficient[ Expand[(1 + Sqrt[2] + Sqrt[3])^n], Sqrt[6]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 08 2013 *)

Crossrefs

Cf. A188570, A188571, A188572.

Sequence in context: A223586 A212762 A108327 * A196010 A121071 A092199

Adjacent sequences: A188570 A188571 A188572 * A188574 A188575 A188576

Keyword

nonn

Author

Mateusz Szymański, Dec 28 2012

Extensions

Keyword tabl removed by Michel Marcus, Apr 11 2018

Status

approved