A188570 Coefficients of the absolute term in (1 + sqrt(2) + sqrt(3))^n sequence, denoted as C1(n).

1, 1, 6, 16, 80, 296, 1296, 5216, 21952, 90304, 375936, 1555456, 6456320, 26754560, 110963712, 460015616, 1907494912, 7908659200, 32792076288, 135963148288, 563742310400, 2337417887744, 9691567030272, 40183767891968, 166612591968256, 690819710058496

Offset

0,3

Links

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

Index entries for linear recurrences with constant coefficients, signature (4,4,-16,8).

Formula

Recurrence: a(n) = 4*a(n-1) + 4*a(n-2) - 16*a(n-3) + 8*a(n-4). - Vaclav Kotesovec, Aug 13 2013

a(n) ~ (1+sqrt(3)+sqrt(2))^n/4. - Vaclav Kotesovec, Aug 13 2013

Example

C1(3) is equal to 16, because (1+sqrt(2)+sqrt(3))^3 = 16 + 14*sqrt(2) + 12*sqrt(3) + 6*sqrt(6).

Mathematica

C1[n_] := Sum[Sum[2^(Floor[n/2] - k - j) 3^j Multinomial[2 k + n - 2 Floor[n/2], 2 j, 2 Floor[n/2] - 2 k - 2 j], {j, 0, Floor[n/2] - k}], {k, 0, Floor[n/2]}]; Table[C1[n], {n, 0, 25}]
a[n_] := Expand[(1 + Sqrt[2] + Sqrt[3])^n] /. Sqrt[_] -> 0; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 08 2013 *)
LinearRecurrence[{4, 4, -16, 8}, {1, 1, 6, 16}, 30] (* Harvey P. Dale, Jan 25 2019 *)

Crossrefs

Cf. A188571, A188572, A188573.

Sequence in context: A350649 A211954 A230942 * A009352 A056204 A091148

Adjacent sequences: A188567 A188568 A188569 * A188571 A188572 A188573

Keyword

nonn

Author

Mateusz Szymański, Dec 28 2012

Status

approved