A188572 a(n) = coefficient of sqrt(3) in the expansion of (1 + sqrt(2) + sqrt(3))^n sequence.

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

Author

Mateusz Szymański, Dec 28 2012

Keywords

• nonn

Comments

From Clark Kimberling, Oct 23 2024: (Start)

Conjecture: every prime divides a(n) for infinitely many n, and if K(p) = (k(1), k(2),...) is the maximal subsequence of indices n such that p divides a(n), then the difference sequence of K(p) is eventually periodic; indeed, K(p) is purely periodic for the first 6 primes, with respective period lengths 6,8,5,4,4,8 and these periods:

p = 2: (3, 3, 1, 2, 2, 1)

p = 3: (4, 2, 6, 6, 1, 1, 3, 1)

p = 5: (20, 20, 9, 10, 1)

p = 7: (18, 1, 16, 1)

p = 11: (32, 1, 30, 1)

p = 13: (28, 14, 1, 10, 3, 17, 10, 1)

See A377109 for a guide to related sequences. (End)

Cf. A377109.

Examples

			a(3) = 12 because (1+sqrt(2)+sqrt(3))^3 = 16 + 14*sqrt(2) + 12*sqrt(3) + 6*sqrt(6).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
• Sela Fried, On the coefficients of (r + sqrt(p) + sqrt(q))^n
• Index entries for linear recurrences with constant coefficients, signature (4,4,-16,8).

Crossrefs

Cf. A188570, A188571, A188573, A377109.

Programs

Mathematica

a[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[a[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 *)

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)

The conjectures by Mathar are true. See link. - Sela Fried, Jan 01 2025

Extensions

Edited by Clark Kimberling, Oct 20 2024