A188573 a(n) = coefficient of the sqrt(6) term in (1 + sqrt(2) + sqrt(3))^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

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 6,5,14,4,4,8 and these periods:

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

p = 3: (1, 4, 3, 8, 8)

p = 5: (1, 5, 4, 1, 1, 6, 6, 6, 6, 6, 6, 2, 4, 6)

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

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

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

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

Examples

			a(3) = 6, 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, A188572, A377109.

Programs

Mathematica

a[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[a[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 *)

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)

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

Extensions

Keyword tabl removed by Michel Marcus, Apr 11 2018

Edited by Clark Kimberling, Oct 23 2024