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a(n) = coefficient of the sqrt(6) term in (1 + sqrt(2) + sqrt(3))^n.
4

%I #30 Jan 09 2025 11:39:42

%S 0,0,2,6,32,120,528,2128,8960,36864,153472,635008,2635776,10922496,

%T 45300736,187800576,778731520,3228696576,13387309056,55506722816,

%U 230146834432,954246856704,3956565671936,16404954546176,68019305840640,282025965649920

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

%C From _Clark Kimberling_, Oct 23 2024: (Start)

%C 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:

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

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

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

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

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

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

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

%H G. C. Greubel, <a href="/A188573/b188573.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)

%H Sela Fried, <a href="/A188571/a188571_1.pdf">On the coefficients of (r + sqrt(p) + sqrt(q))^n</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,4,-16,8).

%F From _G. C. Greubel_, Apr 10 2018: (Start)

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

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

%F The conjectures by Greubel are true. See link. - _Sela Fried_, Jan 01 2025

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

%t 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}]

%t a[n_] := Coefficient[ Expand[(1 + Sqrt[2] + Sqrt[3])^n], Sqrt[6]]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jan 08 2013 *)

%Y Cf. A188570, A188571, A188572, A377109.

%K nonn

%O 0,3

%A _Mateusz Szymański_, Dec 28 2012

%E Keyword tabl removed by _Michel Marcus_, Apr 11 2018

%E Edited by _Clark Kimberling_, Oct 23 2024