This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A188573 #30 Jan 09 2025 11:39:42
%S A188573 0,0,2,6,32,120,528,2128,8960,36864,153472,635008,2635776,10922496,
%T A188573 45300736,187800576,778731520,3228696576,13387309056,55506722816,
%U A188573 230146834432,954246856704,3956565671936,16404954546176,68019305840640,282025965649920
%N A188573 a(n) = coefficient of the sqrt(6) term in (1 + sqrt(2) + sqrt(3))^n.
%C A188573 From _Clark Kimberling_, Oct 23 2024: (Start)
%C A188573 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 A188573 p = 2: (1, 2, 2, 1, 3, 3)
%C A188573 p = 3: (1, 4, 3, 8, 8)
%C A188573 p = 5: (1, 5, 4, 1, 1, 6, 6, 6, 6, 6, 6, 2, 4, 6)
%C A188573 p = 7: (1, 16, 1, 18)
%C A188573 p = 11: (1, 30, 1, 32)
%C A188573 p = 13: (1, 10, 3, 17, 10, 1, 28, 14)
%C A188573 See A377109 for a guide to related sequences. (End)
%H A188573 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 A188573 Sela Fried, <a href="/A188571/a188571_1.pdf">On the coefficients of (r + sqrt(p) + sqrt(q))^n</a>
%H A188573 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,4,-16,8).
%F A188573 From _G. C. Greubel_, Apr 10 2018: (Start)
%F A188573 Empirical: a(n) = 4*a(n-1) + 4*a(n-2) - 16*a(n-3) + 8*a(n-4).
%F A188573 Empirical: G.f.: 2*x^2*(1-x)/(1 - 4*x - 4*x^2 + 16*x^3 - 8*x^4). (End)
%F A188573 The conjectures by Greubel are true. See link. - _Sela Fried_, Jan 01 2025
%e A188573 a(3) = 6, because (1+sqrt(2)+sqrt(3))^3 = 16 + 14 sqrt(2) + 12 sqrt(3) + 6 sqrt(6).
%t A188573 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 A188573 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 A188573 Cf. A188570, A188571, A188572, A377109.
%K A188573 nonn
%O A188573 0,3
%A A188573 _Mateusz Szymański_, Dec 28 2012
%E A188573 Keyword tabl removed by _Michel Marcus_, Apr 11 2018
%E A188573 Edited by _Clark Kimberling_, Oct 23 2024