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A377115 a(n) = coefficient of sqrt(3) in the expansion of (3 + sqrt(2) + sqrt(3))^n.

%I A377115 #12 Aug 09 2025 04:00:26
%S A377115 0,1,6,36,216,1304,7920,48320,295680,1812672,11124864,68320000,
%T A377115 419719680,2579051008,15849305088,97406521344,598661038080,
%U A377115 3679444570112,22614556631040,138994100486144,854291341737984,5250689954316288,32272093691707392,198352703517884416
%N A377115 a(n) = coefficient of sqrt(3) in the expansion of (3 + sqrt(2) + sqrt(3))^n.
%C A377115 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 4 primes, with respective period lengths 4,8,10,10 and these periods:
%C A377115  p = 2: (2, 1, 1, 2)
%C A377115  p = 3: (6, 1, 1, 3, 1, 4, 2, 6)
%C A377115  p = 5: (6, 2, 7, 3, 10, 2, 12, 3, 9, 6)
%C A377115  p = 7: (14, 4, 4, 2, 12, 1, 11, 5, 1, 18)
%C A377115 See A377109 for a guide to related sequences.
%H A377115 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (12,-44,48,8).
%F A377115 a(n) = 12*a(n-1) - 44*a(n-2) + 48*a(n-3) + 8*a(n-4), with a(0)=0, a(1)=1, a(3)=6, a(4)=36.
%F A377115 G.f.: x*(-1 + 6*x - 8*x^2)/(-1 + 12*x - 44*x^2 + 48*x^3 + 8*x^4).
%e A377115 (3 + sqrt(2) + sqrt(3))^3 = 14 + 6*sqrt(2) + 6*sqrt(3) + 2*sqrt(6), so a(3) = 6.
%t A377115 (* Program 1 generates sequences A377113-A377116. *)
%t A377115 tbl = Table[Expand[(3 + Sqrt[2] + Sqrt[3])^n], {n, 0, 24}];
%t A377115 u = MapApply[{#1/#2, #2} /. {1, #} -> {{1}, {#}} &,
%t A377115    Map[({#1, #1 /. _^_ -> 1} &), Map[(Apply[List, #1] &), tbl]]];
%t A377115 {s1,s2,s3,s4}=Transpose[(PadRight[#1,4]&)/@Last/@u][[1;;4]];
%t A377115 s3  (* _Peter J. C. Moses_, Oct 16 2024 *)
%t A377115 (* Program 2 generates this sequence. *)
%t A377115 LinearRecurrence[{12, -44, 48, 8}, {0, 1, 6, 36}, 25]
%Y A377115 Cf. A377090, A377113, A377114, A377116.
%K A377115 nonn
%O A377115 0,3
%A A377115 _Clark Kimberling_, Oct 21 2024