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A377110 a(n) = coefficient of sqrt(2) in the expansion of (2 + sqrt(2) + sqrt(3))^n.

%I A377110 #19 Aug 09 2025 03:59:49
%S A377110 0,1,4,23,120,629,3260,16843,86832,447241,2302516,11851487,60995880,
%T A377110 313912637,1615504748,8313878227,42785563488,220186165393,
%U A377110 1133137017700,5831424517415,30010056528600,154439694647429,794787521046812,4090186754982235,21049182488180880
%N A377110 a(n) = coefficient of sqrt(2) in the expansion of (2 + sqrt(2) + sqrt(3))^n.
%C A377110 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 7 primes, with respective period lengths 1,8,18,17,3,11,50 and these periods:
%C A377110  p = 2: (4)
%C A377110  p = 3: (4,2,6,6,1,1,3,1)
%C A377110  p = 5: (12,2,10,7,3,2,3,9,12,1,8,2,1,12,12,5,7,12)
%C A377110  p = 7: (5, 16, 26, 22, 13, 2, 1, 24, 16, 57, 2, 30, 9, 19, 6, 28, 12)
%C A377110  p = 11: (29, 70, 61)
%C A377110  p = 13: (21, 3, 24, 15, 9, 24, 9, 15, 24, 3, 21)
%C A377110  p = 17: (19, 11, 30, 4, 21, 2, 3, 22, 8, 7, 23, 24, 6, 30, 30, 11, 10, 9, 26, 4, 30, 18, 12, 30, 30, 15, 15, 30, 30, 8, 4, 11, 7, 30, 30, 9, 7, 14, 30, 5, 9, 16, 6, 24, 30, 30, 3, 27, 30, 30)
%C A377110 See A377109 for a guide to related sequences.
%H A377110 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-14,-8,23).
%F A377110 a(n) = 8*a(n-1) - 14*a(n-2) - 8*a(n-3) + 23*a(n-4), with a(0)=0, a(1)=1, a(3)=4, a(4)=23.
%F A377110 G.f.: x*(-1 + 4*x - 5*x^2)/(-1 + 8*x - 14*x^2 - 8*x^3 + 23*x^4).
%F A377110 a(n) = ((2 + sqrt(2) + sqrt(3))^n - (2 - sqrt(2) - sqrt(3))^n + (2 + sqrt(2) - sqrt(3))^n - (2 - sqrt(2) + sqrt(3))^n) / 2^(5/2). - _Vaclav Kotesovec_, Oct 21 2024
%e A377110 (2 + sqrt(2) + sqrt(3))^3 = 9 + 4*sqrt(2) + 4*sqrt(3) + 2*sqrt(6), so a(3) = 4.
%t A377110 (* Program 1 generates sequences A377109-A377112. *)
%t A377110 tbl = Table[Expand[(2 + Sqrt[2] + Sqrt[3])^n], {n, 0, 24}];
%t A377110 u = MapApply[{#1/#2, #2} /. {1, #} -> {{1}, {#}} &,
%t A377110    Map[({#1, #1 /. _^_ -> 1} &), Map[(Apply[List, #1] &), tbl]]];
%t A377110 {s1,s2,s3,s4}=Transpose[(PadRight[#1,4]&)/@Last/@u][[1;;4]];
%t A377110 s2  (* _Peter J. C. Moses_, Oct 16 2024 *)
%t A377110 (* Program 2 generates this sequence. *)
%t A377110 LinearRecurrence[{8, -14, -8, 23}, {0, 1, 4, 23}, 25]
%Y A377110 Cf. A377109.
%K A377110 nonn,easy
%O A377110 0,3
%A A377110 _Clark Kimberling_, Oct 20 2024