A377111 - OEIS

%I #16 Oct 23 2024 01:04:16

%S 0,1,4,21,104,529,2700,13845,71120,365697,1881236,9679605,49809720,

%T 256324433,1319090972,6788338869,34934465440,179781713537,

%U 925203573540,4761340669269,24503114321416,126099496024593,648941324534188,3339623572751061,17186585699725680

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

%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 7 primes, with respective period lengths 6,8,7,28,16,14,50:

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

%C  p = 3: (6, 1, 1, 3, 1, 4, 2, 6)

%C  p = 5: (32, 21, 20, 10, 20, 9, 8)

%C  p = 7: (18, 18, 8, 10, 18, 18, 11, 7, 18, 18, 1, 3, 13, 1, 5, 5, 8, 18, 14, 4, 18, 13, 5, 2, 16, 7, 9, 2)

%C  p = 11: (16, 16, 16, 16, 12, 4, 1, 15, 7, 8, 1, 4, 12, 16, 9, 7)

%C  p = 13: (24, 18, 6, 24, 12, 1, 11, 24, 5, 1, 17, 1, 7, 17)

%C  p = 17: (15, 30, 30, 8, 4, 11, 7, 30, 30, 9, 7, 14, 30, 5, 9, 16, 6, 24, 30, 30, 3, 27, 30, 30, 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).

%C See A377109 for a guide to related sequences.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-14,-8,23).

%F 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)=21.

%F G.f.: (x (-1 + 4 x - 3 x^2))/(-1 + 8 x - 14 x^2 - 8 x^3 + 23 x^4).

%F 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) / (4*sqrt(3)). - _Vaclav Kotesovec_, Oct 21 2024

%e (2 + sqrt(2) + sqrt(3))^3 = 9 + 4*sqrt(2) + 4*sqrt(3) + 2*sqrt(6), so a(3) = 4.

%t (* Program 1 generates sequences A377109-A377112. *)

%t tbl = Table[Expand[(2 + Sqrt[2] + Sqrt[3])^n], {n, 0, 24}];

%t u = MapApply[{#1/#2, #2} /. {1, #} -> {{1}, {#}} &,

%t    Map[({#1, #1 /. _^_ -> 1} &), Map[(Apply[List, #1] &), tbl]]];

%t {s1,s2,s3,s4}=Transpose[(PadRight[#1,4]&)/@Last/@u][[1;;4]];

%t s3  (* _Peter J. C. Moses_, Oct 16 2024 *)

%t (* Program 2 generates this sequence. *)

%t LinearRecurrence[{8, -14, -8, 23}, {0, 1, 4, 21}, 15].

%Y Cf. A377109.

%K nonn

%O 0,3

%A _Clark Kimberling_, Oct 20 2024