A377111 a(n) = coefficient of sqrt(3) in the expansion of (2 + sqrt(2) + sqrt(3))^n.

0, 1, 4, 21, 104, 529, 2700, 13845, 71120, 365697, 1881236, 9679605, 49809720, 256324433, 1319090972, 6788338869, 34934465440, 179781713537, 925203573540, 4761340669269, 24503114321416, 126099496024593, 648941324534188, 3339623572751061, 17186585699725680

Offset: 0

Author

Clark Kimberling, Oct 20 2024

Keywords

• nonn
• easy

Comments

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:

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

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

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

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)

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

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

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).

See A377109 for a guide to related sequences.

Examples

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

Links

• Index entries for linear recurrences with constant coefficients, signature (8,-14,-8,23).

Crossrefs

Cf. A377109.

Programs

Mathematica

(* Program 1 generates sequences A377109-A377112. *)
tbl = Table[Expand[(2 + Sqrt[2] + Sqrt[3])^n], {n, 0, 24}];
u = MapApply[{#1/#2, #2} /. {1, #} -> {{1}, {#}} &,
   Map[({#1, #1 /. ^ -> 1} &), Map[(Apply[List, #1] &), tbl]]];
{s1,s2,s3,s4}=Transpose[(PadRight[#1,4]&)/@Last/@u][[1;;4]];
s3  (* Peter J. C. Moses, Oct 16 2024 *)
(* Program 2 generates this sequence. *)
LinearRecurrence[{8, -14, -8, 23}, {0, 1, 4, 21}, 25]

Formula

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.

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

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