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A377115
a(n) = coefficient of sqrt(3) in the expansion of (3 + sqrt(2) + sqrt(3))^n.
4
0, 1, 6, 36, 216, 1304, 7920, 48320, 295680, 1812672, 11124864, 68320000, 419719680, 2579051008, 15849305088, 97406521344, 598661038080, 3679444570112, 22614556631040, 138994100486144, 854291341737984, 5250689954316288, 32272093691707392, 198352703517884416
OFFSET
0,3
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 4 primes, with respective period lengths 4,8,10,10 and these periods:
p = 2: (2, 1, 1, 2)
p = 3: (6, 1, 1, 3, 1, 4, 2, 6)
p = 5: (6, 2, 7, 3, 10, 2, 12, 3, 9, 6)
p = 7: (14, 4, 4, 2, 12, 1, 11, 5, 1, 18)
See A377109 for a guide to related sequences.
FORMULA
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.
G.f.: (x (-1 + 6 x - 8 x^2))/(-1 + 12 x - 44 x^2 + 48 x^3 + 8 x^4).
EXAMPLE
(3 + sqrt(2) + sqrt(3))^3 = 14 + 6*sqrt(2) + 6*sqrt(3) + 2*sqrt(6), so a(3) = 6.
MATHEMATICA
(* Program 1 generates sequences A377113-A37716. *)
tbl = Table[Expand[(3 + 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[{12, -44, 48, 8}, {0, 1, 6, 36}, 15].
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Oct 21 2024
STATUS
approved