A377314 a(n) = coefficient of the term that is independent of 2^(1/3) and 2^(2/3) in the expansion of (1 + 2^(1/3) + 2^(2/3))^n.

1, 1, 5, 19, 73, 281, 1081, 4159, 16001, 61561, 236845, 911219, 3505753, 13487761, 51891761, 199644319, 768096001, 2955112721, 11369270485, 43741245619, 168286661033, 647452990441, 2490960200041, 9583526232479, 36870912288001, 141854275761481

Offset

0,3

Comments

See A377109 for a guide to related sequences.

Links

Table of n, a(n) for n=0..25.

Index entries for linear recurrences with constant coefficients, signature (3,3,1).

Formula

a(n) = 3*a(n-1) + 3*a(n-2) + a(n-3), with a(0)=1, a(1)=1, a(3)=5. [Corrected by Jianing Song, Oct 31 2024]

G.f.: (-1 + 2 x + x^2)/(-1 + 3 x + 3 x^2 + x^3).

Example

((1 + 2^(1/3) + 2^(2/3)))^3 = 19 + 15 2^(1/3) + 12 2^(2/3), so a(3) = 19.

Mathematica

(* Program 1 generates sequences A377314-A377315 and A108368. *)
tbl = Table[Expand[(1 + 2^(1/3) + 2^(2/3))^n], {n, 0, 24}];
Take[tbl, 6]
u = MapApply[{#1/#2, #2} /. {1, #} -> {{1}, {#}} &,
   Map[({#1, #1 /. _^_ -> 1} &), Map[(Apply[List, #1] &), tbl]]];
{s1, s2, s3} = Transpose[(PadRight[#1, 3] &) /@ Last /@ u][[1 ;; 3]];
s1  (* Peter J. C. Moses, Oct 16 2024 *)
(* Program 2 generates (a(n)) for n>=1. *)
LinearRecurrence[{3, 3, 1}, {1, 1, 5}, 15]

Crossrefs

Cf. A377109, A377117, A377315, A108368 (coefficients of 2^(2/3)).

Sequence in context: A034548 A255455 A255444 * A287805 A129166 A149763

Adjacent sequences: A377311 A377312 A377313 * A377315 A377316 A377317

Keyword

nonn

Author

Clark Kimberling, Oct 26 2024

Status

approved