cp's OEIS Frontend

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.

%I A377314 #20 May 30 2025 10:40:18
%S A377314 1,1,5,19,73,281,1081,4159,16001,61561,236845,911219,3505753,13487761,
%T A377314 51891761,199644319,768096001,2955112721,11369270485,43741245619,
%U A377314 168286661033,647452990441,2490960200041,9583526232479,36870912288001,141854275761481
%N 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.
%C A377314 See A377109 for a guide to related sequences.
%H A377314 Michael De Vlieger, <a href="/A377314/b377314.txt">Table of n, a(n) for n = 0..1709</a>
%H A377314 Sela Fried, <a href="https://arxiv.org/abs/2505.14196">Even-up words and their variants</a>, arXiv:2505.14196 [math.CO], 2025. See p. 4.
%H A377314 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,1).
%F A377314 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]
%F A377314 G.f.: (-1 + 2 x + x^2)/(-1 + 3 x + 3 x^2 + x^3).
%e A377314 ((1 + 2^(1/3) + 2^(2/3)))^3 = 19 + 15 2^(1/3) + 12 2^(2/3), so a(3) = 19.
%t A377314 (* Program 1 generates sequences A377314-A377315 and A108368. *)
%t A377314 tbl = Table[Expand[(1 + 2^(1/3) + 2^(2/3))^n], {n, 0, 24}];
%t A377314 Take[tbl, 6]
%t A377314 u = MapApply[{#1/#2, #2} /. {1, #} -> {{1}, {#}} &,
%t A377314    Map[({#1, #1 /. _^_ -> 1} &), Map[(Apply[List, #1] &), tbl]]];
%t A377314 {s1, s2, s3} = Transpose[(PadRight[#1, 3] &) /@ Last /@ u][[1 ;; 3]];
%t A377314 s1  (* _Peter J. C. Moses_, Oct 16 2024 *)
%t A377314 (* Program 2 generates (a(n)) for n>=1. *)
%t A377314 LinearRecurrence[{3,3,1}, {1, 1, 5}, 15]
%Y A377314 Cf. A377109, A377117, A377315, A108368 (coefficients of 2^(2/3)).
%K A377314 nonn
%O A377314 0,3
%A A377314 _Clark Kimberling_, Oct 26 2024