cp's OEIS Frontend

A377315 a(n) = coefficient of 2^(1/3) in the expansion of (1 + 2^(1/3) + 2^(2/3))^n.

%I A377315 #9 Oct 31 2024 01:47:01
%S A377315 0,1,4,15,58,223,858,3301,12700,48861,187984,723235,2782518,10705243,
%T A377315 41186518,158457801,609638200,2345474521,9023795964,34717449655,
%U A377315 133569211378,513883779063,1977076420978,7606449811501,29264462476500,112589813284981,433169277095944
%N A377315 a(n) = coefficient of 2^(1/3) in the expansion of (1 + 2^(1/3) + 2^(2/3))^n.
%C A377315 See A377109 for a guide to related sequences.
%H A377315 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,1).
%F A377315 a(n) = 3*a(n-1) + 3*a(n-2) + a(n-3), with a(0)=0, a(1)=1, a(3)=4.
%F A377315 G.f.: -((x (1 + x))/(-1 + 3 x + 3 x^2 + x^3)).
%e A377315 (1 + 2^(1/3) + 2^(2/3))^3 = 19 + 15 2^(1/3) + 12 2^(2/3), so a(3) = 15.
%t A377315 (* Program 1 generates sequences A377314-A377315 and A108368. *)
%t A377315 tbl = Table[Expand[(1 + 2^(1/3) + 2^(2/3))^n], {n, 0, 24}];
%t A377315 Take[tbl, 6]
%t A377315 u = MapApply[{#1/#2, #2} /. {1, #} -> {{1}, {#}} &,
%t A377315    Map[({#1, #1 /. _^_ -> 1} &), Map[(Apply[List, #1] &), tbl]]];
%t A377315 {s1, s2, s3} = Transpose[(PadRight[#1, 3] &) /@ Last /@ u][[1 ;; 3]];
%t A377315 s2  (* _Peter J. C. Moses_, Oct 16 2024 *)
%t A377315 (* Program 2 generates (a(n)) for n>=1. *)
%t A377315 LinearRecurrence[{3,3,1}, {0, 1, 4}, 15]
%Y A377315 Cf. A377109, A377117, A377314, A108368 (coefficients of 2^(2/3)).
%K A377315 nonn,easy
%O A377315 0,3
%A A377315 _Clark Kimberling_, Oct 26 2024