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A329404 Interleave 2*n*(3*n-1), (2*n+1)*(6*n+1) for n >= 0.

%I A329404 #34 Dec 27 2024 08:45:27
%S A329404 0,1,4,21,20,65,48,133,88,225,140,341,204,481,280,645,368,833,468,
%T A329404 1045,580,1281,704,1541,840,1825,988,2133,1148,2465,1320,2821,1504,
%U A329404 3201,1700,3605,1908,4033,2128,4485,2360,4961
%N A329404 Interleave 2*n*(3*n-1), (2*n+1)*(6*n+1) for n >= 0.
%C A329404 a(n) + a(n+3) = 21, 21, 69, 69, 153, 153, ...
%C A329404 Hexagonal spiral for A026741:
%C A329404 .
%C A329404                    33--17--35--18
%C A329404                    /
%C A329404                  16   8--17---9--19
%C A329404                  /   /             \
%C A329404                31  15   5---3---7  10
%C A329404                /   /   /         \   \
%C A329404              15   7   2   0===1===4==21==>
%C A329404                \   \   \     /   /   /
%C A329404                29  13   3---1   9  11
%C A329404                  \   \         /   /
%C A329404                  14   6--11---5  23
%C A329404                    \             /
%C A329404                    27--13--25--12
%C A329404 .
%C A329404 a(n) is the horizontal sequence from 0.
%H A329404 Colin Barker, <a href="/A329404/b329404.txt">Table of n, a(n) for n = 0..1000</a>
%H A329404 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).
%F A329404 a(n) = n * A165355(n-1).
%F A329404 From _Colin Barker_, Nov 13 2019: (Start)
%F A329404 G.f.: x*(1 + 4*x + 18*x^2 + 8*x^3 + 5*x^4) / ((1 - x)^3*(1 + x)^3).
%F A329404 a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n > 5.
%F A329404 a(n) = (1/4)*(-1)*((-3 + (-1)^n)*n*(-2+3*n)). (End)
%F A329404 From _Amiram Eldar_, Dec 27 2024: (Start)
%F A329404 Sum_{n>=1} 1/a(n) = Pi/(8*sqrt(3)) + 9*log(3)/8.
%F A329404 Sum_{n>=1} (-1)^(n+1)/a(n) = 5*Pi/(8*sqrt(3)) - 3*log(3)/8. (End)
%t A329404 LinearRecurrence[{0,3,0,-3,0,1},{0,1,4,21,20,65},100] (* _Paolo Xausa_, Nov 13 2023 *)
%o A329404 (PARI) concat(0, Vec(x*(1 + 4*x + 18*x^2 + 8*x^3 + 5*x^4) / ((1 - x)^3*(1 + x)^3) + O(x^45))) \\ _Colin Barker_, Nov 13 2019
%Y A329404 Cf. A005449, A014641, A026741, A033579, A165355.
%K A329404 nonn,easy
%O A329404 0,3
%A A329404 _Paul Curtz_, Nov 13 2019