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A357837 a(n) is the sum of the lengths of all the segments used to draw a square of side n representing a fishbone pattern using symmetric L-shaped tiles with side length 2.

%I A357837 #43 Jan 25 2023 09:20:56
%S A357837 0,4,10,20,32,46,64,84,106,132,160,190,224,260,298,340,384,430,480,
%T A357837 532,586,644,704,766,832,900,970,1044,1120,1198,1280,1364,1450,1540,
%U A357837 1632,1726,1824,1924,2026,2132,2240,2350,2464,2580,2698,2820,2944,3070,3200,3332
%N A357837 a(n) is the sum of the lengths of all the segments used to draw a square of side n representing a fishbone pattern using symmetric L-shaped tiles with side length 2.
%H A357837 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F A357837 a(n) = 2*(ceiling(2*(n+1)^2/3) - 1).
%F A357837 a(n) = 2*(A071619(n+1) - 1).
%F A357837 a(n) = 2*(1 + n^2 - 2*(n - 2)*floor((n - 1)/3) + 3*floor((n - 1)/3)^2) for n > 0.
%F A357837 a(n) = Sum_{k=1..n} A047410(k+1) for n > 0.
%F A357837 a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n > 4.
%F A357837 O.g.f.: 2*x*(2 + x + 2*x^2 - x^3)/((1 - x)^3*(1 + x + x^2)).
%F A357837 E.g.f.: 2*exp(-x/2)*(exp(3*x/2)*(6*x*(3 + x) - 1) + cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/9.
%e A357837 Illustrations for n = 1..8:
%e A357837         _           _ _          _ _ _
%e A357837        |_|         |  _|        |  _|_|
%e A357837                    |_|_|        |_|  _|
%e A357837                                 |_|_|_|
%e A357837     a(1) = 4     a(2) = 10     a(3) = 20
%e A357837      _ _ _ _     _ _ _ _ _    _ _ _ _ _ _
%e A357837     |  _|_| |   |  _|_|  _|  |  _|_|  _|_|
%e A357837     |_|  _|_|   |_|  _|_| |  |_|  _|_|  _|
%e A357837     |_|_|  _|   |_|_|  _|_|  |_|_|  _|_| |
%e A357837     |_ _|_|_|   |  _|_|  _|  |  _|_|  _|_|
%e A357837                 |_|_ _|_|_|  |_|  _|_|  _|
%e A357837                              |_|_|_ _|_|_|
%e A357837     a(4) = 32    a(5) = 46     a(6) = 64
%e A357837       _ _ _ _ _ _ _      _ _ _ _ _ _ _ _
%e A357837      |  _|_|  _|_| |    |  _|_|  _|_|  _|
%e A357837      |_|  _|_|  _|_|    |_|  _|_|  _|_| |
%e A357837      |_|_|  _|_|  _|    |_|_|  _|_|  _|_|
%e A357837      |  _|_|  _|_| |    |  _|_|  _|_|  _|
%e A357837      |_|  _|_|  _|_|    |_|  _|_|  _|_| |
%e A357837      |_|_|  _|_|  _|    |_|_|  _|_|  _|_|
%e A357837      |_ _|_|_ _|_|_|    |  _|_|  _|_|  _|
%e A357837                         |_|_ _|_|_ _|_|_|
%e A357837         a(7) = 84           a(8) = 106
%t A357837 Table[2(Ceiling[2(n+1)^2/3]-1),{n,0,49}]
%Y A357837 Cf. A002264, A002522, A005843, A047410 (first differences), A071619, A211547.
%Y A357837 Cf. A345118.
%K A357837 nonn,easy
%O A357837 0,2
%A A357837 _Stefano Spezia_, Oct 17 2022