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A345118 a(n) is the sum of the lengths of all the segments used to draw a square of side n representing a basketweave pattern where all the multiple strands are of unit width, the horizontal ones appearing as 1 X 3 rectangles, while the vertical ones as unit area squares.

%I A345118 #29 Jan 31 2024 16:47:34
%S A345118 0,4,11,20,34,50,69,92,116,144,175,208,246,286,329,376,424,476,531,
%T A345118 588,650,714,781,852,924,1000,1079,1160,1246,1334,1425,1520,1616,1716,
%U A345118 1819,1924,2034,2146,2261,2380,2500,2624,2751,2880,3014,3150,3289,3432,3576,3724
%N A345118 a(n) is the sum of the lengths of all the segments used to draw a square of side n representing a basketweave pattern where all the multiple strands are of unit width, the horizontal ones appearing as 1 X 3 rectangles, while the vertical ones as unit area squares.
%H A345118 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,-1,3,-3,1).
%F A345118 O.g.f.: x*(4 - x - x^2 + 3*x^3 + x^4)/((1 - x)^3*(1 + x^4)).
%F A345118 E.g.f.: (exp(x)*x*(8 + 3*x) + (-1)^(1/4)*(sinh((-1)^(1/4)*x) - sin((-1)^(1/4)*x)))/2.
%F A345118 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) - a(n-4) + 3*a(n-5) - 3*a(n-6) + a(n-7) for n > 6.
%F A345118 a(n) = (n*(5 + 3*n) - (1 - (-1)^n)*sin((n-1)*Pi/4))/2.
%F A345118 a(n) = A211014(n/2) - A000035(n)*A056594((n-3)/2).
%F A345118 a(2*n) = A211014(n).
%F A345118 a(k) = A115067(k+1) for k not congruent to 3 mod 4 (A004773).
%F A345118 From _Helmut Ruhland_, Jan 29 2024: (Start)
%F A345118 For n > 1: a(n) - (2 * A368052(n+2) + A368052(n+3)) * 2 is periodic for n mod 8, i.e. a(n) = (2 * A368052(n+2) + A368052(n+3)) * 2 + f8(n) with
%F A345118 n mod 8 =   0   1   2   3   4   5   6   7
%F A345118 f8(n)   =   0   0  -3  -2  -2  -2   1   0   (End)
%e A345118 Illustrations for n = 1..8:
%e A345118         _           _ _          _ _ _
%e A345118        |_|         |_|_|        |_ _ _|
%e A345118                    |_ _|        |_|_|_|
%e A345118                                 |_ _ _|
%e A345118     a(1) = 4     a(2) = 11     a(3) = 20
%e A345118      _ _ _ _     _ _ _ _ _    _ _ _ _ _ _
%e A345118     |_ _|_|_|   |_ _|_|_ _|  |_|_|_ _ _|_|
%e A345118     |_|_ _ _|   |_|_ _ _|_|  |_ _ _|_|_ _|
%e A345118     |_ _|_|_|   |_ _|_|_ _|  |_|_|_ _ _|_|
%e A345118     |_|_ _ _|   |_|_ _ _|_|  |_ _ _|_|_ _|
%e A345118                 |_ _|_|_ _|  |_|_|_ _ _|_|
%e A345118                              |_ _ _|_|_ _|
%e A345118     a(4) = 34    a(5) = 50     a(6) = 69
%e A345118       _ _ _ _ _ _ _      _ _ _ _ _ _ _ _
%e A345118      |_|_|_ _ _|_|_|    |_ _|_|_ _ _|_|_|
%e A345118      |_ _ _|_|_ _ _|    |_|_ _ _|_|_ _ _|
%e A345118      |_|_|_ _ _|_|_|    |_ _|_|_ _ _|_|_|
%e A345118      |_ _ _|_|_ _ _|    |_|_ _ _|_|_ _ _|
%e A345118      |_|_|_ _ _|_|_|    |_ _|_|_ _ _|_|_|
%e A345118      |_ _ _|_|_ _ _|    |_|_ _ _|_|_ _ _|
%e A345118      |_|_|_ _ _|_|_|    |_ _|_|_ _ _|_|_|
%e A345118                         |_|_ _ _|_|_ _ _|
%e A345118         a(7) = 92           a(8) = 116
%t A345118 LinearRecurrence[{3,-3,1,-1,3,-3,1},{0,4,11,20,34,50,69},50]
%t A345118 a[ n_] := (3*n^2 + 5*n)/2 - (-1)^Floor[n/4]*Boole[Mod[n, 4] == 3]; (* _Michael Somos_, Jan 25 2024 *)
%o A345118 (PARI) concat(0, Vec(x*(4 - x - x^2 + 3*x^3 + x^4)/((1 - x)^3*(1 + x^4)) + O(x^40))) \\ _Felix Fröhlich_, Jun 09 2021
%o A345118 (PARI) {a(n) = (3*n^2 + 5*n)/2 - (-1)^(n\4)*(n%4==3)}; /* _Michael Somos_, Jan 25 2024 */
%Y A345118 Cf. A000035, A004773, A056594, A115067, A211014, A316316, A316317, A368052.
%K A345118 nonn,easy
%O A345118 0,2
%A A345118 _Stefano Spezia_, Jun 08 2021