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.
0, 4, 10, 20, 32, 46, 64, 84, 106, 132, 160, 190, 224, 260, 298, 340, 384, 430, 480, 532, 586, 644, 704, 766, 832, 900, 970, 1044, 1120, 1198, 1280, 1364, 1450, 1540, 1632, 1726, 1824, 1924, 2026, 2132, 2240, 2350, 2464, 2580, 2698, 2820, 2944, 3070, 3200, 3332
Offset: 0
Examples
Illustrations for n = 1..8:
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a(1) = 4 a(2) = 10 a(3) = 20
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a(4) = 32 a(5) = 46 a(6) = 64
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a(7) = 84 a(8) = 106
Links
- Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).
Crossrefs
Programs
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Mathematica
Table[2(Ceiling[2(n+1)^2/3]-1),{n,0,49}]
Formula
a(n) = 2*(ceiling(2*(n+1)^2/3) - 1).
a(n) = 2*(A071619(n+1) - 1).
a(n) = 2*(1 + n^2 - 2*(n - 2)*floor((n - 1)/3) + 3*floor((n - 1)/3)^2) for n > 0.
a(n) = Sum_{k=1..n} A047410(k+1) for n > 0.
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n > 4.
O.g.f.: 2*x*(2 + x + 2*x^2 - x^3)/((1 - x)^3*(1 + x + x^2)).
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.