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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.
0
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
0,2
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.
EXAMPLE
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
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _
| _|_| _|_| | | _|_| _|_| _|
|_| _|_| _|_| |_| _|_| _|_| |
|_|_| _|_| _| |_|_| _|_| _|_|
| _|_| _|_| | | _|_| _|_| _|
|_| _|_| _|_| |_| _|_| _|_| |
|_|_| _|_| _| |_|_| _|_| _|_|
|_ _|_|_ _|_|_| | _|_| _|_| _|
|_|_ _|_|_ _|_|_|
a(7) = 84 a(8) = 106
MATHEMATICA
Table[2(Ceiling[2(n+1)^2/3]-1), {n, 0, 49}]
CROSSREFS
Cf. A002264, A002522, A005843, A047410 (first differences), A071619, A211547.
Cf. A345118.
Sequence in context: A100439 A301193 A301180 * A008128 A008094 A301286
KEYWORD
nonn,easy
AUTHOR
Stefano Spezia, Oct 17 2022
STATUS
approved