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

0,2

Links

Table of n, a(n) for n=0..49.

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

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:
        _           _ _          _ _ _
       |_|         |  _|        |  _|_|
                   |_|_|        |_|  _|
                                |_|_|_|
    a(1) = 4     a(2) = 10     a(3) = 20
     _ _ _ _     _ _ _ _ _    _ _ _ _ _ _
    |  _|_| |   |  _|_|  _|  |  _|_|  _|_|
    |_|  _|_|   |_|  _|_| |  |_|  _|_|  _|
    |_|_|  _|   |_|_|  _|_|  |_|_|  _|_| |
    |_ _|_|_|   |  _|_|  _|  |  _|_|  _|_|
                |_|_ _|_|_|  |_|  _|_|  _|
                             |_|_|_ _|_|_|
    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

Adjacent sequences: A357834 A357835 A357836 * A357838 A357839 A357840

Keyword

nonn,easy

Author

Stefano Spezia, Oct 17 2022

Status

approved