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.

0, 4, 11, 20, 34, 50, 69, 92, 116, 144, 175, 208, 246, 286, 329, 376, 424, 476, 531, 588, 650, 714, 781, 852, 924, 1000, 1079, 1160, 1246, 1334, 1425, 1520, 1616, 1716, 1819, 1924, 2034, 2146, 2261, 2380, 2500, 2624, 2751, 2880, 3014, 3150, 3289, 3432, 3576, 3724

Offset

0,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (3,-3,1,-1,3,-3,1).

Formula

O.g.f.: x*(4 - x - x^2 + 3*x^3 + x^4)/((1 - x)^3*(1 + x^4)).

E.g.f.: (exp(x)*x*(8 + 3*x) + (-1)^(1/4)*(sinh((-1)^(1/4)*x) - sin((-1)^(1/4)*x)))/2.

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.

a(n) = (n*(5 + 3*n) - (1 - (-1)^n)*sin((n-1)*Pi/4))/2.

a(n) = A211014(n/2) - A000035(n)*A056594((n-3)/2).

a(2*n) = A211014(n).

a(k) = A115067(k+1) for k not congruent to 3 mod 4 (A004773).

From Helmut Ruhland, Jan 29 2024: (Start)

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

n mod 8 = 0 1 2 3 4 5 6 7

f8(n) = 0 0 -3 -2 -2 -2 1 0 (End)

Example

Illustrations for n = 1..8:
        _           _ _          _ _ _
       |_|         |_|_|        |_ _ _|
                   |_ _|        |_|_|_|
                                |_ _ _|
    a(1) = 4     a(2) = 11     a(3) = 20
     _ _ _ _     _ _ _ _ _    _ _ _ _ _ _
    |_ _|_|_|   |_ _|_|_ _|  |_|_|_ _ _|_|
    |_|_ _ _|   |_|_ _ _|_|  |_ _ _|_|_ _|
    |_ _|_|_|   |_ _|_|_ _|  |_|_|_ _ _|_|
    |_|_ _ _|   |_|_ _ _|_|  |_ _ _|_|_ _|
                |_ _|_|_ _|  |_|_|_ _ _|_|
                             |_ _ _|_|_ _|
    a(4) = 34    a(5) = 50     a(6) = 69
      _ _ _ _ _ _ _      _ _ _ _ _ _ _ _
     |_|_|_ _ _|_|_|    |_ _|_|_ _ _|_|_|
     |_ _ _|_|_ _ _|    |_|_ _ _|_|_ _ _|
     |_|_|_ _ _|_|_|    |_ _|_|_ _ _|_|_|
     |_ _ _|_|_ _ _|    |_|_ _ _|_|_ _ _|
     |_|_|_ _ _|_|_|    |_ _|_|_ _ _|_|_|
     |_ _ _|_|_ _ _|    |_|_ _ _|_|_ _ _|
     |_|_|_ _ _|_|_|    |_ _|_|_ _ _|_|_|
                        |_|_ _ _|_|_ _ _|
        a(7) = 92           a(8) = 116

Mathematica

LinearRecurrence[{3, -3, 1, -1, 3, -3, 1}, {0, 4, 11, 20, 34, 50, 69}, 50]
a[ n_] := (3*n^2 + 5*n)/2 - (-1)^Floor[n/4]*Boole[Mod[n, 4] == 3]; (* Michael Somos, Jan 25 2024 *)

Prog

(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
(PARI) {a(n) = (3*n^2 + 5*n)/2 - (-1)^(n\4)*(n%4==3)}; /* Michael Somos, Jan 25 2024 */

Crossrefs

Cf. A000035, A004773, A056594, A115067, A211014, A316316, A316317, A368052.

Sequence in context: A212959 A046279 A301074 * A090541 A301156 A301086

Adjacent sequences: A345115 A345116 A345117 * A345119 A345120 A345121

Keyword

nonn,easy

Author

Stefano Spezia, Jun 08 2021

Status

approved