A316316 -id:A316316 - cp's OEIS frontend

A316357 Partial sums of A316316.

1, 5, 13, 21, 33, 53, 73, 93, 121, 153, 185, 221, 261, 305, 353, 401, 453, 513, 573, 633, 701, 773, 845, 921, 1001, 1085, 1173, 1261, 1353, 1453, 1553, 1653, 1761, 1873, 1985, 2101, 2221, 2345, 2473, 2601, 2733, 2873, 3013, 3153, 3301, 3453, 3605, 3761, 3921

Offset: 0

Rémy Sigrist, Jun 30 2018

nonn

Cf. A316316.

Conjectures from Colin Barker, Dec 13 2018: (Start)
G.f.: (1 + 3*x + 5*x^2 + 2*x^3 + 5*x^4 + 3*x^5 + x^6) / ((1 - x)^3*(1 + x^2)*(1 + x + x^2)).
a(n) = 2*a(n-1) - 2*a(n-2) + 3*a(n-3) - 3*a(n-4) + 2*a(n-5) - 2*a(n-6) + a(n-7) for n>6.
(End)

A316317 Coordination sequence for trivalent node in chamfered version of square grid.

Original entry on oeis.org

1, 3, 6, 11, 14, 15, 20, 25, 26, 29, 34, 37, 40, 43, 46, 51, 54, 55, 60, 65, 66, 69, 74, 77, 80, 83, 86, 91, 94, 95, 100, 105, 106, 109, 114, 117, 120, 123, 126, 131, 134, 135, 140, 145, 146, 149, 154, 157, 160, 163, 166, 171, 174, 175, 180, 185, 186, 189, 194

Offset: 0

N. J. A. Sloane, Jun 29 2018

nonn

Rémy Sigrist, Table of n, a(n) for n = 0..5000
Michel Deza and Mikhail Shtogrin, Isometric embedding of mosaics into cubic lattices, Discrete mathematics 244.1-3 (2002): 43-53. See Fig. 2.
Michel Deza and Mikhail Shtogrin, Isometric embedding of mosaics into cubic lattices, Discrete mathematics 244.1-3 (2002): 43-53. [Annotated scan of page 52 only]
Rémy Sigrist, PARI program for A316317
Rémy Sigrist, Illustration of first terms
N. J. A. Sloane, Initial terms of coordination sequence for trivalent node
N. J. A. Sloane, "Basketweave" tiling by 3X1 rectangles which is equivalent (as far as the graph and coordination sequences are concerned) to this tiling

See A316316 for tetravalent node.
See A250120 for links to thousands of other coordination sequences.
Cf. A316358 (partial sums).
Cf. A056594, A102283.

PARI
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See Links section.
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Apparently, a(n + 12) = a(n) + 40 for any n > 0. - Rémy Sigrist, Jun 30 2018
This can surely be proved by the Coloring Book Method, although I have not worked out the details. See A316316 for the corresponding proof for a tetravalent node. - N. J. A. Sloane, Jun 30 2018
G.f. (assuming above conjecture): (1+x)^2*(1+3*x^2+x^4)/((1-x)^2*(1+x+x^2)*(1+x^2)). - Robert Israel, Jul 01 2018
a(n) = (30*n - 9*A056594(n-1) + 6*A102283(n))/9 for n > 0. - Conjectured by Stefano Spezia, Jun 12 2021

More terms from Rémy Sigrist, Jun 30 2018

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.

Original entry on oeis.org

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

Stefano Spezia, Jun 08 2021

			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

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

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

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 *)

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

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

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)

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A316357 Partial sums of A316316.

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A316317 Coordination sequence for trivalent node in chamfered version of square grid.

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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.

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