Johan Nilsson - cp's OEIS frontend

A375874 Number of distinct n X n patterns in the squiral tiling.

1, 2, 14, 70, 126, 270, 438, 630, 790, 958, 1134, 1542, 1974, 2430, 2910, 3414, 3942, 4494, 5070, 5670, 6142, 6622, 7110, 7606, 8110, 8622, 9142, 9670, 10206, 11406, 12630, 13878, 15150, 16446, 17766, 19110, 20478, 21870, 23286, 24726, 26190, 27678, 29190

Offset: 0

Johan Nilsson, Sep 01 2024

nonn

The squiral tiling, can be obtained as the limit pattern under the binary block substitution 0 -> [[1,0,1],[0,0,0][1,0,1]] and 1 -> [[0,1,0],[1,1,1][0,1,0]], when starting with the seed 0.

			a(1) = 2, since there are 2 different 1X1 patterns in the squiral tiling; namely 0 and 1.
a(2) = 14, since there are 14 different 2X2 patterns in the squiral tiling; namely all 16 2X2 binary matrices except [[0,0],[0,0]] and [[1,1],[1,1]].

M. Baake, and U. Grimm, Aperiodic Order. Volume 1: A Mathematical Invitation, Encyclopedia of Mathematics and its Applications No. 149 Cambridge University Press, Cambridge (2013).
B. Grünbaum and F. C. Shephard, Tilings and Patterns, W.H. Freeman 1987, MR0857454.

Johan Nilsson, Table of n, a(n) for n = 0..10000
Tilings Encyclopedia, Squiral
Johan Nilsson The Pattern Complexity of the Squiral Tiling

a:= n-> `if`(n<3, [1, 2, 14][n+1], ((A, B)-> (4+8*A-8*B)*(n-1)^2+
    (12*3^A+24*3^B)*(n-1)-18*9^A)(ilog[3](n-2), ilog[3]((n-2)/2))):
seq(a(n), n=0..42);  # Alois P. Heinz, Sep 18 2024

a[n_] := If[n<3, {1, 2, 14}[[n+1]], With[{A = Floor@ Log[3, n-2], B = Floor@ Log[3, (n-2)/2]}, (4+8*A-8*B)*(n-1)^2+(12*3^A+24*3^B)*(n-1)-18*9^A]];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Mar 27 2025, after Alois P. Heinz *)

a(n)=if(n<4, [1,2,14,70][n+1], my(A=logint(n-2,3), B=logint((n-2)\2,3)); (4 + 8*A - 8*B)*(n - 1)^2 + (12 * 3^A + 24 * 3^B) * (n - 1) - 18 * 9^A) \\ Andrew Howroyd, Sep 18 2024

from sympy import integer_log
def A375874(n):
    if n<4: return (1,2,14,70)[n]
    a, b = integer_log(n-2,3)[0]+1, integer_log((n>>1)-1,3)[0]+1
    return (n-1)*((1+(a-b<<1))*(n-1)+((c:=3**a)+(3**b<<1))<<1)-c**2<<1 # Chai Wah Wu, Sep 18 2024

a(n) = (4 + 8*A - 8*B)*(n - 1)^2 + (12 * 3^A + 24 * 3^B) * (n - 1) - 18 * 9^A, for n>=4 where A = floor(log3(n-2)), B = floor(log3((n-2)/2)), and log3 is the logarithm in base 3.
For n>=2;
a(3*n-2) = 9*a(n),
a(9*n-7) = 5*a(3*n+1) - 16*a(3*n) + 20*a(3*n-1),
a(9*n-4) = - a(3*n+1) + 5*a(3*n) + 5*a(3*n-1),
a(9*n-1) = 2*a(3*n+1) + 8*a(3*n) - a(3*n-1),
a(3*n) = a(3*n-1) + 3*a(n+1) - 3*a(n).

A271081 Number of ways to tile an n X n X n cube with 1 X 1 X 1 and 2 X 2 X 2 tiles.

Original entry on oeis.org

1, 2, 9, 2089, 3144692, 2748613397101, 107008949868167431857

Offset: 1

Johan Nilsson, Mar 30 2016

			There are 9 ways to tile a cube of side length 3 with cubes of side length 1 and 2, so a(3) = 9.

Cf. A063443 (a 2D version of this tiling count).

A257555 Number of distinct ways to dissect a cube into n rectangular boxes of equal volume, up to reflection and rotation.

Original entry on oeis.org

1, 1, 2, 8, 35, 198

Offset: 1

Johan Nilsson, Apr 29 2015

The boxes may have different shapes, as long as they all have the same volume and right angle corners.

			There are eight ways to dissect a cube into four boxes of equal volume, so a(4)=8.

Johan Nilsson, Illustrations for terms up to a(5).

Cf. A108066 is a 2D version of the dissection count.

A195206 Number of 1s in the first 10^n entries of the Kolakoski sequence, A000002.

Original entry on oeis.org

1, 5, 49, 502, 4996, 49972, 499986, 5000046, 50000675, 500001223, 4999997671, 50000001587, 500000050701, 5000000008159, 50000000316237, 500000000977421, 4999999994637728, 49999999977479348, 499999999944465105, 4999999999725703450, 49999999999090850760

Offset: 0

Johan Nilsson, Sep 13 2011

nonn

			The first entries of the Kolakoski sequence, A000002, are 1221121221... From this we see that a(0)=1, since the first letter is 1, and a(1)=5 since among the first 10 letters 5 of them are 1s.

J. Nilsson, A Space Efficient Algorithm for the Calculation of the Digit Distribution in the Kolakoski Sequence, arXiv preprint arXiv:1110.4228 [math.CO], 2011, J. Int. Seq. 15 (2012) #12.6.7
J. Nilsson, Letter Frequencies in the Kolakoski Sequence, Acta Physica Polonica A, 126 (2014), 549-552.
Michael Rao, Trucs et bidules sur la séquence de Kolakoski, 2012, in French.
Ed Wynn, C program to calculate A195206 (and A195211 etc)

Cf. A000002.

a(14) from Ed Wynn, Jun 24 2014
a(15)-a(19) from Richard P. Brent, Jul 02 2017
a(20) from Richard P. Brent, Mar 01 2018

A195211 Number of 2s in the first 10^n entries of the generalized Kolakoski-(2,3) sequence A071820.

Original entry on oeis.org

1, 5, 51, 502, 4995, 49999, 499980, 4999995, 50000202, 499999731, 5000005565, 50000013114, 499999997503, 4999999971938, 49999999974390, 499999999976909, 4999999999414101, 50000000022964476, 500000000029433861, 4999999999986496894

Offset: 0

Johan Nilsson, Sep 13 2011

nonn

			The first entries of the Kolakoski-(2,3) sequence, A071820, are 2233222333... From this we see that a(0)=1, since the first letter is 2, and a(1) = 5 since among the first 10 letters 5 of them are 2s.

J. Nilsson, A Space Efficient Algorithm for the Calculation of the Digit Distribution in the Kolakoski Sequence, arXiv preprint arXiv:1110.4228 [math.CO], 2011, J. Int. Seq. 15 (2012) #12.6.7

Cf. A000002, A071820.

/* See A195206. - Ed Wynn, Jun 24 2014 */

a(13)-a(14) from Ed Wynn, Jun 24 2014

a(15)-a(19) from Hiroaki Yamanouchi, Jul 25 2017

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User: Johan Nilsson

A375874 Number of distinct n X n patterns in the squiral tiling.

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A271081 Number of ways to tile an n X n X n cube with 1 X 1 X 1 and 2 X 2 X 2 tiles.

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A257555 Number of distinct ways to dissect a cube into n rectangular boxes of equal volume, up to reflection and rotation.

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A195206 Number of 1s in the first 10^n entries of the Kolakoski sequence, A000002.

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A195211 Number of 2s in the first 10^n entries of the generalized Kolakoski-(2,3) sequence A071820.

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