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a(n)=2^8=256 for n>=8. We observe that this sequence is the transform of A171442 by T such that: T(u_0,u_1,u_2,u_3,u_4,u_5,...)=(u_0,u_0+u_1,u_1+u_2,u_2+u_3,u_3+u_4,...).

a(n)=2^6=64 for n>=6. We observe that this sequence is the transform of A171440 by T such that: T(u_0,u_1,u_2,u_3,u_4,u_5,...)=(u_0,u_0+u_1,u_1+u_2,u_2+u_3,u_3+u_4,...).

Also continued fraction expansion of 1+(1233212607598+5*sqrt(41))/8688482797079. - Bruno Berselli, Sep 23 2011

a(n)=2^7=128 for n>=7. We observe that this sequence is the transform of A171441 by T such that: T(u_0,u_1,u_2,u_3,u_4,u_5,...)=(u_0,u_0+u_1,u_1+u_2,u_2+u_3,u_3+u_4,...).

For the case when the cuts are at 45 degrees to the grid lines, see A329504.

See A329508, A329512, and A329515 for coordination sequences for cylinders formed by rolling up the hexagonal grid ("carbon nanotubes").

The g.f.s for the rows can easily be found using the "trunks and branches" method (see Goodman-Strauss and Sloane). In the illustration for n=5, there are two trunks (blue) and ten branches (red).

The width of the strip is a little harder to define here. In the illustration for n=2, the strip is four hexagons wide if measured along hexagons that touch edge-to-edge. A path joining two vertices to be identified when the cylinder is formed has length 4n edges (8 edges in the illustration for n=2).

For the case when the strip is 2*n+1 hexagons wide see A329515.

For the case when the cuts are parallel to the grid lines, see A329508.

See A329501 and A329504 for coordination sequences for cylinders formed by rolling up the square grid.

From Gary W. Adamson, Dec 16 2009: (Start)

Let M = an infinite lower triangular matrix with (1, 3, 4, 4, 4, ...) in every column shifted down twice, with the rest zeros:

1;

3, 0;

4, 1, 0;

4, 3, 0, 0;

4, 4, 1, 0, 0;

4, 4, 3, 0, 0, 0;

...

A131205 = lim_{n->infinity} M^n, the left-shifted vector considered as a sequence. (End)

The subsequence of primes in this sequence begins with 5 in a row: 3, 7, 13, 23, 37, 83, 647, 1867, 2707, 88873, 388837, 655121, 754903, 928621, 1062443. - Jonathan Vos Post, Apr 25 2010

See A054871 for definitions and key links.

Also, decimal expansion of 31/90. - Bruno Berselli, Mar 18 2015

Essentially the same as A010709, A040002, A113311, A123932, and A151798. - R. J. Mathar, Mar 20 2015

Remainder of the Euclidean division when 10^(10^n) is divided by 7 (proof by induction for n >= 1) [see reference Julien Freslon & Jérôme Poineau]; example: 10^(10^1) = 1428571428 * 7 + 4. - Bernard Schott, Aug 28 2020

Self-convolution of A019590. Up to a sign the convolutional inverse of the natural numbers sequence. - Tanya Khovanova, Jul 14 2007

Iterated partial sums give the chain A130713 -> A113311 -> A008574 -> A001844 -> A005900 -> A006325 -> A033455 -> A259181, up to index. The k-th term of the n-th partial sums is (n^2-7n+14 + 4k(k+n-4))(k+n-4)!/(k-1)!/(n-1)!, for k > 3-n. Iterating partial sums in reverse (n-th differences with n zeros prepended) gives row (n+3) of A182533, modulo signs and trailing zeros. - Travis Scott, Feb 19 2023

The optimal partition is P(n,k) = ([(n+i)/k] : 0 <= i < k).

The table also appears in the solution of a maximum problem in arithmetic considered by K. Mahler and J. Popken. - J. van de Lune and Juan Arias-de-Reyna, Jan 05 2012

T(n,k) is the number of ways to select k class representatives from the mod k partitioning of {1,2,...,n}. - Dennis P. Walsh, Nov 27 2012

T(n,k) is the maximum number of length-k longest common subsequences of a pair of length-n strings. - Cees H. Elzinga, Jun 08 2014

Row sums are A113311. Diagonal sums are A113312. Inverse is A113313. The family of Riordan arrays ((1+x)/(1-(q-1)x),x/(1+x)) allow one to calculate the weight distribution of MDS codes.