A001960 -id:A001960 - cp's OEIS frontend

A001959 u-pile numbers for the 3-Wythoff game with i=2.

0, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 26, 28, 29, 30, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 82, 84, 85, 86, 88

Offset: 0

N. J. A. Sloane

See Connell (1959) for further information.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Ian G. Connell, A generalization of Wythoff's game, Canad. Math. Bull. 2 (1959) 181-190

Cf. A001954, A001958, A001960, A001963-A001968.

Table[Floor[(n + 2/3)*(Sqrt[13] - 1)/2], {n, 0, 100}] (* T. D. Noe, Aug 17 2012 *)

a(n) = floor( (n+2/3)*(sqrt(13)-1)/2 ). - R. J. Mathar, Feb 14 2011

Edited by Hugo Pfoertner, Dec 27 2021

A001957 u-pile positions in the 3-Wythoff game with i=1.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 42, 43, 44, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 62, 64, 65, 66, 68, 69, 70, 72, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87

Offset: 0

N. J. A. Sloane

See Connell (1959) for further information.

The complement is A001960. - Omar E. Pol, Jan 06 2009

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Ian G. Connell, A generalization of Wythoff's game, Canad. Math. Bull. 2 (1959) 181-190.

Cf. A001954, A001958-A001960, A001963-A001968.

Table[Floor[(n + 1/3)*(Sqrt[13] - 1)/2], {n, 0, 100}] (* T. D. Noe, Aug 17 2012 *)

a(n) = floor((n+1/3)*(sqrt(13)-1)/2). - R. J. Mathar, Feb 14 2011

Edited by N. J. A. Sloane, Dec 27 2021

A295928 Number of triangular matrices T(n,i,k), k <= i <= n, with entries "0" or "1" with the property that each triple {T(n,i,k), T(n,i,k+1), T(n,i-1,k)} containing a single "0" can be successively replaced by {1, 1, 1} until finally no "0" entry remains.

Original entry on oeis.org

1, 3, 16, 122, 1188, 13844, 185448, 2781348, 45868268, 821096828, 15804092592, 324709899276, 7081361097108, 163179784397820, 3958519452109912, 100778473796398524

Offset: 1

Gerhard Kirchner, Nov 30 2017

A triple {T(n,i,k), T(n,i,k+1), T(n,i-1,k)} will be called a primitive triangle. It is easy to see that b(n) = n(n-1)/2 is the number of such triangles. At each step, exactly one primitive triangle is completed (replaced by {1, 1, 1}). So there are b(n) "0"- and n "1"-terms. Thus the starting matrix has no complete primitive triangle. Furthermore, any triangular submatrix T(m,i,k), k <= i <= m < n cannot have more than m "1"-terms because otherwise it would have less "0"-terms than primitive triangles. The replacement of at least one "0"-term would complete more than one primitive triangle. This has been excluded.
So T(n, i, k) is a special case of U(n, i, k), described in A101481: a(n) < A101481(n+1).
A start matrix may serve as a pattern for a number wall used on worksheets for elementary mathematics, see link "Number walls". That is why I prefer the more descriptive name "fill matrix".
The algorithm for the sequence is rather slow because each start matrix is constructed separately. There exists a faster recursive algorithm which produces the same terms and therefore is likely to be correct, but it is based on a conjecture. For the theory of the recurrence, see "Recursive aspects of fill matrices". Probable extension a(10)-a(14): 821096828, 15804092592, 324709899276, 7081361097108, 163179784397820.
The number of fill matrices with n rows and all "1"- terms concentrated on the last two rows, is A001960(n).
See link "Reconstruction of a sequence".
Number of 3-permutations of size n avoiding the patterns (12,12) and (312, 231) (explicit bijection with fill matrices). - Juliette Schabanel, Apr 14 2025

			Example (n=2):    0     1    1
    a(2)=3       1 1   0 1  1 0
Example for completing a 3-matrix (3 bottom terms):
    1        1       1       1
   0 0  ->  1 0 ->  1 1 ->  1 1
  1 1 0    1 1 0   1 1 0   1 1 1
Example for a 3-matrix which cannot be completed:
    1        1
   1 1  or  0 0
  0 0 0    1 0 1

Nicolas Bonichon and Pierre-Jean Morel Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022.
Gerhard Kirchner, Recursive aspects of fill matrices
Gerhard Kirchner, Number walls
Gerhard Kirchner, VB-program
Gerhard Kirchner, Reconstruction of a sequence
Ville Salo, Cutting Corners, arXiv:2002.08730 [math.DS], 2020.
Juliette Schabanel, 3D permutations and triangle solitaire, 2025
Yuan Yao and Fedir Yudin, Fine Mixed Subdivisions of a Dilated Triangle, arXiv:2402.13342 [math.CO], 2024.

Cf. A101481, A001960.

From Juliette Schabanel, Apr 14 2025: (Start)
G.f. satisfies T(x)-1-x=I^3(x)∆^3T(x)-3I^2(x)∆^2T(x)+ 3I(x)(∆T(x)-1) with ∆T(x) = (xT(x))' and T(x)= 1 +I(x)∆T(x) (proved).
a(n) ~ cn!e^(sqrt(12n))n^(5/12) (conjectured). (End)

a(10)-a(16) from Juliette Schabanel, Apr 14 2025

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A001959 u-pile numbers for the 3-Wythoff game with i=2.

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A001957 u-pile positions in the 3-Wythoff game with i=1.

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A295928 Number of triangular matrices T(n,i,k), k <= i <= n, with entries "0" or "1" with the property that each triple {T(n,i,k), T(n,i,k+1), T(n,i-1,k)} containing a single "0" can be successively replaced by {1, 1, 1} until finally no "0" entry remains.

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