A258445 -id:A258445 - cp's OEIS frontend

A259052 Sum of Pascal triples.

3, 4, 4, 4, 5, 6, 8, 6, 5, 6, 8, 13, 12, 13, 8, 6, 7, 10, 19, 20, 26, 20, 19, 10, 7, 8, 12, 26, 30, 45, 40, 45, 30, 26, 12, 8, 9, 14, 34, 42, 71, 70, 90, 70, 71, 42, 34, 14, 9, 10, 16, 43, 56, 105, 112, 161, 140, 161, 112, 105, 56, 43, 16, 10

Offset: 1

Wolfdieter Lang, Jun 27 2015

The sequence of row lengths of this irregular triangle is A005408(n-1) = 2*n - 1.
This entry is motivated by A258445 from Craig Knecht. There the minima of the Pascal triples are given.
A Pascal triple PT(n, k) for n >= 1, k = 1, 2, ..., 2*n-1 is defined for even k by (P(n-1, k/2-1), P(n-1, k/2), P(n, k/2)) with P(n, k) = A007318(n, k) = binomial(n, k), and for odd k by (P(n-1, (k-1)/2), P(n, (k-1)/2), P(n, (k+1)/2)).
The strip S_n between row n-1 and n of Pascal's triangle (written as symmetric equilateral triangle) is divided into 2*n-1 small equilateral up - down triangles connecting neighboring entries of Pascal's triangle. For odd k these triangles have their base on row n of Pascal's triangle (up triangles), and for even n their base is on row n-1 (down triangles). There are n up triangles and n-1 down triangles in strip S_n.
The present irregular triangle gives the sum of the Pascal triples.
This is motivated by the idea (see A258445) of considering equal touching cylinders (closed only with a bottom disk) with centers at the corners of the small up and down triangles and radius r/2 if the side of each triangle has length r. They are filled with a liquid to a height h with h/r given by the Pascal entry at the center of the bottom of the cylinder. If, for each of the three pairs from a triple of touching cylinders a hole on the bottom of the vertical touching line is opened, then the new height H of the liquid for such a triple will be the arithmetic mean of the three original heights of the three touching cylinders. The ratio H/r will be 1/3 of the corresponding irregular triangle entry for this Pascal triple.
The row sums of this irregular triangle give 3*A033484(n-1), n >= 1.

			The irregular triangle T(n, k) starts:
  n\k 1  2  3  4  5  6  7  8  9 10 11 12 13 ...
  1:  3
  2:  4  4  4
  3:  5  6  8  6  5
  4:  6  8 13 12 13  8  6
  5:  7 10 19 20 26 20 19 10  7
  6:  8 12 26 30 45 40 45 30 26 12  8
  7:  9 14 34 42 71 70 90 70 71 42 34 14  9
  ...
T(3, 2) = 6 from the sum of the Pascal triple (1, 2, 3) (from the first down triangle in Pascal's triangle strip S_3).
The height ratio H/r for this Pascal triple PT(3, 2) = (1, 2, 3) is (1 + 2 + 3)/3 = T(3, 2)/3 = 2.

Craig Knecht, Pascal triples
Craig Knecht, Sum of triples table

Cf. A007318, A258445, A005408, 3*A033484.

tabl(nn) = {for (n=1, nn, for (k=1, 2*n-1, kk = (k+1)\2; if (k%2, v = binomial(n-1, kk-1) + binomial(n, kk-1) + binomial(n, kk), v = binomial(n, kk) + binomial(n-1, kk-1) + binomial(n-1, kk)); print1(v, ", ");); print(););} \\ Michel Marcus, Jun 27 2015

T(n, 2*m) = P(n-1, m-1) + P(n-1, m) + P(n, m) with P(n, k) = A007318(n, k) = binomial(n, k), for m = 1..n-1, and
T(n, 2*m-1) = P(n-1, m-1) + P(n, m-1) + P(n, m) for m = 1..n.
T(n, 2*m-1) = A028263(n-1, m-1), n >= 1, m = 1..n, and T(n, 2*m) = 2*A014410(n, m), n >= 2, m = 1..n-1.

A204259 Matrix given by f(i,j) = 1 + [(2i+j) mod 3], by antidiagonals.

Original entry on oeis.org

1, 2, 3, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 3, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 3, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3

Offset: 1

Clark Kimberling, Jan 14 2012

This data is used to specify the height of hexagonally packed cylinders in a triangle with open boundaries. Three cylinders that touch each other define a "triple" and water can be retained between these cylinders. A257594, A258445 and A259052 give a classification for such spaces. The links below ignore the inter-cylinder space retention and only consider the water retention above solid cylinders. - Craig Knecht, Jul 16 2015

			Northwest corner:
1 2 3 1 2 3 1 2
3 1 2 3 1 2 3 1
2 3 1 2 3 1 2 3
1 2 3 1 2 3 1 2
3 1 2 3 1 2 3 1
2 3 1 2 3 1 2 3

Craig Knecht, Water retention triple.
Craig Knecht, Row sums of numbers completely surrounded by larger numbers (water retention) in A204259 = A008611.
Wikipedia, Water retention on mathematical surfaces

Cf. A204260.

f[i_, j_] := 1 + Mod[2 i + j, 3];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 12}, {i, 1, n}]]   (* A204259 *)
Permanent[m_] :=
  With[{a = Array[x, Length[m]]},
   Coefficient[Times @@ (m.a), Times @@ a]];
Table[Permanent[m[n]], {n, 1, 20}]     (* A204258 *)

A164991 Number of triangular involutions of n. A triangular involution is a square involution with at most three faces.

Original entry on oeis.org

1, 1, 3, 6, 13, 26, 54, 108, 221, 442, 898, 1796, 3634, 7268, 14668, 29336, 59101, 118202, 237834, 475668, 956198, 1912396, 3841588, 7683176, 15425138, 30850276, 61908564, 123817128, 248377156, 496754312

Offset: 1

Simone Rinaldi (rinaldi(AT)unisi.it), Sep 04 2009

The sequence 2^(n+1) - binomial(n, floor(n/2)), which begins 1,3,6,... has Hankel transform (-1)^n*(2*n+1) (A157142). - Paul Barry, Nov 03 2010

For n >= 2 also row sums of A258445. - Wolfdieter Lang, Jun 27 2015

F. Disanto, A. Frosini, S. Rinaldi, Square Involutions, Proceedings of Permutation Patterns, July, 13-17 2009, Florence.
T. Mansour, S. Severini, Grid polygons from permutations and their enumeration by the kernel method, 19th Conference on Formal Power Series and Algebraic Combinatorics, Tianjin, China, July 2-6, 2007.

F. Disanto, A. Frosini, S. Rinaldi, Square involutions, J. Int. Seq. 14 (2011) # 11.3.5.
T. Mansour, S. Severini, Grid polygons from permutations and their enumeration by the kernel method, arXiv:math/0603225 [math.CO], 2006.

Cf. A128652, A128650, A258445.

Join[{1},Table[2^(n-1)-Binomial[n-2,Floor[(n-2)/2]],{n,2,30}]] (* Harvey P. Dale, Dec 26 2015 *)

a(n) =  2^(n-1) - binomial(n-2, (n-2)\2) \\ Michel Marcus, May 27 2013

a(n) = 2^(n-1) - binomial(n-2, floor((n-2)/2)) for n>1, a(1)=1.
From Wolfdieter Lang, Jun 27 2015: (Start)
a(n) = Sum_{k = 1..2*n-3} A258445(n-1, k), n >= 2.
a(2*k+1) = 4*Sum_{j = 0..(k-2)} binomial(2*k-1,j) + 3*binomial(2*k-1,k-1), k >= 1.
a(2*k) = 4*Sum_{j = 0..(k-2)} binomial(2*(k-1),j) + binomial(2*(k-1),k-1), k >= 1. (End)
(-n+1)*a(n) + 2*(n-1)*a(n-1) + 4*(n-4)*a(n-2) + 8*(-n+4)*a(n-3) = 0. - R. J. Mathar, Aug 09 2017

A259549 Triangle T(n,k) with rows of length 2n-1 filled with consecutive integers, each appearing twice except for the last term, T(n,2n-1) = n(n+1)/2.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 37, 37, 38, 38

Offset: 1

Craig Knecht, Jun 30 2015

This sequence is derived from the water retention heights between a group of hexagonally packed, vertically oriented cylinders. The cylinders fill an equilateral triangle. The cylinder height is specified by sequential natural numbers.
This entry is motivated by A258445 and A259052. The water retention on mathematical surfaces concept uses the integer values to specify the height of cylinders. Three touching cylinders define the retention unit and thus a relationship between 3 integers. Taking the minimum, maximum or sum of these three integers provides the data points to construct a new triangle. The water retention between three cylinders is the minimum of the three heights.
The row sums for the minimum of three terms in this triangle are A081436.
The retention values between rows can be thought of having a wave form with peak and trough values. These peak and trough values correspond to A006002 and A006003. This water retention business shows that A081436(n) = A006002(n) + A006003(n+1).
In general any 2D arrangement of numbers can be subdivided by up and down retention triples.
The terms of the "flattened" sequence (concatenation of the rows) are the positive integers, repeated twice except for (1, 3, 6, 10, ...) = triangular numbers A000217, which are only listed once. - M. F. Hasler, Aug 11 2015

			The irregular triangle T(n,k) starts:
n/k 1  2  3  4  5  6  7  8  9 10 11
1:  1
2:  2  2  3
3:  4  4  5  5  6
4:  7  7  8  8  9  9 10
5: 11 11 12 12 13 13 14 14 15
6: 16 16 17 17 18 18 19 19 20 20 21

Craig Knecht, Micro and macro equilateral triangles.
Craig Knecht, Wave peak and trough.
Wikipedia, Water Retention on Mathematical Surfaces

Partial sums of A071028.

a259549(nmax) = { /* Give the first nmax rows */
  my(L=List(), t);
  for(n=1, nmax,
    t=(n^2-n+2)/2;
    for(k=0, n-2,
      listput(L, t+k); listput(L, t+k)
    );
    listput(L, n*(n+1)/2)
  );
  Vec(L)
}
a259549(6) \\ Colin Barker, Jul 04 2015

A259549(n,k)=(n^2-n+k+1)\2 \\ M. F. Hasler, Aug 11 2015

a(n) = (1/4)*(2*t(n) + 1 - (-1)^t(n)), where t(n) = n + floor(sqrt(n)). - Ridouane Oudra, Jun 08 2019

More terms from Colin Barker, Jul 04 2015

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A259052 Sum of Pascal triples.

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A204259 Matrix given by f(i,j) = 1 + [(2i+j) mod 3], by antidiagonals.

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A164991 Number of triangular involutions of n. A triangular involution is a square involution with at most three faces.

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A259549 Triangle T(n,k) with rows of length 2n-1 filled with consecutive integers, each appearing twice except for the last term, T(n,2n-1) = n(n+1)/2.

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