A259052 -id:A259052 - cp's OEIS frontend

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

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 *)

A263536 Row sum of an equilateral triangle tiled with the 3,4,5 Pythagorean triple.

Original entry on oeis.org

5, 7, 12, 17, 19, 24, 29, 31, 36, 41, 43, 48, 53, 55, 60, 65, 67, 72, 77, 79, 84, 89, 91, 96, 101, 103, 108, 113, 115, 120, 125, 127, 132, 137, 139, 144, 149, 151, 156, 161, 163, 168, 173, 175, 180, 185, 187, 192, 197, 199, 204, 209, 211, 216, 221, 223, 228

Offset: 1

Craig Knecht, Oct 20 2015

Maximum number of Pythagorean triples in an equilateral triangle.
Two rules are used to construct this equilateral triangle: #1. Start with the number 5 at the top. #2. Require every "triple" to contain the Pythagorean triple 3, 4, 5 (see link below).
Up and down Pythagorean triples consist of two terms below and one above when k is odd (an up triple), and two terms above and one below when k is even (a down triple). Three adjacent terms in a straight line within the triangle form a linear triple.

			Triangle T(n,k):           Row sum
  5;                          5
  3, 4;                       7
  4, 5, 3;                   12
  5, 3, 4, 5;                17
  3, 4, 5, 3, 4;             19
  4, 5, 3, 4, 5, 3;          24

Colin Barker, Table of n, a(n) for n = 1..1000
Craig Knecht, Equilateral triangle tiled with 3,4,5 Pythagorean triples.
Craig Knecht, Interlocked up/down Pythagorean pairs.
Craig Knecht, Linear and triangular triples.
Craig Knecht, Incarcerated numbers.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A136289 (every triple contains 1,2,3), A008854 (every triple contains 1,2,2), A259052 (sum of Pascal triples).

Vec(x*(5*x^2+2*x+5)/((x-1)^2*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Oct 26 2015

From Colin Barker, Oct 26 2015: (Start)
a(n) = a(n-1)+a(n-3)-a(n-4) for n>4.
G.f.: x*(5*x^2+2*x+5) / ((x-1)^2*(x^2+x+1)).
(End)

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

cp's OEIS Frontend

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

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A263536 Row sum of an equilateral triangle tiled with the 3,4,5 Pythagorean triple.

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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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