A033586 -id:A033586 - cp's OEIS frontend

A322181 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = A321768(n, k) + A321769(n, k) + A321770(n, k).

12, 30, 70, 40, 56, 176, 126, 208, 408, 198, 154, 234, 84, 90, 330, 260, 546, 1026, 476, 456, 736, 286, 418, 1218, 828, 1178, 2378, 1188, 800, 1160, 390, 340, 900, 570, 644, 1364, 714, 374, 494, 144, 132, 532, 442, 1044, 1924, 874, 918, 1518, 608, 1116, 3196

Offset: 1

Rémy Sigrist, Nov 30 2018

This sequence gives the perimeters of the primitive Pythagorean triangles corresponding to the primitive Pythagorean triples in the tree described in A321768.

If we order the terms in this sequence and keep duplicates then we obtain A024364.

			The first rows are:
   12
   30, 70, 40
   56, 176, 126, 208, 408, 198, 154, 234, 84
T(1,1) corresponds to the perimeter of the triangle with sides 3, 4, 5; hence T(1, 1) = 3 + 4 + 5 = 12.

Rémy Sigrist, Rows n = 1..9, flattened
Index entries related to Pythagorean Triples

Cf. A001542, A002939, A024364, A033586, A321768, A321769, A321770.

M = [[1, -2, 2; 2, -1, 2; 2, -2, 3], [1, 2, 2; 2, 1, 2; 2, 2, 3], [-1, 2, 2; -2, 1, 2; -2, 2, 3]];
T(n, k) = my (t=[3; 4; 5], d=digits(3^(n-1)+k-1, 3)); for (i=2, #d, t = M[d[i]+1] * t); return (t[1, 1] + t[2, 1] + t[3, 1])

Empirically:
- T(n, 1) = A002939(n+1),
- T(n, (3^(n-1) + 1)/2) = A001542(n+1),
- T(n, 3^(n-1)) = A033586(n).

A067875 One player's total legal chess moves by piece type on standard chessboard.

Original entry on oeis.org

668, 1964, 2434, 3248, 5152, 8400

Offset: 1

Rick L. Shepherd, Feb 25 2002

The terms are given in order of increasing numbers of total moves for the six piece types; that is, a(1) Pawn, a(2) Knight, a(3) King, a(4) Bishop, a(5) Rook and a(6) Queen.
The sum of these six terms is 21866, the total number of moves available to White or to Black. Hence 43732 moves, the answer to a question raised in the link below, are available to both players.
Notes: (1) Capturing, for example, a Knight on a particular square counts as a different move from capturing, for example, a Rook on the same square.
(2) Moving a piece to a square without capturing is counted separately from captures by the piece on that square.
(3) The two castling moves are counted as King moves only.
(4) The 14 en passant captures are included in the Pawn moves.
(5) Two-square initial Pawn moves are included.
(6) Pawn promotion on a particular square to a Bishop, for example, counts as a different move from promotion to a Queen on the same square.
(7) Pieces of the same type and color are considered indistinguishable.
(8) Moves causing check, discovered check, double check, checkmate, or stalemate are not distinguished from other moves.
Valid boards and moves require (these two somewhat subtle realizations):
(9) A King cannot capture Pawns on their original squares when they would be attacking the King; this would require the King to have made an illegal move earlier by walking into check.
(10) A King cannot diagonally capture Bishops on their home corner squares - again this would require the King to have made an illegal move earlier. However, a King can diagonally capture Bishops on the other two corners as the Bishop can be in position *after* the King is - via Pawn promotion in this case.

Inspired by a question posed by Tim Krabbé.

T. Krabbé, Open Chess Diary, Item 105, Mar 29 2001: Never in a trillion

Cf. A035005 - A035008, A033586 (count the moves per piece type differently).

a(6) = a(4) + a(5) (Queen moves equal sum of Bishop and Rook moves). Generalizing all terms for n X n chessboards other than 8 x 8 requires defining how many pieces and how many types of pieces are originally on the board and/or can be promoted to, especially because of the way captures are counted.

A135443 Number of maximal directed trails in the labeled n-ladder graph P_2 X P_n.

Original entry on oeis.org

2, 8, 12, 40, 84, 144, 220, 312, 420, 544, 684, 840, 1012, 1200, 1404, 1624, 1860, 2112, 2380, 2664, 2964, 3280, 3612, 3960, 4324, 4704, 5100, 5512, 5940, 6384, 6844, 7320, 7812, 8320, 8844, 9384, 9940, 10512, 11100, 11704, 12324, 12960, 13612, 14280

Offset: 1

Yasutoshi Kohmoto, Feb 18 2008

			For n = 4 the graph is
  .__.__.__.
  |__|__|__|
Names of nodes:
  1 2 3 4
  a b c d
Maximal directed paths which start from node 3:
  34dcba123c
  34dc32ba12
  34dc321ab2
  34dc321abc
  3cd432ba12
  3cd4321ab2
  3cd4321abc
  3cba1234dc
  321abc34dc
  321abcd43c
There are also paths from nodes c,b,2. So a(4) = 4*10 = 40.

Apart from initial terms sequence is the same as A033586.

Cf. A003682, A089243.

For n > 2, a(n) = 4 * (n-2) * (2*n - 3) = A033586(n-2). - Max Alekseyev, May 04 2023

Edited and extended by Max Alekseyev, May 04 2023

cp's OEIS Frontend

A322181 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = A321768(n, k) + A321769(n, k) + A321770(n, k).

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A067875 One player's total legal chess moves by piece type on standard chessboard.

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A135443 Number of maximal directed trails in the labeled n-ladder graph P_2 X P_n.

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