A236630 Irregular triangle T(n,k) of alternating sums of squares of entries in the rows in the triangle of A235791, read by rows.

1, 4, 9, 8, 16, 15, 25, 21, 36, 32, 33, 49, 40, 41, 64, 55, 56, 81, 65, 69, 100, 84, 88, 87, 121, 96, 100, 99, 144, 119, 128, 127, 169, 133, 142, 141, 196, 160, 169, 165, 225, 176, 192, 188, 189, 256, 207, 223, 219, 220, 289, 225, 241, 237, 238

Offset: 1

Author

Omar E. Pol, Jan 29 2014

Keywords

• nonn
• tabf

Comments

The original name was: Number of "ON" cells at n-th stage in a cellular automaton (or pseudo cellular automaton) related to sigma (see Comments for precise definition).

Examples

			Triangle begins:
    1;
    4;
    9,   8;
   16,  15;
   25,  21;
   36,  32,  33;
   49,  40,  41;
   64,  55,  56;
   81,  65,  69;
  100,  84,  88,  87;
  121,  96, 100,  99;
  144, 119, 128, 127;
  169, 133, 142, 141;
  196, 160, 169, 165;
  225, 176, 192, 188, 189;
  256, 207, 223, 219, 220;
  289, 225, 241, 237, 238;
  ...
From _Omar E. Pol_, Apr 20 2024: (Start)
Illustration of the 6th row as the area of a polygon (or the number of cells) in the fourth quadrant:
.     _ _ _ _ _ _       _ _ _ _ _ _       _ _ _ _ _ _
.    |           |     |           |     |           |
.    |           |     |           |     |           |
.    |           |     |           |     |           |
.    |           |     |        _ _|     |          _|
.    |           |     |       |         |        _|
.    |_ _ _ _ _ _|     |_ _ _ _|         |_ _ _ _|
.
.          36           36 - 4 = 32     36 - 4 + 1 = 33
.
(End)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10075 (rows n = 1..500, flattened)
• Hartmut F. W. Hoft, Proof that right border of triangle is A024916

Crossrefs

Row n has length A003056(n).

The first element of column k is in row A000217(k).

Column 1 gives the positive terms of A000290.

Right border gives A024916.

Row n is the alternating sum of entries in row n of A236104.

Cf. A003056, A000203, A004125, A026914, A071561, A196020, A235791, A236631, A237048, A237591, A237593, A237270, A249223, A264116, A280851.

Programs

Mathematica

Map[Accumulate, Table[(-2 Boole[EvenQ[k]] + 1)*Ceiling[(n + 1)/k - (k + 1)/2]^2, {n, 20}, {k, Floor[(Sqrt[8*n + 1] - 1)/2]}]] // Flatten (* Michael De Vlieger, Apr 30 2024, after Hartmut F. W. Hoft at A235791 *)

Formula

From Hartmut F. W. Hoft, Apr 30 2024: (Start)

T(n, k) = Sum_{j = 1 .. k} (-1)^(j + 1) * S(n, j)^2, n >= 0, 1 <= k <= A003056(n), where S(n, j) is the j-th entry in the n-th row of the triangle of A235791.

T(n, k) = Sum_{j = 1 .. k} (-1)^(j+1) * S(n, j), n >= 0, 1 <= k <= A003056(n), where S(n, j) is the j-th entry in the n-th row of the triangle of A236104. (End)

Extensions

New name from Hartmut F. W. Hoft, Apr 27 2024

0 removed, offset changed and minor edits from Omar E. Pol, Apr 28 2024