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A236630
Irregular triangle T(n,k) of alternating sums of squares of entries in the rows in the triangle of A235791, read by rows.
3
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
1,2
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).
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10075 (rows n = 1..500, flattened)
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)
EXAMPLE
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)
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 *)
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.
Sequence in context: A248390 A075570 A308289 * A351381 A342551 A359681
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Jan 29 2014
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
STATUS
approved