A236631 Triangle read by rows: T(j,k), j>=1, k>=1, in which column k lists the positive squares repeated k-1 times, except the column 1 which is A123327. The elements of the even-indexed columns are multiplied by -1. The first element of column k is in row k(k+1)/2.

1, 3, 5, -1, 8, -1, 10, -4, 15, -4, 1, 16, -9, 1, 23, -9, 1, 25, -16, 4, 31, -16, 4, -1, 34, -25, 4, -1, 45, -25, 9, -1, 42, -36, 9, -1, 55, -36, 9, -4, 60, -49, 16, -4, 1, 67, -49, 16, -4, 1, 69, -64, 16, -4, 1, 86, -64, 25, -9, 1, 84, -81, 25, -9, 1, 103

Offset

1,2

Comments

T(j,k) which row j has length A003056(j) hence the first element of column k is in row A000217(j).

Row sums give A000203.

Interpreted as a sequence with index n this is also the first differences of A236630. If a(n) is positive then a(n) is the number of cells turned ON at n-th stage in the structure of A236630. If a(n) is negative then a(n) is the number of cells turned OFF at n-th stage in the structure of A236630.

Links

Table of n, a(n) for n=1..66.

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Index entries for sequences related to sigma(n)

Formula

T(n,1) = A000203(n) + A004125(n).

Example

Written as an irregular triangle the sequence begins:
1;
3;
5,     -1;
8,     -1;
10,    -4;
15,    -4,    1;
16,    -9,    1;
23,    -9,    1;
25,   -16,    4;
31,   -16,    4,   -1;
34,   -25,    4,   -1;
45,   -25,    9,   -1;
42,   -36,    9,   -1;
55,   -36,    9,   -4;
60,   -49,   16,   -4,   1;
67,   -49,   16,   -4,   1;
69,   -64,   16,   -4,   1;
86,   -64,   25,   -9,   1;
84,   -81,   25,   -9,   1;
103,  -81,   25,   -9,   4;
102, -100,   36,   -9,   4,  -1;
113, -100,   36,  -16,   4,  -1;
122, -121,   36,  -16,   4,  -1;
145, -121,   49,  -16,   4,  -1;
...
For j = 15 the divisors of 15 are 1, 3, 5, 15, therefore the sum of divisors of 15 is 1 + 3 + 5 + 15 = 24. On the other hand the 15th row of triangle is 60, -49, 16, -4, 1, therefore the row sum is 60 - 49 + 16 - 4 + 1 = 24, equalling the sum of divisors of 15.

Crossrefs

Cf. A000203, A000217, A000290, A003056, A004125, A024916, A008794, A123327, A196020, A211547, A236104, A236630, A237593.

Sequence in context: A367743 A242390 A065395 * A302800 A367067 A340529

Adjacent sequences: A236628 A236629 A236630 * A236632 A236633 A236634

Keyword

sign,tabf

Author

Omar E. Pol, Jan 29 2014

Status

approved