A236631 - OEIS

%I #27 May 20 2022 08:57:19

%S 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,

%T -25,4,-1,45,-25,9,-1,42,-36,9,-1,55,-36,9,-4,60,-49,16,-4,1,67,-49,

%U 16,-4,1,69,-64,16,-4,1,86,-64,25,-9,1,84,-81,25,-9,1,103

%N 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.

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

%C Row sums give A000203.

%C 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.

%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

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

%e Written as an irregular triangle the sequence begins:

%e 1;

%e 3;

%e 5,     -1;

%e 8,     -1;

%e 10,    -4;

%e 15,    -4,    1;

%e 16,    -9,    1;

%e 23,    -9,    1;

%e 25,   -16,    4;

%e 31,   -16,    4,   -1;

%e 34,   -25,    4,   -1;

%e 45,   -25,    9,   -1;

%e 42,   -36,    9,   -1;

%e 55,   -36,    9,   -4;

%e 60,   -49,   16,   -4,   1;

%e 67,   -49,   16,   -4,   1;

%e 69,   -64,   16,   -4,   1;

%e 86,   -64,   25,   -9,   1;

%e 84,   -81,   25,   -9,   1;

%e 103,  -81,   25,   -9,   4;

%e 102, -100,   36,   -9,   4,  -1;

%e 113, -100,   36,  -16,   4,  -1;

%e 122, -121,   36,  -16,   4,  -1;

%e 145, -121,   49,  -16,   4,  -1;

%e ...

%e 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.

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

%K sign,tabf

%O 1,2

%A _Omar E. Pol_, Jan 29 2014