This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A236106 #46 Nov 05 2024 05:39:55
%S A236106 2,6,10,2,14,0,18,6,22,0,2,26,10,0,30,0,0,34,14,6,38,0,0,2,42,18,0,0,
%T A236106 46,0,10,0,50,22,0,0,54,0,0,6,58,26,14,0,2,62,0,0,0,0,66,30,0,0,0,70,
%U A236106 0,18,10,0,74,34,0,0,0,78,0,0,0,6,82,38,22,0,0,2
%N A236106 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the twice odd numbers (A016825) interleaved with k-1 zeros, and the first element of column k is in row k(k+1)/2.
%C A236106 Gives an identity for the twice sigma function (A074400), the sum of the even divisors of 2n.
%C A236106 Alternating sum of row n equals A074400(n), i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = 2*A000203(n) = A074400(n).
%C A236106 Row n has length A003056(n) hence the first element of column k is in row A000217(k).
%C A236106 The number of positive terms in row n is A001227(n).
%C A236106 For more information see A196020.
%F A236106 T(n,k) = 2*A196020(n,k).
%e A236106 Triangle begins:
%e A236106 2;
%e A236106 6;
%e A236106 10, 2;
%e A236106 14, 0;
%e A236106 18, 6;
%e A236106 22, 0, 2;
%e A236106 26, 10, 0;
%e A236106 30, 0, 0;
%e A236106 34, 14, 6;
%e A236106 38, 0, 0, 2;
%e A236106 42, 18, 0, 0;
%e A236106 46, 0, 10, 0;
%e A236106 50, 22, 0, 0;
%e A236106 54, 0, 0, 6;
%e A236106 58, 26, 14, 0, 2;
%e A236106 62, 0, 0, 0, 0;
%e A236106 66, 30, 0, 0, 0;
%e A236106 70, 0, 18, 10, 0;
%e A236106 74, 34, 0, 0, 0;
%e A236106 78, 0, 0, 0, 6;
%e A236106 82, 38, 22, 0, 0, 2;
%e A236106 86, 0, 0, 14, 0, 0;
%e A236106 90, 42, 0, 0, 0, 0;
%e A236106 94, 0, 26, 0, 0, 0;
%e A236106 ...
%e A236106 For n = 9 the divisors of 2*9 = 18 are 1, 2, 3, 6, 9, 18, therefore the sum of the even divisors of 18 is 2 + 6 + 18 = 26. On the other hand the 9th row of triangle is 34, 14, 6, therefore the alternating row sum is 34 - 14 + 6 = 26, equaling the sum of the even divisors of 18.
%e A236106 If n is even then the alternating sum of the n-th row of triangle is simpler than the sum of the even divisors of 2n. Example: for n = 12 the sum of the even divisors of 2*12 = 24 is 2 + 4 + 6 + 8 + 12 + 24 = 56, and the alternating sum of the 12th row of triangle is 46 - 0 + 10 - 0 = 56.
%Y A236106 Cf. A000203, A000217, A001227, A003056, A016825, A074400, A196020, A211343, A228813, A231345, A231347, A235791, A235794, A236104, A236112.
%K A236106 nonn,tabf
%O A236106 1,1
%A A236106 _Omar E. Pol_, Jan 23 2014