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A236112 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists k+1 copies of the squares in nondecreasing order, and the first element of column k is in row k(k+1)/2.

%I A236112 #35 Nov 05 2024 05:40:24
%S A236112 0,0,1,0,1,0,4,0,4,1,0,9,1,0,9,1,0,16,4,0,16,4,1,0,25,4,1,0,25,9,1,0,
%T A236112 36,9,1,0,36,9,4,0,49,16,4,1,0,49,16,4,1,0,64,16,4,1,0,64,25,9,1,0,81,
%U A236112 25,9,1,0,81,25,9,4,0,100,36,9,4,1,0,100,36,16,4,1,0,121,36,16,4,1,0,121,49,16,4,1,0
%N A236112 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists k+1 copies of the squares in nondecreasing order, and the first element of column k is in row k(k+1)/2.
%C A236112 Gives an identity for the sum of remainders of n mod k, for k = 1,2,3,...,n. Alternating sum of row n equals A004125(n), i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A004125(n).
%C A236112 Row n has length A003056(n) hence the first element of column k is in row A000217(k).
%e A236112 Triangle begins:
%e A236112     0;
%e A236112     0;
%e A236112     1,   0;
%e A236112     1,   0;
%e A236112     4,   0;
%e A236112     4,   1,   0;
%e A236112     9,   1,   0;
%e A236112     9,   1,   0;
%e A236112    16,   4,   0;
%e A236112    16,   4,   1,   0;
%e A236112    25,   4,   1,   0;
%e A236112    25,   9,   1,   0;
%e A236112    36,   9,   1,   0;
%e A236112    36,   9,   4,   0;
%e A236112    49,  16,   4,   1,  0;
%e A236112    49,  16,   4,   1,  0;
%e A236112    64,  16,   4,   1,  0;
%e A236112    64,  25,   9,   1,  0;
%e A236112    81,  25,   9,   1,  0;
%e A236112    81,  25,   9,   4,  0;
%e A236112   100,  36,   9,   4,  1,  0;
%e A236112   100,  36,  16,   4,  1,  0;
%e A236112   121,  36,  16,   4,  1,  0;
%e A236112   121,  49,  16,   4,  1,  0;
%e A236112   ...
%e A236112 For n = 24 the 24th row of triangle is 121, 49, 16, 4, 1, 0 therefore the alternating row sum is 121 - 49 + 16 - 4 + 1 - 0 = 85 equaling A004125(24).
%Y A236112 Cf. A000203, A000217, A000290, A003056, A004125, A120444, A196020, A211343, A228813, A231345, A231347, A235791, A235794, A236104, A236106, A237048, A237591, A237593, A261699.
%K A236112 nonn,tabf
%O A236112 1,7
%A A236112 _Omar E. Pol_, Jan 23 2014