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A264116 Irregular triangle read by rows: T(n,k), n>=1, k>=1, of the alternating sum of entries 1 through k in the n-th row of A235791; the first element of column k is in row k(k+1)/2.

%I A264116 #17 Jan 06 2017 08:23:45
%S A264116 1,2,3,2,4,3,5,3,6,4,5,7,4,5,8,5,6,9,5,7,10,6,8,7,11,6,8,7,12,7,10,9,
%T A264116 13,7,10,9,14,8,11,9,15,8,12,10,11,16,9,13,11,12,17,9,13,11,12,18,10,
%U A264116 15,12,13,19,10,15,12,13,20,11,16,13,15,21,11,17,14,16,15,22,12,18,14,16,15
%N A264116 Irregular triangle read by rows: T(n,k), n>=1, k>=1, of the alternating sum of entries 1 through k in the n-th row of A235791; the first element of column k is in row k(k+1)/2.
%C A264116 The numbers in the n-th row of the triangle are the coordinates on the diagonal in the first quadrant of the polygons constructed by alternately adding and subtracting squares taken from the n-th row of A236104. The boundary from (0,n) to (n,0) of the final polygon is the Dyck path as defined in the n-th row of A237593. Therefore, using the arguments in A196020, A236104 and A071561, sigma(n) equals the area of its symmetric representation, for all n>=1.
%C A264116 The right border gives A240542.
%C A264116 For an image of the construction process of the Dyck path for sigma(15) see the image file in the Links section.
%C A264116 The length of the n-th row is A003056(n). - _Omar E. Pol_, Nov 03 2015
%H A264116 Hartmut F. W. Hoft, <a href="/A264116/a264116.pdf">Construction process for sigma(15)</a>
%H A264116 Hartmut F. W. Hoft, <a href="/A264116/a264116_1.pdf">Sigma(n) equals area of its symmetric representation</a>
%F A264116 T(n, k) = Sum_{i=1..k} (-1)^(i+1) A235791(n,i), for n>=1 and 1<=k<=floor((sqrt(8n+1) - 1)/2).
%e A264116 The data in form of the irregular triangle T(n,k):
%e A264116 1;
%e A264116 2;
%e A264116 3,   2;
%e A264116 4,   3;
%e A264116 5,   3;
%e A264116 6,   4,  5;
%e A264116 7,   4,  5;
%e A264116 8,   5,  6;
%e A264116 9,   5,  7;
%e A264116 10,  6,  8,  7;
%e A264116 11,  6,  8,  7;
%e A264116 12,  7, 10,  9;
%e A264116 13,  7, 10,  9;
%e A264116 14,  8, 11,  9;
%e A264116 15,  8, 12, 10, 11;
%e A264116 16,  9, 13, 11, 12;
%e A264116 17,  9, 13, 11, 12;
%e A264116 18, 10, 15, 12, 13;
%e A264116 19, 10, 15, 12, 13;
%e A264116 20, 11, 16, 13, 15;
%e A264116 21, 11, 17, 14, 16, 15;
%e A264116 22, 12, 18, 14, 16, 15;
%t A264116 a264116[n_, k_] := Sum[(-1)^(i+1)*Ceiling[(n+1)/i - (i+1)/2], {i,k}]
%t A264116 a264116[n_] := Map[a264116[n, #]&, Range[Floor[(Sqrt[8*n+1] - 1)/2]]]
%t A264116 Flatten[Map[a264116,Range[22]]] (* data *)
%Y A264116 Cf. A000217, A003056, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A071561.
%K A264116 nonn,tabf
%O A264116 1,2
%A A264116 _Hartmut F. W. Hoft_, Nov 03 2015