|
%I #17 Jan 06 2017 08:23:45
%S 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 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 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 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 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 The right border gives A240542.
%C For an image of the construction process of the Dyck path for sigma(15) see the image file in the Links section.
%C The length of the n-th row is A003056(n). - _Omar E. Pol_, Nov 03 2015
%H Hartmut F. W. Hoft, <a href="/A264116/a264116.pdf">Construction process for sigma(15)</a>
%H Hartmut F. W. Hoft, <a href="/A264116/a264116_1.pdf">Sigma(n) equals area of its symmetric representation</a>
%F 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 The data in form of the irregular triangle T(n,k):
%e 1;
%e 2;
%e 3, 2;
%e 4, 3;
%e 5, 3;
%e 6, 4, 5;
%e 7, 4, 5;
%e 8, 5, 6;
%e 9, 5, 7;
%e 10, 6, 8, 7;
%e 11, 6, 8, 7;
%e 12, 7, 10, 9;
%e 13, 7, 10, 9;
%e 14, 8, 11, 9;
%e 15, 8, 12, 10, 11;
%e 16, 9, 13, 11, 12;
%e 17, 9, 13, 11, 12;
%e 18, 10, 15, 12, 13;
%e 19, 10, 15, 12, 13;
%e 20, 11, 16, 13, 15;
%e 21, 11, 17, 14, 16, 15;
%e 22, 12, 18, 14, 16, 15;
%t a264116[n_, k_] := Sum[(-1)^(i+1)*Ceiling[(n+1)/i - (i+1)/2], {i,k}]
%t a264116[n_] := Map[a264116[n, #]&, Range[Floor[(Sqrt[8*n+1] - 1)/2]]]
%t Flatten[Map[a264116,Range[22]]] (* data *)
%Y Cf. A000217, A003056, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A071561.
%K nonn,tabf
%O 1,2
%A _Hartmut F. W. Hoft_, Nov 03 2015
|
