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A338758 a(n) is the sum of even-indexed terms (of every row) of first n rows of the triangle A237591.

%I A338758 #34 Dec 23 2020 07:28:04
%S A338758 0,0,1,2,4,5,7,9,11,14,18,21,25,30,34,38,43,48,54,59,65,72,80,87,94,
%T A338758 102,111,119,128,136,145,154,164,175,185,195,206,218,231,243,256,268,
%U A338758 281,295,308,322,337,351,365,379,394,410,427,443,460,476,493,511,530,548,567,587,606,625
%N A338758 a(n) is the sum of even-indexed terms (of every row) of first n rows of the triangle A237591.
%F A338758 a(n) = A000217(n) - A338204(n).
%e A338758 Illustration of a(16) = 38 in two ways:
%e A338758 .
%e A338758 Level
%e A338758 1
%e A338758 2                            _                                      _
%e A338758 3                           |_|                                   _|_|
%e A338758 4                          _|_|                                 _|_|
%e A338758 5                         |_ _|                               _|_ _|
%e A338758 6                        _|_|                               _|_|
%e A338758 7                       |_ _|                             _|_ _|
%e A338758 8                      _|_ _|                           _|_ _|
%e A338758 9                     |_ _|  _                        _|_ _|
%e A338758 10                   _|_ _| |_|                     _|_ _|_|
%e A338758 11                  |_ _ _| |_|                   _|_ _ _|_|
%e A338758 12                 _|_ _|   |_|                 _|_ _|_|
%e A338758 13                |_ _ _|  _|_|               _|_ _ _|_|
%e A338758 14               _|_ _ _| |_ _|             _|_ _ _|_ _|
%e A338758 15              |_ _ _|   |_|             _|_ _ _|_|
%e A338758 16              |_ _ _|   |_|            |_ _ _|_|
%e A338758 ...
%e A338758                     Figure 1.                       Figure 2.
%e A338758 .
%e A338758 For n = 16, figure 1 shows the illustration of a(16) taken from the isosceles triangle of A237593. There are 38 cells in the first 16 levels of the diagram, so a(16) = 38.
%e A338758 Figure 2 shows the illustration of a(16) taken from an octant of the pyramid described in A244050 and A245092. There are 38 cells in the first 16 levels of the diagram, so a(16) = 38.
%o A338758 (PARI) row235791(n) = vector((sqrtint(8*n+1)-1)\2, i, 1+(n-(i*(i+1)/2))\i);
%o A338758 row237591(n) = {my(orow = concat(row235791(n), 0)); vector(#orow -1, i, orow[i] - orow[i+1]); }
%o A338758 a003056(n) = floor((sqrt(1+8*n)-1)/2);
%o A338758 a322141(n) = my(row=row237591(n)); sum(k=1, a003056(n), if (!(k%2), row[k]));
%o A338758 a(n) = sum(k=1, n, a322141(k)); \\ _Michel Marcus_, Dec 22 2020
%Y A338758 Partial sums of A322141.
%Y A338758 Cf. A000217, A067742, A237591, A237593, A240542, A259177, A286001, A338204.
%K A338758 nonn
%O A338758 1,4
%A A338758 _Omar E. Pol_, Dec 21 2020