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A322141 a(n) is also the sum of the even-indexed terms of the n-th row of the triangle A237591.

%I A322141 #31 Dec 23 2020 07:27:28
%S A322141 0,0,1,1,2,1,2,2,2,3,4,3,4,5,4,4,5,5,6,5,6,7,8,7,7,8,9,8,9,8,9,9,10,
%T A322141 11,10,10,11,12,13,12,13,12,13,14,13,14,15,14,14,14,15,16,17,16,17,16,
%U A322141 17,18,19,18,19,20,19,19,20,19,20,21,22,21,22,20,21
%N A322141 a(n) is also the sum of the even-indexed terms of the n-th row of the triangle A237591.
%H A322141 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr02.jpg">Illustration of A237593 as an isosceles triangle (rows: 1..28)</a>
%H A322141 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the pyramid (first 16 levels)</a>
%F A322141 a(n) = n - A240542(n).
%e A322141 Illustration of initial terms in two ways:
%e A322141 .
%e A322141 n    a(n)
%e A322141 1      0
%e A322141 2      0                     _                                      _
%e A322141 3      1                    |_|                                   _|_|
%e A322141 4      1                   _|_|                                 _|_|
%e A322141 5      2                  |_ _|                               _|_ _|
%e A322141 6      1                 _|_|                               _|_|
%e A322141 7      2                |_ _|                             _|_ _|
%e A322141 8      2               _|_ _|                           _|_ _|
%e A322141 9      2              |_ _|  _                        _|_ _|
%e A322141 10     3             _|_ _| |_|                     _|_ _|_|
%e A322141 11     4            |_ _ _| |_|                   _|_ _ _|_|
%e A322141 12     3           _|_ _|   |_|                 _|_ _|_|
%e A322141 13     4          |_ _ _|  _|_|               _|_ _ _|_|
%e A322141 14     5         _|_ _ _| |_ _|             _|_ _ _|_ _|
%e A322141 15     4        |_ _ _|   |_|             _|_ _ _|_|
%e A322141 16     4        |_ _ _|   |_|            |_ _ _|_|
%e A322141 ...
%e A322141                     Figure 1.                       Figure 2.
%e A322141 .
%e A322141 Figure 1 shows the illustration of initial terms taken from the isosceles triangle of A237593 (see link). For n = 16 there are (3 + 1) = 4 cells in the 16th row of the diagram, so a(16) = 4.
%e A322141 Figure 2 shows the illustration of initial terms taken from an octant of the pyramid described in A244050 and A245092 (see link). For n = 16 there are (3 + 1) = 4 cells in the 16th row of the diagram, so a(16) = 4.
%e A322141 Note that if we fold each level (or row) of that isosceles triangle of A237593 we essentially obtain the structure of the pyramid described in A245092 whose terraces at the n-th level have a total area equal to sigma(n) = A000203(n).
%o A322141 (PARI) row235791(n) = vector((sqrtint(8*n+1)-1)\2, i, 1+(n-(i*(i+1)/2))\i);
%o A322141 row237591(n) = {my(orow = concat(row235791(n), 0)); vector(#orow -1, i, orow[i] - orow[i+1]); }
%o A322141 a003056(n) = floor((sqrt(1+8*n)-1)/2);
%o A322141 a(n) = my(row=row237591(n)); sum(k=1, a003056(n), if (!(k%2), row[k])); \\ _Michel Marcus_, Dec 22 2020
%Y A322141 Cf. A000203, A000217, A067742, A237591, A237593, A240542, A244050, A245092, A259177, A286001, A338204, A338758.
%K A322141 nonn
%O A322141 1,5
%A A322141 _Omar E. Pol_, Dec 21 2020