%I #31 Dec 23 2020 07:27:28

%S 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 11,10,10,11,12,13,12,13,12,13,14,13,14,15,14,14,14,15,16,17,16,17,16,

%U 17,18,19,18,19,20,19,19,20,19,20,21,22,21,22,20,21

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

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr02.jpg">Illustration of A237593 as an isosceles triangle (rows: 1..28)</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the pyramid (first 16 levels)</a>

%F a(n) = n - A240542(n).

%e Illustration of initial terms in two ways:

%e .

%e n a(n)

%e 1 0

%e 2 0 _ _

%e 3 1 |_| _|_|

%e 4 1 _|_| _|_|

%e 5 2 |_ _| _|_ _|

%e 6 1 _|_| _|_|

%e 7 2 |_ _| _|_ _|

%e 8 2 _|_ _| _|_ _|

%e 9 2 |_ _| _ _|_ _|

%e 10 3 _|_ _| |_| _|_ _|_|

%e 11 4 |_ _ _| |_| _|_ _ _|_|

%e 12 3 _|_ _| |_| _|_ _|_|

%e 13 4 |_ _ _| _|_| _|_ _ _|_|

%e 14 5 _|_ _ _| |_ _| _|_ _ _|_ _|

%e 15 4 |_ _ _| |_| _|_ _ _|_|

%e 16 4 |_ _ _| |_| |_ _ _|_|

%e ...

%e Figure 1. Figure 2.

%e .

%e 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 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 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 (PARI) row235791(n) = vector((sqrtint(8*n+1)-1)\2, i, 1+(n-(i*(i+1)/2))\i);

%o row237591(n) = {my(orow = concat(row235791(n), 0)); vector(#orow -1, i, orow[i] - orow[i+1]); }

%o a003056(n) = floor((sqrt(1+8*n)-1)/2);

%o a(n) = my(row=row237591(n)); sum(k=1, a003056(n), if (!(k%2), row[k])); \\ _Michel Marcus_, Dec 22 2020

%Y Cf. A000203, A000217, A067742, A237591, A237593, A240542, A244050, A245092, A259177, A286001, A338204, A338758.

%K nonn

%O 1,5

%A _Omar E. Pol_, Dec 21 2020