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

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, 11, 10, 10, 11, 12, 13, 12, 13, 12, 13, 14, 13, 14, 15, 14, 14, 14, 15, 16, 17, 16, 17, 16, 17, 18, 19, 18, 19, 20, 19, 19, 20, 19, 20, 21, 22, 21, 22, 20, 21

Offset: 1

Author

Omar E. Pol, Dec 21 2020

Keywords

• nonn

Examples

			Illustration of initial terms in two ways:
.
n    a(n)
1      0
2      0                     _                                      _
3      1                    |_|                                   _|_|
4      1                   _|_|                                 _|_|
5      2                  |_ _|                               _|_ _|
6      1                 _|_|                               _|_|
7      2                |_ _|                             _|_ _|
8      2               _|_ _|                           _|_ _|
9      2              |_ _|  _                        _|_ _|
10     3             _|_ _| |_|                     _|_ _|_|
11     4            |_ _ _| |_|                   _|_ _ _|_|
12     3           _|_ _|   |_|                 _|_ _|_|
13     4          |_ _ _|  _|_|               _|_ _ _|_|
14     5         _|_ _ _| |_ _|             _|_ _ _|_ _|
15     4        |_ _ _|   |_|             _|_ _ _|_|
16     4        |_ _ _|   |_|            |_ _ _|_|
...
                    Figure 1.                       Figure 2.
.
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.
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.
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).
		

Links

• Omar E. Pol, Illustration of A237593 as an isosceles triangle (rows: 1..28)
• Omar E. Pol, Perspective view of the pyramid (first 16 levels)

Crossrefs

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

Programs

PARI

row235791(n) = vector((sqrtint(8*n+1)-1)\2, i, 1+(n-(i*(i+1)/2))\i);
row237591(n) = {my(orow = concat(row235791(n), 0)); vector(#orow -1, i, orow[i] - orow[i+1]); }
a003056(n) = floor((sqrt(1+8*n)-1)/2);
a(n) = my(row=row237591(n)); sum(k=1, a003056(n), if (!(k%2), row[k])); \\ Michel Marcus, Dec 22 2020

Formula

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