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

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, 102, 111, 119, 128, 136, 145, 154, 164, 175, 185, 195, 206, 218, 231, 243, 256, 268, 281, 295, 308, 322, 337, 351, 365, 379, 394, 410, 427, 443, 460, 476, 493, 511, 530, 548, 567, 587, 606, 625

Offset: 1

Author

Omar E. Pol, Dec 21 2020

Keywords

• nonn

Examples

			Illustration of a(16) = 38 in two ways:
.
Level
1
2                            _                                      _
3                           |_|                                   _|_|
4                          _|_|                                 _|_|
5                         |_ _|                               _|_ _|
6                        _|_|                               _|_|
7                       |_ _|                             _|_ _|
8                      _|_ _|                           _|_ _|
9                     |_ _|  _                        _|_ _|
10                   _|_ _| |_|                     _|_ _|_|
11                  |_ _ _| |_|                   _|_ _ _|_|
12                 _|_ _|   |_|                 _|_ _|_|
13                |_ _ _|  _|_|               _|_ _ _|_|
14               _|_ _ _| |_ _|             _|_ _ _|_ _|
15              |_ _ _|   |_|             _|_ _ _|_|
16              |_ _ _|   |_|            |_ _ _|_|
...
                    Figure 1.                       Figure 2.
.
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.
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.
		

Crossrefs

Partial sums of A322141.

Cf. A000217, A067742, A237591, A237593, A240542, A259177, A286001, A338204.

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);
a322141(n) = my(row=row237591(n)); sum(k=1, a003056(n), if (!(k%2), row[k]));
a(n) = sum(k=1, n, a322141(k)); \\ Michel Marcus, Dec 22 2020

Formula

a(n) = A000217(n) - A338204(n).