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

1,4

Links

Table of n, a(n) for n=1..64.

Formula

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

Example

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.

Prog

(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

Crossrefs

Partial sums of A322141.

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

Sequence in context: A058577 A356075 A071917 * A051060 A116650 A140205

Adjacent sequences: A338755 A338756 A338757 * A338759 A338760 A338761

Keyword

nonn

Author

Omar E. Pol, Dec 21 2020

Status

approved