A338204 a(n) is the sum of odd-indexed terms (of every row) of the first n rows of the triangle A237591.

1, 3, 5, 8, 11, 16, 21, 27, 34, 41, 48, 57, 66, 75, 86, 98, 110, 123, 136, 151, 166, 181, 196, 213, 231, 249, 267, 287, 307, 329, 351, 374, 397, 420, 445, 471, 497, 523, 549, 577, 605, 635, 665, 695, 727, 759, 791, 825, 860, 896, 932, 968, 1004, 1042, 1080, 1120, 1160, 1200, 1240, 1282

Offset

1,2

Links

Michel Marcus, Table of n, a(n) for n = 1..5000

Formula

a(n) = A000217(n) - A338758(n).

Example

Illustration of a(16) = 98 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 98 cells in the first 16 rows of the diagram, so a(16) = 98.
Figure 2 shows the illustration of a(16) taken from an octant of the pyramid described in A244050 and A245092. There are 98 cells in the first 16 rows of the diagram, so a(16) = 98.

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);
f(n) = my(row=row237591(n)); sum(k=1, a003056(n), if ((k%2), row[k])); \\ A240542
a(n) = sum(k=1, n, f(k)); \\ Michel Marcus, Dec 22 2020

Crossrefs

Partial sums of A240542.

Cf. A000217, A067742, A235791, A237591, A237593, A259176, A286001, A322141, A338758.

Sequence in context: A344010 A281333 A371154 * A244031 A194803 A284835

Adjacent sequences: A338201 A338202 A338203 * A338205 A338206 A338207

Keyword

nonn

Author

Omar E. Pol, Dec 21 2020

Status

approved