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A338758
a(n) is the sum of even-indexed terms (of every row) of first n rows of the triangle A237591.
3
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
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
KEYWORD
nonn
AUTHOR
Omar E. Pol, Dec 21 2020
STATUS
approved