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

%I A338204 #46 Dec 23 2020 17:18:34
%S A338204 1,3,5,8,11,16,21,27,34,41,48,57,66,75,86,98,110,123,136,151,166,181,
%T A338204 196,213,231,249,267,287,307,329,351,374,397,420,445,471,497,523,549,
%U A338204 577,605,635,665,695,727,759,791,825,860,896,932,968,1004,1042,1080,1120,1160,1200,1240,1282
%N A338204 a(n) is the sum of odd-indexed terms (of every row) of the first n rows of the triangle A237591.
%H A338204 Michel Marcus, <a href="/A338204/b338204.txt">Table of n, a(n) for n = 1..5000</a>
%F A338204 a(n) = A000217(n) - A338758(n).
%e A338204 Illustration of a(16) = 98 in two ways:
%e A338204 .
%e A338204 Level                                 _              _
%e A338204 1                                   _|_|            |_|_
%e A338204 2                                 _|_ _|            |_ _|
%e A338204 3                               _|_ _|              |_ _|_
%e A338204 4                             _|_ _ _|              |_ _ _|
%e A338204 5                           _|_ _ _|  _             |_ _ _|_ _
%e A338204 6                         _|_ _ _ _| |_|            |_ _ _ _|_|
%e A338204 7                       _|_ _ _ _|   |_|            |_ _ _ _|_|_
%e A338204 8                     _|_ _ _ _ _|  _|_|            |_ _ _ _ _|_|_
%e A338204 9                   _|_ _ _ _ _|   |_ _|            |_ _ _ _ _|_ _|
%e A338204 10                _|_ _ _ _ _ _|   |_|              |_ _ _ _ _ _|_|
%e A338204 11              _|_ _ _ _ _ _|    _|_|              |_ _ _ _ _ _|_|_ _
%e A338204 12            _|_ _ _ _ _ _ _|   |_ _|              |_ _ _ _ _ _ _|_ _|
%e A338204 13          _|_ _ _ _ _ _ _|     |_ _|              |_ _ _ _ _ _ _|_ _|
%e A338204 14        _|_ _ _ _ _ _ _ _|    _|_|  _             |_ _ _ _ _ _ _ _|_|_ _
%e A338204 15      _|_ _ _ _ _ _ _ _|     |_ _| |_|            |_ _ _ _ _ _ _ _|_ _|_|_
%e A338204 16     |_ _ _ _ _ _ _ _ _|     |_ _| |_|            |_ _ _ _ _ _ _ _ _|_ _|_|
%e A338204 ...
%e A338204                   Figure 1.                                 Figure 2.
%e A338204 .
%e A338204 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.
%e A338204 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.
%o A338204 (PARI) row235791(n) = vector((sqrtint(8*n+1)-1)\2, i, 1+(n-(i*(i+1)/2))\i);
%o A338204 row237591(n) = {my(orow = concat(row235791(n), 0)); vector(#orow -1, i, orow[i] - orow[i+1]); }
%o A338204 a003056(n) = floor((sqrt(1+8*n)-1)/2);
%o A338204 f(n) = my(row=row237591(n)); sum(k=1, a003056(n), if ((k%2), row[k])); \\ A240542
%o A338204 a(n) = sum(k=1, n, f(k)); \\ _Michel Marcus_, Dec 22 2020
%Y A338204 Partial sums of A240542.
%Y A338204 Cf. A000217, A067742, A235791, A237591, A237593, A259176, A286001, A322141, A338758.
%K A338204 nonn
%O A338204 1,2
%A A338204 _Omar E. Pol_, Dec 21 2020