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A352116
Partial sums of the odd triangular numbers (A014493).
2
1, 4, 19, 40, 85, 140, 231, 336, 489, 660, 891, 1144, 1469, 1820, 2255, 2720, 3281, 3876, 4579, 5320, 6181, 7084, 8119, 9200, 10425, 11700, 13131, 14616, 16269, 17980, 19871, 21824, 23969, 26180, 28595, 31080, 33781, 36556, 39559, 42640, 45961, 49364, 53019, 56760, 60765
OFFSET
1,2
FORMULA
a(n) = Sum_{k=1..n} A014493(k) = Sum_{k=1..n} (2*k-1)(2*k-1-(-1)^k)/2.
a(n) = A352115(n-1) + (-1)^(n-1)*n.
a(n) = A000447(n) - A352115(n-1).
From Stefano Spezia, Mar 05 2022: (Start)
a(n) = n*(4*n^2 - 1 - 3*(-1)^n)/6.
G.f.: x*(1 + 2*x + 10*x^2 + 2*x^3 + x^4)/((1 - x)^4*(1 + x)^2). (End)
EXAMPLE
a(1) = 1 because 1 is the first odd term in A000027.
a(2) = 1 + 3 = 4, the sum of the first two odd terms in A000027, and so on.
MATHEMATICA
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {1, 4, 19, 40, 85, 140}, 50] (* Amiram Eldar, Mar 05 2022 *)
PROG
(PARI) to(n) = (2*n-1)*(2*n-1-(-1)^n)/2; \\ A014493
a(n) = sum(k=1, n, to(k)); \\ Michel Marcus, Mar 05 2022
(Python)
def A352116(n): return n*((n-1)<<1)*(n+1)//3 + n*(n&1) # Chai Wah Wu, Feb 12 2023
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Michel Marcus, Mar 05 2022
STATUS
approved