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A126587
a(n) is the number of integer lattice points inside the right triangle with legs 3n and 4n (and hypotenuse 5n).
10
3, 17, 43, 81, 131, 193, 267, 353, 451, 561, 683, 817, 963, 1121, 1291, 1473, 1667, 1873, 2091, 2321, 2563, 2817, 3083, 3361, 3651, 3953, 4267, 4593, 4931, 5281, 5643, 6017, 6403, 6801, 7211, 7633, 8067, 8513, 8971, 9441, 9923, 10417, 10923, 11441
OFFSET
1,1
COMMENTS
Row sums of triangle A193832. - Omar E. Pol, Aug 22 2011
FORMULA
a(n) = A186424(2*n-1).
By Pick's theorem, a(n) = 6*n^2 - 4*n + 1. - Nick Hobson, Mar 13 2007
O.g.f.: x*(3+8*x+x^2)/(1-x)^3 = -1 - 12/(-1+x)^3 - 11/(-1+x) - 22/(-1+x)^2. - R. J. Mathar, Dec 10 2007
E.g.f.: exp(x)*(1 + 2*x + 6*x^2) - 1. - Stefano Spezia, May 09 2021
a(n) = (A000326(2n-1) + A000326(2n))/2. - Charlie Marion, Apr 17 2024
EXAMPLE
At n=1, three lattice points (1,1), (1,2) and (2,1) are inside the triangle with vertices at the points (0,0), (3n,0) and (0,4n); hence a(1)=3.
MATHEMATICA
nip[a_, b_]:=Sum[Floor[b-b*i/a-10^-6], {i, a-1}] Table[nip[3k, 4k], {k, 100}]
Table[6*n^2-4*n+1, {n, 1, 50}] (* G. C. Greubel, Mar 06 2018 *)
PROG
(Magma) [6*n^2 - 4*n + 1: n in [1..50] ]; // Vincenzo Librandi, May 23 2011
(PARI) a(n)=6*n^2-4*n+1 \\ Charles R Greathouse IV, Jun 17 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Zak Seidov, Jan 05 2007
STATUS
approved