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A126587 a(n) = number of integer lattice points inside the right-angle triangle with legs 3n and 4n (and hypotenuse 5n). 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Row sums of triangle A193832. - Omar E. Pol, Aug 22 2011

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Zak Seidov Inside points

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = A186424(2*n-1).

By Pick's theorem, a(n) = 6*n^2 - 4*n + 1. - Nick Hobson (nickh(AT)qbyte.org), 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

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}]

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

Sequence in context: A226492 A092347 A215429 * A108126 A106256 A091624

Adjacent sequences:  A126584 A126585 A126586 * A126588 A126589 A126590

KEYWORD

nonn,easy

AUTHOR

Zak Seidov, Jan 05 2007

STATUS

approved

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Last modified August 23 19:19 EDT 2017. Contains 291021 sequences.