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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).

%I A126587 #51 Apr 18 2024 17:56:50
%S A126587 3,17,43,81,131,193,267,353,451,561,683,817,963,1121,1291,1473,1667,
%T A126587 1873,2091,2321,2563,2817,3083,3361,3651,3953,4267,4593,4931,5281,
%U A126587 5643,6017,6403,6801,7211,7633,8067,8513,8971,9441,9923,10417,10923,11441
%N A126587 a(n) is the number of integer lattice points inside the right triangle with legs 3n and 4n (and hypotenuse 5n).
%C A126587 Row sums of triangle A193832. - _Omar E. Pol_, Aug 22 2011
%H A126587 Vincenzo Librandi, <a href="/A126587/b126587.txt">Table of n, a(n) for n = 1..10000</a>
%H A126587 Zak Seidov <a href="http://web.archive.org/web/20091026222954/http://geocities.com/zseidov/InsidePoints.html">Inside points</a>
%H A126587 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A126587 a(n) = A186424(2*n-1).
%F A126587 By Pick's theorem, a(n) = 6*n^2 - 4*n + 1. - _Nick Hobson_, Mar 13 2007
%F A126587 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
%F A126587 E.g.f.: exp(x)*(1 + 2*x + 6*x^2) - 1. - _Stefano Spezia_, May 09 2021
%F A126587 a(n) = (A000326(2n-1) + A000326(2n))/2. - _Charlie Marion_, Apr 17 2024
%e A126587 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.
%t A126587 nip[a_,b_]:=Sum[Floor[b-b*i/a-10^-6],{i,a-1}] Table[nip[3k,4k],{k,100}]
%t A126587 Table[6*n^2-4*n+1, {n,1,50}] (* _G. C. Greubel_, Mar 06 2018 *)
%o A126587 (Magma) [6*n^2 - 4*n + 1: n in [1..50] ]; // _Vincenzo Librandi_, May 23 2011
%o A126587 (PARI) a(n)=6*n^2-4*n+1 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A126587 Cf. A000326, A186424, A193832.
%K A126587 nonn,easy
%O A126587 1,1
%A A126587 _Zak Seidov_, Jan 05 2007