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%I #60 Dec 29 2021 17:47:19
%S 1,5,9,13,17,29,33,37,41,45,57,61,65,77,81,93,97,109,113,117,129,133,
%T 137,141,145,165,177,181,185,197,209,213,217,221,233,245,249,261,265,
%U 277,289,301,305,309,313,325,329,333,337,341,361,373,385,397,401,413,417,421,433,437,449
%N The number of pseudo-Pythagorean triples (which allow negative or 0 sides) on a 2D lattice that are on or inside a circle of radius n.
%C Consider a 2D lattice, where the Cartesian coordinates x and y are legs of the Pythagorean triangle. Thus the notion of Pythagorean triple is extended to the cases when sides x, y are in Z (i.e., sides also include negative integers and zero). The sequence gives the number of such triples on or inside a circle of radius n.
%C Partial sums of A046109.
%H Alexander Kritov, <a href="/A349538/a349538_1.c.txt">Source code</a>
%F a(n) = (A211432(n) + 1)/2.
%F a(n) = a(n-1) + 4 + 8*A046080(n).
%e Sides (coordinates) a(n)
%e ------------------------------------------------------------------------------
%e (0,0) 1
%e (-1,0)(0,-1)(0,1)(1,0) 5
%e (-2,0)(0,-2)(0,2)(2,0) 9
%e (-3,0)(0,-3)(0,3)(3,0) 13
%e (-4,0)(0,-4)(0,4)(4,0) 17
%e (-5,0)(-4,-3)(-4,3)(-3,-4)(-3,4)(0,-5)(0,5)(3,-4)(3,4)(4,-3)(4,3)(5,0) 29
%e (-6,0)(0,-6)(0,6)(6,0) 33
%e (-7,0)(0,-7)(0,7)(7,0) 37
%e (-8,0)(0,-8)(0,8)(8,0) 41
%e (-9,0)(0,-9)(0,9)(9,0) 45
%e (-10,0)(-8,-6)(-8,6)(-6,-8)(-6,8)(0,-10)(0,10)(6,-8)(6,8)(8,-6)(8,6)(10,0) 57
%e (-11,0)(0,-11)(0,11)(11,0) 61
%e (-12,0)(0,-12)(0,12)(12,0) 65
%o (C) See links.
%o (PARI) f(n) = if(n==0, return(1)); my(f=factor(n)); 4*prod(i=1, #f~, if(f[i, 1]%4==1, 2*f[i, 2]+1, 1)); \\ A046109
%o a(n) = sum(k=0, n, f(k)); \\ _Michel Marcus_, Nov 27 2021
%Y Cf. A046080, A211432, A046109 (first differences), A349536 (in 1/8 sector).
%K nonn
%O 0,2
%A _Alexander Kritov_, Nov 21 2021
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