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If n is even, (n^2-2*n+2)/2, otherwise (n^2-n+2)/2.
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%I #29 Jul 07 2024 07:58:04

%S 1,1,1,4,5,11,13,22,25,37,41,56,61,79,85,106,113,137,145,172,181,211,

%T 221,254,265,301,313,352,365,407,421,466,481,529,545,596,613,667,685,

%U 742,761,821,841,904,925,991,1013,1082,1105,1177,1201,1276,1301,1379,1405,1486,1513,1597,1625,1712,1741,1831,1861,1954

%N If n is even, (n^2-2*n+2)/2, otherwise (n^2-n+2)/2.

%C Total number of circles in A371373 and A371253, if in the later all the circular arcs are completed to form full circles.

%C The sequence also gives the number of vertices created from circle intersections when a circle of radius r is drawn around each of n equally spaced points on the circumference of a circle of radius r. The number of regions in these constructions is A093005(n) and the number of edges is A183207(n). See the attached images. - _Scott R. Shannon_, Jul 06 2024.

%H Scott R. Shannon, <a href="/A370980/a370980.jpg">Image for n = 3</a>. In this and other images the center of each circle of shown as a white dot.

%H Scott R. Shannon, <a href="/A370980/a370980_1.jpg">Image for n = 4</a>.

%H Scott R. Shannon, <a href="/A370980/a370980_2.jpg">Image for n = 5</a>

%H Scott R. Shannon, <a href="/A370980/a370980_3.jpg">Image for n = 10</a>.

%H Scott R. Shannon, <a href="/A370980/a370980_4.jpg">Image for n = 20</a>.

%F a(n) = A183207(n) - A093005(n) + 1, by Euler's formula. - _Scott R. Shannon_, Jul 07 2024

%e a(n) = 1+n*floor((n-1)/2) = 1+n*A004526(n-1). - _Chai Wah Wu_, Mar 23 2024

%o (Python)

%o def A370980(n): return n*(n-1>>1)+1 # _Chai Wah Wu_, Mar 23 2024

%Y Bisections are A001844 and A084849.

%Y Cf. A371373, A371374, A371375.

%Y Cf. A004526, A371253, A371254, A371253, A183297, A093005, A183207.

%K nonn

%O 0,4

%A _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 23 2024