A272370 Number of geometrically inscriptible regular polygons with fewer than 2^n + 1 sides.

0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189, 209, 230, 252, 275, 299, 324, 350, 377, 405, 434, 464, 495, 527, 559, 591, 623, 655, 687, 719, 751, 783, 815, 847, 879, 911, 943, 975, 1007, 1039, 1071, 1103, 1135, 1167, 1199, 1231, 1263, 1295

Offset

1,2

Comments

a(n) is the number of terms of A003401, except its first two degenerate case terms, that are less than 2^n + 1.

Links

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Raymond Clare Archibald, Remarks on Klein's "Famous Problems of Elementary Geometry", The American Mathematical Monthly, Vol. 21, No. 8 (Oct., 1914), pp. 247-259.

Leonard E. Dickson, On the number of inscriptible regular polygons, Bull. Amer. Math. Soc. 3 (1894), 123-125.

Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m

Formula

a(n) = (n-1)*(n+2)/2 if n < 32, otherwise 32*n-497 if n < 2^33.

Example

For n=2, there are 2 such polygons, those with 3 and 4 sides, below 2^2+1 = 5.

Prog

(PARI) a(n) = if(n < 32, (n-1)*(n+2)/2, if(n < 2^33, 32*n-497));
(PARI) a(n) = {i = 2^n; j = 2*n - 2; k = 1; while(i > A045544(k) && k < 31, k++; j+=floor(log(i/A045544(k))/log(2))+1); j; } \\ Jinyuan Wang, Jul 29 2019

Crossrefs

Cf. A000051, A003401, A004169, A019434, A045544.

Sequence in context: A132315 A079509 A132336 * A212342 A080956 A132337

Adjacent sequences: A272367 A272368 A272369 * A272371 A272372 A272373

Keyword

nonn

Author

Michel Marcus, Apr 28 2016

Status

approved