A280926 Least k such that the first n digits of the decimal expansion of the ratio of the perimeter of a regular k-gon to its diameter match those of Pi.

5, 7, 29, 47, 119, 699, 1407, 4911, 18971, 46803, 119951, 363209, 1276197, 3722389, 19973297, 73605289, 183273481, 390720475, 1671075265, 4541314567, 22107473795, 44810965685, 172567099183, 617945607281, 1835952288687, 3938674815741, 19847928172101

Offset

1,1

Comments

By definition, the diameter of a regular k-gon is the length of its longest diagonal.

All terms are odd; see Formula section. - Jon E. Schoenfield, Mar 29 2021

Links

Table of n, a(n) for n=1..27.

Formula

a(n) = 1 + 2*floor((1/2)*(1 + sqrt((Pi^3/24)/(Pi-floor(Pi*10^(n-1))/10^(n-1))))). - Jon E. Schoenfield, Mar 28 2021

Example

An equilateral triangle (k=3) has no diagonals, and a square (k=4) has perimeter/diameter = sqrt(8) = 2.828427..., but a regular pentagon (k=5) has perimeter/diameter = (5/2)*(sqrt(5) - 1) = 3.090169..., whose first digit (3) matches that of Pi = 3.141592..., so a(1)=5. - Jon E. Schoenfield, Mar 31 2021
This ratio for a regular 7-gon (heptagon) is 3.115293... (A280533), where 3.1 equals the first two digits of Pi's decimal expansion. Because the first two digits are not 3.1 for k < 7, a(2) = 7.

Crossrefs

Cf. A000796, A280533.

Sequence in context: A018776 A104683 A153121 * A070153 A293943 A171619

Adjacent sequences: A280923 A280924 A280925 * A280927 A280928 A280929

Keyword

nonn,base,more

Author

Rick L. Shepherd, Jan 10 2017

Extensions

a(13)-a(27) from Jon E. Schoenfield, Mar 28 2021

Status

approved