A262353 a(n) = ceiling((3-sqrt(5))*10^(2*n+1)).

8, 764, 76394, 7639321, 763932023, 76393202251, 7639320225003, 763932022500211, 76393202250021031, 7639320225002103036, 763932022500210303591, 76393202250021030359083, 7639320225002103035908264, 763932022500210303590826332, 76393202250021030359082633127

Offset

0,1

Comments

a(n) is a special family of 2nd-order base-10 grafting integers, because every integer generated by ceiling((3-sqrt(5))*10^(2*n+1)) is a grafting integer.

A grafting number is a number whose digits, represented in base b, appear before or directly after the decimal point of its r-th root. Numbers of the simplest type deal with square roots in the decimal system.

The constant x = 3-sqrt(5) is a solution of the general grafting equation (x*b^a)^(1/r) = x + c with corresponding values r = 2, b = 10, a = 1, c = 2 (where r >= 2 is the grafting root, b >= 2 is the base in which the numbers are represented, a >= 0 is the number of places the decimal point is shifted, and c >= 0 is the constant added to the front of the result).

References

Matt Parker, Things to make and do in the Fourth Dimension, New York (Ferrar, Strauss and Giroux), 2014, p. 62-63.

Links

Table of n, a(n) for n=0..14.

Formula

a(n) = ceiling((3-sqrt(5))*10^(2*n+1)).

Example

sqrt(8) = 2.828427...,
sqrt(764) = 27.6405...,
sqrt(76394) = 276.39464...

Maple

Digits:=2000: a:=n->ceil((3-sqrt(5))*10^(2*n+1)); seq(a(n), n=0..14);

Mathematica

Table[Ceiling[(3 - Sqrt@ 5) 10^(2 n + 1)], {n, 14}] (* Michael De Vlieger, Mar 24 2016 *)

Prog

(PARI) a(n) = ceil((3-sqrt(5))*10^(2*n+1)); \\ Altug Alkan, Mar 24 2016
(PARI) a(n) = 30*100^n - sqrtint(10^(4*n+2)*5) \\ Charles R Greathouse IV, Jan 20 2017
(Magma) [Ceiling((3-Sqrt(5))*10^(2*n+1)):n in [0..14]]; // Marius A. Burtea, Aug 08 2019

Crossrefs

Subsequence of A232087.

Cf. A187799.

Sequence in context: A181153 A184974 A060183 * A268148 A145415 A371595

Adjacent sequences: A262350 A262351 A262352 * A262354 A262355 A262356

Keyword

nonn,base

Author

Martin Renner, Mar 24 2016

Extensions

a(0) = 8 prepended by Robert Tanniru, Aug 06 2019

Status

approved