A065622 Numerator of 1 - (3/4)^n - frac((3/2)^n), where frac(x) = x - floor(x).

0, -1, 3, 13, 159, 173, 1767, 12789, 17759, 126237, 292183, 1930245, 3724303, 23940141, 14206087, 99585429, 640559295, 12562430525, 7042526903, 43417422885, 813747135599, 494896655693, 3000760993767, 18098709141429, 249612172740383

Offset: 0

Author

Henry Bottomley, Dec 03 2001

Keywords

• frac
• sign

Comments

The presumption that the fraction is positive for n > 1 underlies the presumed solution to Waring's problem.

Examples

			a(3) = 13 since 1 - (3/4)^3 - frac((3/2)^3) = 1 - 27/64 - frac(27/8) = 1 - 27/64 - 3/8 = (64 - 27 - 24)/64 = 13/64.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..200
• Eric Weisstein's World of Mathematics, Waring's Problem

Crossrefs

Denominator is A000302. Cf. A002804.

Programs

Mathematica

Table[1 - (3/4)^n - FractionalPart[(3/2)^n], {n, 0, 24}] // Numerator (* Jean-François Alcover, Apr 26 2016 *)

PARI

{ for (n=0, 200, a=numerator(1 - (3/4)^n - frac((3/2)^n)); write("b065622.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 24 2009

Formula

a(n) = 4^n*(1 + floor((3/2)^n)) - 3^n - 6^n = A005061(n) - A002380(n)*A000079(n) = A000302(n)*(1 + A002379(n)) - A000244(n) - A000400(n).