A064628 a(n) = floor((4/3)^n).

1, 1, 1, 2, 3, 4, 5, 7, 9, 13, 17, 23, 31, 42, 56, 74, 99, 133, 177, 236, 315, 420, 560, 747, 996, 1328, 1771, 2362, 3149, 4199, 5599, 7466, 9954, 13273, 17697, 23596, 31462, 41950, 55933, 74577, 99437, 132583, 176777, 235703, 314271, 419028, 558704

Offset: 0

Author

Labos Elemer, Oct 01 2001

Keywords

• nonn
• easy

Comments

a(n) is the perimeter of a hexaflake (rounded down) after n iterations. The total number of holes = A000420(n) - 1. The total number of irregular polygon holes = A000420(n-1) - 1. The total number of triangle holes = 6*A000420(n-1). - Kival Ngaokrajang, Apr 18 2014

a(n) is composite infinitely often (Forman and Shapiro). More exactly, a(n) is divisible by at least one of 2, 3, 5 infinitely often (Dubickas and Novikas). - Tomohiro Yamada, Apr 15 2017

References

• R. K. Guy, Unsolved Problems in Number Theory, E19.

Links

• Harry J. Smith, Table of n, a(n) for n=0,...,400
• Arturas Dubickas and Aivaras Novikas, Integer parts of powers of rational numbers, Math. Z. 251 (2005), 635--648, available from the first author's page.
• W. Forman and H. N. Shapiro, An arithmetic property of certain rational powers, Comm. Pure. Appl. Math. 20 (1967), 561-573.
• Kival Ngaokrajang, Illustration of hexaflake for n = 0..3.
• Eric Weisstein's World of Mathematics, Power Floors.
• Wikipedia, n-flake.

Crossrefs

Cf. A002379, A002380, A060692.

Cf. A094969 - A094500.

Cf. A046038, A070761, A070762, A067905 (Composites and Primes).

Programs

Maple

A064628:=n->floor(4^n/3^n); seq(A064628(n), n=0..30); # Wesley Ivan Hurt, Apr 19 2014

Mathematica

Table[Floor[(4/3)^n], {n, 0, 30}] (* Robert G. Wilson v *)

PARI

{ f=t=1; for (n=0, 400, write("b064628.txt", n, " ", f\t); f*=4; t*=3 ) } \\ Harry J. Smith, Sep 20 2009

Sage

def A064628(n): return floor((4/3)^n)
[A064628(n) for n in range(0, 46)] # Stefano Spezia, Oct 13 2024

Extensions

More terms from Robert G. Wilson v, May 26 2004

OFFSET changed from 1 to 0 by Harry J. Smith, Sep 20 2009