|
|
A039724
|
|
a(n) is the negabinary expansion of n, that is, the expansion of n in base -2.
|
|
47
|
|
|
0, 1, 110, 111, 100, 101, 11010, 11011, 11000, 11001, 11110, 11111, 11100, 11101, 10010, 10011, 10000, 10001, 10110, 10111, 10100, 10101, 1101010, 1101011, 1101000, 1101001, 1101110, 1101111, 1101100, 1101101, 1100010, 1100011, 1100000, 1100001, 1100110, 1100111, 1100100
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
The numbers written in base -2.
a(A007583(n)) are the only terms with all 1s digits; the number of digits = 2n + 1. - Bob Selcoe, Aug 21 2016
|
|
REFERENCES
|
M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 101.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 189.
|
|
LINKS
|
Roberto Avanzi, Gerhard Frey, Tanja Lange, and Roger Oyono, On using expansions to the base of -2, International Journal of Computer Mathematics, 81:4 (2004), pp. 403-406. arXiv:math/0312060 [math.NT], 2003.
|
|
FORMULA
|
G.f. g(x) satisfies g(x) = (x + 10*x^2 + 11*x^3)/(1 - x^4) + 100(1 + x + x^2 + x^3)*g(x^4)/x^2. - Robert Israel, Feb 24 2016
|
|
EXAMPLE
|
2 = 4 + (-2) + 0 = 110_(-2), 3 = 4 + (-2) + 1 = 111_(-2), ..., 6 = 16 + (-8) + 0 + (-2) + 0 = 11010_(-2).
|
|
MAPLE
|
f:= proc(n) option remember; 10*floor((n mod 4)/2) + (n mod 2) + 100*procname(round(n/4)) end proc:
f(0):= 0:
|
|
MATHEMATICA
|
ToNegaBases[ i_Integer, b_Integer ] := FromDigits[ Rest[ Reverse[ Mod[ NestWhileList[ (#1 - Mod[ #1, b ])/-b &, i, #1 != 0 & ], b ] ] ] ]; Table[ ToNegaBases[ n, 2 ], {n, 0, 31} ]
|
|
PROG
|
(Haskell)
a039724 0 = 0
a039724 n = a039724 n' * 10 + m where
(n', m) = if r < 0 then (q + 1, r + 2) else (q, r)
where (q, r) = quotRem n (negate 2)
(Python)
s, q = '', n
while q >= 2 or q < 0:
q, r = divmod(q, -2)
if r < 0:
q += 1
r += 2
s += str(r)
|
|
CROSSREFS
|
Cf. A212529 (negative numbers in base -2).
|
|
KEYWORD
|
base,nice,nonn,easy
|
|
AUTHOR
|
Robert Lozyniak (11(AT)onna.com)
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|