A084544 Alternate number system in base 4.

1, 2, 3, 4, 11, 12, 13, 14, 21, 22, 23, 24, 31, 32, 33, 34, 41, 42, 43, 44, 111, 112, 113, 114, 121, 122, 123, 124, 131, 132, 133, 134, 141, 142, 143, 144, 211, 212, 213, 214, 221, 222, 223, 224, 231, 232, 233, 234, 241, 242, 243, 244, 311, 312, 313, 314, 321

Offset

1,2

Links

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

EMIS, Mirror site for Southwest Journal of Pure and Applied Mathematics

R. R. Forslund, A logical alternative to the existing positional number system, Southwest Journal of Pure and Applied Mathematics, Vol. 1 1995 pp. 27-29.

R. R. Forslund, Positive Integer Pages [Broken link]

James E. Foster, A Number System without a Zero-Symbol, Mathematics Magazine, Vol. 21, No. 1. (1947), pp. 39-41.

Index entries for 10-automatic sequences.

Formula

From Hieronymus Fischer, Jun 06 and Jun 08 2012: (Start)

The formulas are designed to calculate base-10 numbers only using the digits 1..4.

a(n) = Sum_{j=0..m-1} (1 + b(j) mod 4)*10^j,

where m = floor(log_4(3*n+1)), b(j) = floor((3*n+1-4^m)/(3*4^j)).

Special values:

a(k*(4^n-1)/3) = k*(10^n-1)/9, k = 1,2,3,4.

a((7*4^n-4)/3) = (13*10^n-4)/9 = 10^n + 4*(10^n-1)/9.

a((4^n-1)/3 - 1) = 4*(10^(n-1)-1)/9, n > 1.

Inequalities:

a(n) <= (10^log_4(3*n+1)-1)/9, equality holds for n=(4^k-1)/3, k>0.

a(n) > (4/10)*(10^log_4(3*n+1)-1)/9, n > 0.

Lower and upper limits:

lim inf a(n)/10^log_4(3*n) = 2/45, for n --> infinity.

lim sup a(n)/10^log_4(3*n) = 1/9, for n --> infinity.

G.f.: g(x) = (x^(1/3)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(4/3)*(1 - 5z(j)^4 + 4z(j)^5)/((1-z(j))(1-z(j)^4)), where z(j) = x^4^j.

Also: g(x) = (1/(1-x)) Sum_{j>=0} (1-5(x^4^j)^4 + 4(x^4^j)^5)*x^4^j*f_j(x)/(1-x^4^j), where f_j(x) = 10^j*x^((4^j-1)/3)/(1-(x^4^j)^4). The f_j obey the recurrence f_0(x) = 1/(1-x^4), f_(j+1)(x) = 10x*f_j(x^4).

Also: g(x) = (1/(1-x))* (h_(4,0)(x) + h_(4,1)(x) + h_(4,2)(x) + h_(4,3)(x) - 4*h_(4,4)(x)), where h_(4,k)(x) = Sum_{j>=0} 10^j*x^((4^(j+1)-1)/3) * (x^4^j)^k/(1-(x^4^j)^4).

(End)

a(n) = A045926(n) / 2. - Reinhard Zumkeller, Jan 01 2013

Example

From Hieronymus Fischer, Jun 06 2012: (Start)
a(100)  = 1144.
a(10^3) = 33214.
a(10^4) = 2123434.
a(10^5) = 114122134.
a(10^6) = 3243414334.
a(10^7) = 211421121334.
a(10^8) = 11331131343334.
a(10^9) = 323212224213334. (End)

Prog

(Python)
def A084544(n):
    m = (3*n+1).bit_length()-1>>1
    return int(''.join((str(((3*n+1-(1<<(m<<1)))//(3<<((m-1-j)<<1))&3)+1) for j in range(m)))) # Chai Wah Wu, Feb 08 2023

Crossrefs

Cf. A007931, A007932, A052382, A084545, A046034, A089581, A084984, A001742, A001743, A001744, A202267, A202268, A014261, A014263.

Sequence in context: A108467 A265565 A265549 * A268236 A039023 A110918

Adjacent sequences: A084541 A084542 A084543 * A084545 A084546 A084547

Keyword

nonn,base

Author

Robert R. Forslund (forslund(AT)tbaytel.net), Jun 27 2003

Extensions

Offset set to 1 according to A007931, A007932 by Hieronymus Fischer, Jun 06 2012

Status

approved