Define Phi = (1+sqrt(5))/2. The unique Stolarsky representation uses the first column of the Stolarsky array 
A(n) = floor[n*(1+Phi)-Phi/2] and its compliment B(n)=round(n*Phi) given in A007064 and A007067.

These two sequences are used to create the Stolarsky representation of a positive integer z. z appears in
exactly one of the complementary sequences above. 

The rules are as follows:
If z = 1, the representation is the empty sequence of digits.
Otherwise, if z = A(n), the representation is '0' preceded by the representation of n.
Otherwise, if z = B(n), the representation is '1' preceded by the representation of n.

Stolarsky representations S(n) of n = 1 ... 150 (0 for A, 1 for B):


n	S(n)		n	S(n)		n	S(n)

1	0		51	111100		101	1001110
2	1		52	1111011		102	10011111
3	11		53	1111101		103	1011010
4	10		54	1111110		104	1011001
5	111		55	11111111	105	10110111
6	101		56	10000		106	1011100
7	110		57	101001		107	10111011
8	1111		58	100011		108	10111101
9	100		59	101010		109	10111110
10	1011		60	1010111		110	101111111
11	1101		61	100101		111	110000
12	1110		62	100110		112	1101001
13	11111		63	1001111		113	1100011
14	1010		64	101100		114	1101010
15	1001		65	1011011		115	11010111
16	10111		66	1011101		116	1100101
17	1100		67	1011110		117	1100110
18	11011		68	10111111	118	11001111
19	11101		69	110100		119	1101100
20	11110		70	110001		120	11011011
21	111111		71	1101011		121	11011101
22	1000		72	110010		122	11011110
23	10101		73	1100111		123	110111111
24	10011		74	1101101		124	1110100
25	10110		75	1101110		125	1110001
26	101111		76	11011111	126	11101011
27	11010		77	111000		127	1110010
28	11001		78	1110101		128	11100111
29	110111		79	1110011		129	11101101
30	11100		80	1110110		130	11101110
31	111011		81	11101111	131	111011111
32	111101		82	1111010		132	1111000
33	111110		83	1111001		133	11110101
34	1111111		84	11110111	134	11110011
35	10100		85	1111100		135	11110110
36	10001		86	11111011	136	111101111
37	101011		87	11111101	137	11111010
38	10010		88	11111110	138	11111001
39	100111		89	111111111	139	111110111
40	101101		90	101000		140	11111100
41	101110		91	100001		141	111111011
42	1011111		92	1010011		142	111111101
43	11000		93	100010		143	111111110
44	110101		94	1000111		144	1111111111
45	110011		95	1010101		145	100000
46	110110		96	1010110		146	1010001
47	1101111		97	10101111	147	1000011
48	111010		98	100100		148	1010010
49	111001		99	1001011		149	10100111
50	1110111		100	1001101		150	1000101

 

The lengths of this binary Stolarsky code for n = 1 ... 150 are (this sequence A200648):

1, 1, 2, 2, 3, 3, 3, 4, 3, 4, 4, 4, 5, 4, 4, 5, 4, 5, 5, 5, 6, 4, 5, 5, 5, 6, 5, 5, 6, 5, 6, 6, 6, 7, 5, 5, 6, 5, 6, 6, 6, 7, 5, 6, 6, 6, 7, 6, 6, 7, 6, 7, 7, 7, 8, 5, 6, 6, 6, 7, 6, 6, 7, 6, 7, 7, 7, 8, 6, 6, 7, 6, 7, 7, 7, 8, 6, 7, 7, 7, 8, 7, 7, 8, 7, 8, 8, 8, 9, 6, 6, 7, 6, 7, 7, 7, 8, 6, 7, 7


The number of 1's in the Stolarsky representation (the number of applications of the B-sequence 
needed) is, for n = 1 ... 150  (see A200649):

0, 1, 2, 1, 3, 2, 2, 4, 1, 3, 3, 3, 5, 2, 2, 4, 2, 4, 4, 4, 6, 1, 3, 3, 3, 5, 3, 3, 5, 3, 5, 5, 5, 7, 2, 2, 4, 2, 4, 4, 4, 6, 2, 4, 4, 4, 6, 4, 4, 6, 4, 6, 6, 6, 8, 1, 3, 3, 3, 5, 3, 3, 5, 3, 5, 5, 5, 7, 3, 3, 5, 3, 5, 5, 5, 7, 3, 5, 5, 5, 7, 5, 5, 7, 5, 7, 7, 7, 9, 2, 2, 4, 2, 4, 4, 4, 6, 2, 4, 4


The number of  0's in the Stolarskyrepresentation (the number of applications of the A-sequence 
needed) is, for n = 1 ... 150  (see A200650):

1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 1, 0, 2, 2, 1, 2, 1, 1, 1, 0, 3, 2, 2, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 1, 0, 2, 2, 1, 2, 1, 1, 1, 0, 3, 2, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1, 0, 3, 3, 2, 3, 2, 2, 2, 1, 3, 2, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1, 0, 4, 3, 3, 3, 2, 3, 3, 2, 3, 2, 2, 2, 1, 3, 3, 2, 3, 2, 2, 2, 1


The number of equal bit runs (e.g. '000', '1' or '111111') in the Stolarskyrepresentation (the number of applications of the A-sequence 
needed) is, for n = 1 ... 150  (see A200651):

1, 1, 1, 2, 1, 3, 2, 1, 2, 3, 3, 2, 1, 4, 3, 3, 2, 3, 3, 2, 1, 2, 5, 3, 4, 3, 4, 3, 3, 2, 3, 3, 2, 1, 4, 3, 5, 4, 3, 5, 4, 3, 2, 5, 3, 4, 3, 4, 3, 3, 2, 3, 3, 2, 1, 2, 5, 3, 6, 5, 5, 4, 3, 4, 5, 5, 4, 3, 4, 3, 5, 4, 3, 5, 4, 3, 2, 5, 3, 4, 3, 4, 3, 3, 2, 3, 3, 2, 1, 4, 3, 5, 4, 3, 7, 6, 5, 4, 5, 5


See also Wythoff representations http://oeis.org/A007895.