Golden Ratio Digits

The golden ratio has decimal expansion

(OEIS A001622). It can be computed to  digits of precision in 24 CPU-minutes on modern hardware and was computed to  decimal digits by A. J. Yee on Jul. 8, 2010.

The Earls sequence (starting position of  copies of the digit ) for  is given for , 2, ... by 2, 62, 158, 1216, 72618, 2905357, 7446157, 41398949, 1574998166, ... (OEIS A224844).

The digit sequence 0123456789 does not occur in the first  digits of , but 9876543210 does, starting at position  (E. Weisstein, Jul. 22, 2013).

Phi-primes, i.e., -constant primes occur for 7, 13, 255, 280, 97241, ... (OEIS A064119) decimal digits.

The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of  (including the initial 1 and counting it as the first digit) are 5, 1, 20, 6, 12, 23, 2, 11, 4, 8, 232, ... (OEIS A088577).

Scanning the decimal expansion of  until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 5, 55, 515, 0092, 67799, 290503, ... (OEIS A000000), which end at digits 23, 770, 5819, 93910, 1154766, 13192647, ... (OEIS A000000).

It is not known if  is normal, but the following table giving the counts of digits in the first  terms shows that the decimal digits are very uniformly distributed up to at least .

	OEIS	10	100
0	A000000	1	11	100	1020	9986	99805	1001143	10003332	100007840	1000031042
1	A000000	1	9	105	1062	9963	99993	1000118	10000255	99999864	999990982
2	A000000	0	11	116	994	9950	99529	1000776	10002116	100002106	1000005392
3	A000000	2	9	88	1039	10079	99833	999794	9999184	99979352	999978183
4	A000000	0	12	92	976	10041	100151	999367	9998797	99995481	999952470
5	A000000	0	5	84	988	10016	100067	999725	9996151	99999934	1000032985
6	A000000	1	9	104	918	9975	100328	999455	9996149	100004208	1000014191
7	A000000	1	10	113	1025	9988	100160	1000852	9997524	100018237	1000023870
8	A000000	3	15	105	987	10008	100236	1000059	10005419	99995223	999976728
9	A000000	1	9	93	991	9994	99898	998711	10001073	99997755	999994157

REFERENCES:

Sloane, N. J. A. Sequences A/M4046, A064119, A088577, and A224844 in "The On-Line Encyclopedia of Integer Sequences."

Yee, A. J. "y-cruncher - A Multi-Threaded Pi-Program." http://www.numberworld.org/y-cruncher/.