Rightmost digit, leftmost digit, variants
The rightmost digit d of a(n) says: "There are exactly d even digits immediately after me". This should be the lexicographically earliest sequence S of distinct nonnegative numbers with this property.
And you know what? S was a nightmare to compute by hand! But, as we love Füssli, S was also a huge pleasure to explore!
S = 0, 1, 21, 23, 2, 20, 3, 200, 5, 4, 22, 40, 7, 6, 24, 42, 60, 9, 8, 26, 44, 62, 80, 10, 11, 25, 202, 201, 27, 204, 2000, 12, 203, 220, 13, 2001, 29, 226, 64, 82, 205, 222, 207, 224, 20001, 41, 43, 240, 14, 20003, 260, 15, 242, 209, 246, 84, 20005, 262, 221, 45, 282, 223, 280, 16, 2002, 225, 402, 227, 244, 20007, 264, 20009, 266, 2022, 229, 286, 2042, 241, 47, 284, 20021, ...
Explanation
S = 0, 1, 21, 23, 2, 20, 3, 200, 5, 4, 22, 40, 7, 6, 24, 42, 60, 9, 8, ...
(in blue above, the rightmost digit; underlined are the even digits that come after a blue digit)
a(1) = 0, ending in 0, and there is no even digit after this 0 (there is a 1)
a(2) = 1, ending in 1, and there is 1 even digit after this 1 (the 2 of 21)
a(3) = 21, ending in 1, and there is 1 even digit after this 1 (the 2 of 23)
a(4) = 23, ending in 3, and there are 3 even digits after this 3 (2, 2 and 0)
a(5) = 2, ending in 2, and there are 2 even digits after this 2 (2 and 0)
a(6) = 20, ending in 0, and there is no even digit after this 0 (there is a 3)
a(7) = 3, ending in 3, and there are 3 even digits after this 3 (2, 0 and 0)
a(8) = 200, ending in 0, and there is no even digit after this 0 (there is a 5)
a(9) = 5, ending in 5, and there are 5 even digits after this 5 (4, 2, 2, 4 and 0)
a(10) = 4, ending in 4, and there are 4 even digits after this 4 (2, 2, 4 and 0), etc.
I guess S is finite – how many terms does it have, and what is the last one?
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Variants's time – rightmost digit
(1) The rightmost digit d of a(n) says: "There are exactly d odd digits immediately after me". This should be
the lexicographically earliest sequence T of distinct nonnegative numbers with this property. T is finite – how many terms does it have, and what is the last one?
T = 0, 2, 1, 10, 4, 3, 11, 12, 110, 6, 5, 13, 31, 30, 8,
7, 15, 33, 51, 50, 20, 21, 14, 111, 16, 113, 71, 18, 115, 53, 91, 32, 112,
114, 131, 70, 22, 116, 133, 1110, 23, 1130, 24, 151, 34, 171, 36, 153, 1150,
25, 1111, 52, 118, ...
(2) The rightmost digit d of a(n) says: "There are exactly d even digits immediately before me". This should be the lexicographically earliest sequence U of distinct nonnegative numbers with this property. U is finite – how many terms does it have, and what is the last one?
U = 0, 1, 10, 22, 3, 21, 30, 42, 24, 5, 50, 62, 44, 26, 7, 70, 82, 64, 46, 28, 9, 90, 101, 110, 121, 130, 141, 150, 161, 170, 181, 190, 203, 202, 84, 66, 48, 210, 223, 222, 205, 230, 243, 242, 225, 250, 263, 262, 245, 270, 283, 282, 265, 290, ...
(3) The rightmost digit d of a(n) says: "There are exactly d odd digits immediately before me". This should be the lexicographically earliest sequence V of distinct nonnegative numbers with this property. V is finite – how many terms does it have, and what is the last one?
V = 0, 11, 2, 20, 31, 13, 4, 51, 33, 15, 6, 71, 53, 35, 17, 8, 91, 73, 55, 37, 19, 40, 60, 80, 100, 112, 120, 132, 140, 152, 160, 172, 180, 192, 200, 211, 114, 220, 231, 134, 240, 251, 154, 260, 271, 174, 280, 291, 194, 300, 312, 320, 332, 340, 352, 360, 372, 380, 392, 400, ...
(4) The rightmost digit d of a(n) says: "a(n+1) has d digits". This should be the lexicographically earliest sequence W of distinct positive terms with this property.
W is finite – how many terms does it have, and what is the last one?
W = 1, 2, 11, 3, 101, 4, 1001, 5, 10001, 6, 100001, 7, 1000001, 8, 10000001, 9, 100000002, 12, 13, 102, 14, 1002, 15, 10002, 16, 100002, 17, 1000002, 18, 10000002, 19, 100000003, 103, 104, 1003, 105, 10003, 106, 100003, 107, 1000003, 108, 10000003, 109, 100000004, 1004, 1005, ...
(5) The rightmost digit d of a(n) says: "a(n-1) has d digits". This should be the lexicographically earliest sequence X of nonnegative terms with this property.
X is finite – how many terms does it have, and what is the last one?
X = 0, 1, 11, 2, 21, 12, 22, 32, 42, 52, 62, 72, 82, 92, 102, 3, 31, 112, 13, 122, 23, 132, 33, 142, 43, 152, 53, 162, 63, 172, 73, 182, 83, 192, 93, 202, 103, 113, 123, 133, 143, 153, 163, 173, 183, 193, 203, 213, 223, 233, 243, 253, 263, 273, 283, 293, 303, 313, 323, 333, 343, 353, ...
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Variants's time – leftmost digit
(6) The leftmost digit d of a(n) says: "There are exactly d even digits immediately after me". This should be the lexicographically earliest sequence Y of distinct nonnegative numbers with this property. Y is finite – how many terms does it have, and what is the last one?
Y =
(7) The leftmost digit d of a(n) says: "There are exactly d odd digits immediately after me". This should be the lexicographically earliest sequence Z of distinct nonnegative numbers with this property. Z is finite – how many terms does it have, and what is the last one?
Z =
(8) The leftmost digit d of a(n) says: "There are exactly d even digits immediately before me". This should be the lexicographically earliest sequence A of distinct nonnegative numbers with this property. A is finite – how many terms does it have, and what is the last one?
A =
(9) The leftmost digit d of a(n) says: "There are exactly d odd digits immediately before me". This should be the lexicographically earliest sequence B of distinct nonnegative numbers with this property. B is finite – how many terms does it have, and what is the last one?
B =
(10) The leftmost digit d of a(n) says: "a(n+1) has d digits". This should be the lexicographically earliest sequence C of distinct positive terms with this property.
C is finite – how many terms does it have, and what is the last one?
C =
(11) The leftmost digit d of a(n) says: "a(n-1) has d digits". This should be the lexicographically earliest sequence D of distinct nonnegative terms with this property.
D is finite – how many terms does it have, and what is the last one?
D =
(12) The leftmost digit d of a(n) – except for a(1) – says: "There are exactly d even digits immediately to my left" and the rightmost digit f of a(n) says: "There are exactly f odd digits immediately to my right". This should be the lexicographically earliest sequence E of distinct positive terms with this property.
E = 2, 1100, 20, 4, 511100, 202, 5100, 204, 511120, 206, 51111100,
208, 5111111100, 222, 5120, 224, 511140, 226, 51111120, 228, 5111111120, 242, 5140,
244, 511160, 246, 51111140, 248, 5111111140, 262, 5160, 264, 511180, 266, 51111160,
268, 5111111160, 282, 5180, 284, 511300, 286, 51111180, 288, 5111111180, 2000, 6,
71111100, 2020, 60, 8, 9111111100, 2040, 602, 9100, 2060, 604, 911100, 2080, 606,
91111100, 2100, 2120, 2140, 2160, 2180, 2300, 2320, 2340, 2360, 2380, 2500, 2520,
2540, 2560, 2580, 2700, 2720, 2740, 2760, 2780, 2900, 2920, 2940, 2960, 2980, 20002,
7100, 20004, 711100, 20006, 71111120, 20008, 7111111100, 20022, 7120, 20024, 711120,
20026, 71111140, 20028, 7111111120, 20042, 7140, 20044, 711140, 20046, 71111160,
20048, 7111111140, 20062, 7160, 20064, 711160, 20066, 71111180, 20068, 7111111160,
20082, 7180, 20084, 711180, 20086, 71111300, 20088, 7111111180, 20100, 20120, 20140,
20160, 20180, 20300, 20320, 20340, 20360, 20380, 20500, 20520, ...
Highlighted patterns hereunder – blue digits counting even digits immediately on their left, red digits counting odd digits immediately on their right:
A last idea – having nothing to do with the above:
"Lexicographically earliest sequence F of distinct positive terms such that either the concatenation [a(n); n; a(n+1)] or the concatenation [a(n+1); n; a(n)] is a prime number".
F = 1, 2, 3, 4, 9, 6, 19, 8, 7, 10, 21, 11, 17, 13, 5, 39, 16, 29, 14, 31, 20, 27, 24, 57, 23, 32, 33, 26, 53, 12, 49, 35, 37, 15, 43, 22, 67, 30, 59, 18, 73, 28, 61, 41, 51, 70, 47, 25, 69, 34, 89, 42, 79, 38, 71, 52, 77, 48, 93, 45, 91, 44, 63, 58, 99, 36, 83, 50, 87, 55...
211
223
433
449
659
6619
8719
887
1097
101021
111121
111217
131317
51413
51539
161639
161739
...
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