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A336661
Numbers that have decimal expansion c(1)c(2)...c(n) with distinct digits that satisfy c(1) <> 0, c(1) is the largest digit, and for each i in 1..n there is j in 0..2 such that c(i) == 3*c(i-1) + j (mod 10) (with c(0): = c(n)).
3
0, 4, 5, 9, 31, 72, 86, 301, 431, 602, 715, 842, 856, 973, 986, 4301, 6015, 7142, 7302, 7315, 8426, 8572, 8602, 9713, 9726, 9843, 9856, 60142, 71302, 73015, 73142, 84302, 85602, 85726, 97143, 97156, 97286, 98426, 98573, 714302, 715602, 726015, 730142, 843026, 843156, 857142, 857302, 860142, 971426, 972843, 972856, 973026, 973156, 984273, 985713, 985726, 986013
OFFSET
1,2
COMMENTS
This is one of Schuh's examples of a puzzle tree.
Putting the number on a circle and going clockwise, we observe that a 0 is followed by a 1 or 2; a 1 is followed by a 3, 4, or 5; a 2 is followed by a 6, 7, or 8; a 3 is followed by a 0, 1, or 9; a 4 is followed by a 2 or 3; a 5 is followed by a 6 or 7; a 6 is followed by a 0, 8, or 9; a 7 is followed by a 1, 2, or 3; an 8 is followed by a 4, 5, or 6; and a 9 is followed by a 7 or 8. (These observations assume the number has at least two digits.)
Schuh (pp. 25-31) uses the solution to this problem to solve the "trebles puzzle": find all numbers (with no initial 0) that are written with the same digits as their treble (the treble of k is 3*k). These numbers are listed in A023087.
The number 0 has been included here for two reasons: (i) we may assume that it satisfies the conditions of the problem vacuously, and (ii) its inclusion allows Schuh to solve the "treble puzzle". The numbers in A023087 are all permutations of combinations of numbers in this sequence.
REFERENCES
Fred Schuh, The Master Book of Mathematical Recreations, Dover, New York, 1968, pp. 8-10.
LINKS
David A. Corneth, PARI program. [It generates all 141 terms of this finite sequence.]
Wikipedia, Frederik Schuh.
Dutch Wikipedia, Frederik Schuh. [Has more extensive biography in Dutch.]
EXAMPLE
In all the cases below, the first digit must be the largest and all the digits must be distinct.
4 belongs to this list because c(1) = 4 = c(0) and 4 == 3*4 + 2 (mod 10).
31 belongs to this list because c(1) = 3, c(2) = 1 = c(0), 3 == 3*1 (mod 10), and 1 == 3*3 + 2 (mod 10).
301 belongs to this list because 3 == 3*1 (mod 10), 0 == 3*3 + 1 (mod 10), and 1 == 3*0 + 1 (mod 10).
4301 belongs to this list because 4 == 3*1 + 1 (mod 10), 3 == 3*4 + 1 (mod 10), 0 == 3*3 + 1 (mod 10), and 1 == 3*0 + 1 (mod 10).
60142 belongs to this list because 6 == 3*2 (mod 10), 0 == 3*6 + 2 (mod 10), 1 == 3*0 + 1 (mod 10), 4 = 3*1 + 1 (mod 10), and 2 = 3*4 (mod 10).
PROG
(PARI) See Corneth link
CROSSREFS
Cf. A023087.
Sequence in context: A163868 A019148 A360699 * A352509 A222543 A042221
KEYWORD
nonn,base,fini
AUTHOR
Petros Hadjicostas, Jul 28 2020
STATUS
approved