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)).

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

Author

Petros Hadjicostas, Jul 28 2020

Keywords

• nonn
• base
• fini

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.

Examples

			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).
		

References

• Fred Schuh, The Master Book of Mathematical Recreations, Dover, New York, 1968, pp. 8-10.

Links

• Petros Hadjicostas, Table of n, a(n) for n = 1..141
• 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.]

Crossrefs

Cf. A023087.

Programs

PARI

See Corneth link