

A336670


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, 1} such that c(i) == 2*c(i1) + j (mod 10) (with c(0): = c(n)).


2



0, 9, 63, 512, 874, 5012, 7513, 8624, 9874, 62513, 75013, 86374, 98624, 625013, 875124, 986374, 8750124, 9875124, 86251374, 86375124, 87513624, 98750124, 862501374, 863750124, 875013624, 986251374, 986375124, 987513624, 9862501374, 9863750124, 9875013624
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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; a 1 is followed by a 2 or 3; a 2 is followed by a 4 or 5; a 3 is followed by a 6 or 7; a 4 is followed by an 8 or 9; a 5 is followed by a 0 or 1; a 6 is followed by a 2 or 3; a 7 is followed by a 4 or 5; an 8 is followed by a 6 or 7; and a 9 is followed by an 8. (These observations assume the number has at least two digits.)
Schuh (pp. 3135) uses the solution to this problem to solve the "doubles puzzle": find all numbers (with no initial 0) that are written with the same digits as their double (the double of k is 2*k). These numbers are listed in A023086.
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 "doubles puzzle". The numbers in A023086 are all permutations of combinations of numbers in this sequence.


REFERENCES

Fred Schuh, The Master Book of Mathematical Recreations, Dover, New York, 1968, pp. 3135.


LINKS

Table of n, a(n) for n=1..31.
David A. Corneth and Petros Hadjicostas, PARI program.


EXAMPLE

In all the cases below, the first digit must be the largest and all the digits must be distinct.
9 belongs to this list because c(1) = 9 = c(0) and 9 == 2*9 + 1 (mod 10).
63 belongs to this list because c(1) = 6, c(2) = 3 = c(0), 6 == 2*3 (mod 10), and 3 == 2*6 + 1 (mod 10).
512 belongs to this list because 5 == 2*2 + 1 (mod 10), 1 == 2*5 + 1 (mod 10), and 2 == 2*1 (mod 10).
5012 belongs to this list because 5 == 2*2 + 1 (mod 10), 0 == 2*5 (mod 10), 1 == 2*0 + 1 (mod 10), and 2 == 2*1 (mod 10).
62513 belongs to this list because 6 == 2*3 (mod 10), 2 == 2*6 (mod 10), 5 == 2*2 + 1 (mod 10), 1 = 2*5 + 1 (mod 10), and 3 = 2*1 + 1 (mod 10).


PROG

(PARI) See the Corneth and Hadjicostas link. \\ David A. Corneth, Jul 30 2020


CROSSREFS

Cf. A023086, A336661.
Sequence in context: A037508 A037691 A206816 * A065025 A165510 A165749
Adjacent sequences: A336667 A336668 A336669 * A336671 A336672 A336673


KEYWORD

nonn,base,fini,full


AUTHOR

Petros Hadjicostas, Jul 29 2020


STATUS

approved



