This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A336670 #20 Jan 29 2025 19:36:45
%S A336670 0,9,63,512,874,5012,7513,8624,9874,62513,75013,86374,98624,625013,
%T A336670 875124,986374,8750124,9875124,86251374,86375124,87513624,98750124,
%U A336670 862501374,863750124,875013624,986251374,986375124,987513624,9862501374,9863750124,9875013624
%N 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(i-1) + j (mod 10) (with c(0): = c(n)).
%C A336670 This is one of Schuh's examples of a puzzle tree.
%C A336670 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.)
%C A336670 Schuh (pp. 31-35) 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.
%C A336670 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.
%D A336670 Fred Schuh, The Master Book of Mathematical Recreations, Dover, New York, 1968, pp. 31-35.
%H A336670 David A. Corneth and Petros Hadjicostas, <a href="/A336670/a336670.txt">PARI program</a>.
%e A336670 In all the cases below, the first digit must be the largest and all the digits must be distinct.
%e A336670 9 belongs to this list because c(1) = 9 = c(0) and 9 == 2*9 + 1 (mod 10).
%e A336670 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).
%e A336670 512 belongs to this list because 5 == 2*2 + 1 (mod 10), 1 == 2*5 + 1 (mod 10), and 2 == 2*1 (mod 10).
%e A336670 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).
%e A336670 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).
%o A336670 (PARI) \\ See the Corneth and Hadjicostas link. _David A. Corneth_, Jul 30 2020
%Y A336670 Cf. A023086, A336661.
%K A336670 nonn,base,fini,full
%O A336670 1,2
%A A336670 _Petros Hadjicostas_, Jul 29 2020