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 A335502 #13 Jul 11 2020 08:53:36 %S A335502 0,1,1,4,1,1,2,1,1,0,4,1,1,3,1,12,2,1,6,1,4,78,1,1,6,1,3,6,3,1,1,0,1, %T A335502 0,0,0,6,1,1,18,1,4,36,4,1,36,4,1,4,1,8,4,72,1,540,100,1,1,16,1,4,17, %U A335502 0,1,8,4,90,2,1,12,1,4,14,6,1,4,4,240 %N A335502 Triangle read by rows, 0 <= k < n, n >= 1: T(n,k) is the eventual period of the sequence x(j) (or 0 if x(j) never enters a cycle) defined as follows: x(0) = 1 and for j > 1 x(j) is obtained from 2*x(j-1) by deleting all occurrences of the digit k in base n. %C A335502 T(1,0) = 0 is defined in order to make the triangle of numbers regular. %C A335502 One way of getting T(n,k) = 0 is to have x(j) = x(i)*n^e for some j > i and e > 0. For k < n <= 48, this is the only way to get T(n,k) = 0 (but see A335506 for another situation where the x-sequence is not periodic). %C A335502 T(n,k) = 1 whenever k is a power of 2 and k > 1. %C A335502 It seems that k = 0 and k = n-1 often lead to particularly long cycles. %H A335502 Pontus von Brömssen, <a href="/A335502/b335502.txt">Rows n = 1..48, flattened</a> %e A335502 Triangle begins: %e A335502 n\k 0 1 2 3 4 5 6 7 8 9 10 11 %e A335502 --------------------------------------------------- %e A335502 1: 0 %e A335502 2: 1 1 %e A335502 3: 4 1 1 %e A335502 4: 2 1 1 0 %e A335502 5: 4 1 1 3 1 %e A335502 6: 12 2 1 6 1 4 %e A335502 7: 78 1 1 6 1 3 6 %e A335502 8: 3 1 1 0 1 0 0 0 %e A335502 9: 6 1 1 18 1 4 36 4 1 %e A335502 10: 36 4 1 4 1 8 4 72 1 540 %e A335502 11: 100 1 1 16 1 4 17 0 1 8 4 %e A335502 12: 90 2 1 12 1 4 14 6 1 4 4 240 %e A335502 For n = 10 and k = 5, the x-sequence starts 1, 2, 4, 8, 16, 32, 64, 128, 26, 2, and then repeats with a period of 8, so T(10,5) = 8. %e A335502 T(10,0) = 36, because A242350 eventually enters a cycle of length 36. %e A335502 For n=11 and k=7, the x-sequence starts (in base 11) 1, 2, 4, 8, 15, 2A, 59, 10. Disregarding trailing zeros, the sequence then repeats with period 7 and x(i+7*j) = x(i)*11^j for positive i and j. The x-sequence itself is therefore not eventually periodic, so T(11,7)=0. %o A335502 (Python) %o A335502 from sympy.ntheory.factor_ import digits %o A335502 from functools import reduce %o A335502 def drop(x,n,k): %o A335502 # Drop all digits k from x in base n. %o A335502 return reduce(lambda x,j:n*x+j if j!=k else x,digits(x, n)[1:],0) %o A335502 def cycle_length(n,k,m): %o A335502 # Brent's algorithm for finding cycle length. %o A335502 # Note: The function may hang if the sequence never enters a cycle. %o A335502 if (m,n,k)==(5,10,7): %o A335502 return 0 # A little cheating; see A335506. %o A335502 p=1 %o A335502 length=0 %o A335502 tortoise=hare=1 %o A335502 nz=0 %o A335502 while True: %o A335502 hare=drop(m*hare,n,k) %o A335502 while hare and hare%n==0: %o A335502 hare//=n %o A335502 nz+=1 # Keep track of the number of trailing zeros. %o A335502 length+=1 %o A335502 if tortoise==hare: %o A335502 break %o A335502 if p==length: %o A335502 tortoise=hare %o A335502 nz=0 %o A335502 p*=2 %o A335502 length=0 %o A335502 return length if not nz else 0 %o A335502 def A335502(n,k): %o A335502 return cycle_length(n,k,2) if n>1 else 0 %Y A335502 Cf. A242350. %Y A335502 Cf. A335503, A335504, A335505, A335506. %Y A335502 Cf. A243846, A306569, A306773. %K A335502 nonn,base,tabl %O A335502 1,4 %A A335502 _Pontus von Brömssen_, Jun 13 2020