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%I #15 May 10 2024 12:29:23
%S 1,1,1,1,1,1,1,1,1,3,1,1,1,3,3,1,1,1,3,3,1,1,1,1,3,3,3,3,1,1,1,1,3,3,
%T 1,3,1,1,1,1,3,1,1,3,3,1,1,1,1,1,1,1,5,3,3,1,1,1,1,1,1,1,3,1,3,1,1,1,
%U 1,1,1,1,1,1,1,3,1,3,1,1,1,1,1,1,1,1,1,3,1,5,3,1,1,1,1,1,1,1,1,1,3,1,3,5,3
%N Array read by upward antidiagonals: A(n, k) = A265745(A372282(n, k)), n,k >= 1, where A265745(n) is the sum of digits in "Jacobsthal greedy base".
%C Collatz conjecture is equal to the claim that each column will eventually settle to constant 1's, somewhere under its topmost row. This works as only the bisection A002450 of Jacobsthal numbers (A001045) contains numbers of the form 4k+1, while the other bisection contains only numbers of the form 4k+3, which do not occur among the range of A372351. See also the comments in A371094.
%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%e Array begins:
%e n\k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
%e ---+----------------------------------------------------------------------------
%e 1 | 1, 1, 1, 3, 3, 1, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 1,
%e 2 | 1, 1, 1, 3, 3, 3, 1, 3, 3, 3, 1, 5, 5, 5, 3, 5, 3, 3, 3, 5, 5, 3,
%e 3 | 1, 1, 1, 3, 3, 3, 1, 5, 1, 3, 1, 3, 3, 5, 3, 5, 5, 1, 3, 3, 5, 3,
%e 4 | 1, 1, 1, 3, 3, 1, 1, 3, 1, 3, 1, 1, 3, 5, 3, 3, 3, 1, 3, 5, 5, 3,
%e 5 | 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 5, 3, 1, 3, 3, 3, 3,
%e 6 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 5, 3, 1, 1, 5, 5, 3,
%e 7 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 3, 3, 1, 1, 3, 5, 3,
%e 8 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 3, 3, 3,
%e 9 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 3, 5, 1,
%e 10 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 3, 5, 1,
%e 11 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 2155, 1, 1, 1, 1, 5, 1,
%e 12 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 1, 6251, 1,
%e 13 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10347, 1, 5, 1, 1, 1, 1, 5, 1,
%e 14 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 1, 5, 1,
%e 15 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 7, 1, 1, 1, 1, 5, 1,
%e 16 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 1, 7, 1,
%o (PARI)
%o up_to = 105;
%o A130249(n) = (#binary(3*n+1)-1);
%o A001045(n) = (2^n - (-1)^n) / 3;
%o A265745(n) = { my(s=0); while(n,s++; n -= A001045(A130249(n))); (s); };
%o A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
%o A372282sq(n,k) = if(1==n,2*k-1,A371094(A372282sq(n-1,k)));
%o A372288sq(n,k) = A265745(A372282sq(n,k));
%o A372288list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A372288sq((a-(col-1)),col))); (v); };
%o v372288 = A372288list(up_to);
%o A372288(n) = v372288[n];
%Y Cf. A001045, A002450, A265745, A265747, A371094, A372351, A372282, A372283, A372287.
%Y Cf. also array A372561 (formed by columns whose indices in this array are given by A372443).
%K nonn,tabl
%O 1,10
%A _Antti Karttunen_, Apr 28 2024
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