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A156146 Table T(m,n) = round( c(m,n)/2 ), where c(m,n) is the concatenation of all preceding terms in row m, T(m,1)...T(m,n-1) and T(m,1)=m.

%I A156146 #44 Dec 23 2024 14:53:42
%S A156146 1,1,2,6,1,3,58,11,2,4,5829,1056,16,2,5,58292915,10555528,1608,21,3,6,
%T A156146 5829291479146458,1055552805277764,16080804,2111,27,3,7,
%U A156146 58292914791464577914645739573229,10555528052777640527776402638882
%N A156146 Table T(m,n) = round( c(m,n)/2 ), where c(m,n) is the concatenation of all preceding terms in row m, T(m,1)...T(m,n-1) and T(m,1)=m.
%C A156146 Originally, round( c/2 ) was formulated as "rank of c in the sequence of odd resp. even (positive) numbers". Each of the rows has some characteristics reminiscent of Thue-Morse type sequences.
%C A156146 It is interesting that the number of digits of T(1,k) for k>2 equals to 2^(k-3). And for i>1 & k>1 [and i<20 - _M. F. Hasler_] the number of digits of T(i,k) equals to 2^(k-2). - _Farideh Firoozbakht_
%H A156146 Alois P. Heinz, <a href="/A156146/b156146.txt">Table of n, a(n) for n=1..78</a>
%H A156146 Eric Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/ThueMorseRank.htm">Rang dans les Pairs/Impairs</a>
%H A156146 Eric Angelini, <a href="/A156146/a156146.pdf">Rang dans les Pairs/Impairs</a> [Cached copy, with permission]
%H A156146 Eric Angelini et al., <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2009-February/000740.html">Rank of n in the Odd/Even sequence</a> and follow-up messages on the SeqFan list, Feb 03 2009
%e A156146 T(2,2) = 1 since T(2,1) = 2 is the first even number. T(2,3) = 11 since concat(T(2,1),T(2,2)) = 21 is the 11th odd number.
%e A156146 Table begins:
%e A156146   1, 1,  6,   58,     5829,         58292915, ...
%e A156146   2, 1, 11, 1056, 10555528, 1055552805277764, ...
%e A156146   3, 2, 16, 1608, 16080804, 1608080408040402, ...
%e A156146   4, 2, 21, 2111, 21106056, 2110605560553028, ...
%e A156146   5, 3, 27, 2664, 26636332, 2663633213318166, ...
%e A156146   6, 3, 32, 3166, 31661583, 3166158315830792, ...
%p A156146 rank:= n-> `if`(irem(n,2)=0, n/2, (n+1)/2); a:= proc(n,k) option remember; if n=1 then k else rank(parse(cat(seq(a(j,k), j=1..n-1)))) fi end; seq(seq(a(d-k,k), k=1..d-1), d=1..10); # _Alois P. Heinz_
%t A156146 Si[1]=i;Si[n_]:=Si[n]=(v={};Do[v= Join[v,IntegerDigits[Si[k]]],{k,n-1}]; Floor[(1+FromDigits[v])/2]) (* _Farideh Firoozbakht_ *)
%o A156146 (PARI) T(m,n)={ my(t=round(m/2)); n>1 || return(m); while( n-- > 1, t=round(1/2*m=eval(Str(m,t)))); t }
%o A156146 A156146=concat( vector( 12,d,vector( d,k, T(k,d-k+1)))) /* _M. F. Hasler_ */
%Y A156146 Cf. A156147 (first row of the table).
%K A156146 base,easy,nonn,tabl
%O A156146 1,3
%A A156146 _Eric Angelini_, _Alois P. Heinz_, _Farideh Firoozbakht_ and _M. F. Hasler_, Feb 04 2009
%E A156146 Typos fixed by _Charles R Greathouse IV_, Oct 28 2009