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.

1, 1, 2, 6, 1, 3, 58, 11, 2, 4, 5829, 1056, 16, 2, 5, 58292915, 10555528, 1608, 21, 3, 6, 5829291479146458, 1055552805277764, 16080804, 2111, 27, 3, 7, 58292914791464577914645739573229, 10555528052777640527776402638882

Offset: 1

Author

Eric Angelini, Alois P. Heinz, Farideh Firoozbakht and M. F. Hasler, Feb 04 2009

Keywords

• base
• easy
• nonn
• tabl

Comments

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.

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

Examples

			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.
Table begins:
  1, 1,  6,   58,     5829,         58292915, ...
  2, 1, 11, 1056, 10555528, 1055552805277764, ...
  3, 2, 16, 1608, 16080804, 1608080408040402, ...
  4, 2, 21, 2111, 21106056, 2110605560553028, ...
  5, 3, 27, 2664, 26636332, 2663633213318166, ...
  6, 3, 32, 3166, 31661583, 3166158315830792, ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n=1..78
• Eric Angelini, Rang dans les Pairs/Impairs
• Eric Angelini, Rang dans les Pairs/Impairs [Cached copy, with permission]
• Eric Angelini et al., Rank of n in the Odd/Even sequence and follow-up messages on the SeqFan list, Feb 03 2009

Crossrefs

Cf. A156147 (first row of the table).

Programs

Maple

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

Mathematica

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 *)

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 }
A156146=concat( vector( 12,d,vector( d,k, T(k,d-k+1)))) /* M. F. Hasler */

Extensions

Typos fixed by Charles R Greathouse IV, Oct 28 2009