A156147 -id:A156147 - cp's OEIS frontend

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

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

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

			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, ...

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

Cf. A156147 (first row of the table).

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

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 *)

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 */

Typos fixed by Charles R Greathouse IV, Oct 28 2009

A159861 Square array A(m,n), m>=1, n>=1, read by antidiagonals: A(m,1)=1, A(m,n) is the rank with respect to m of the concatenation of all preceding terms in row m, and the rank of S with respect to m is floor ((S+m-1)/m).

Original entry on oeis.org

1, 1, 1, 11, 1, 1, 1111, 6, 1, 1, 11111111, 58, 4, 1, 1, 1111111111111111, 5829, 38, 3, 1, 1, 11111111111111111111111111111111, 58292915, 3813, 29, 3, 1, 1, 1111111111111111111111111111111111111111111111111111111111111111, 5829291479146458, 38127938, 2833, 23, 2, 1, 1

Offset: 1

Eric Angelini and Alois P. Heinz, Apr 24 2009

			A(3,4) = 38, because A(3,1).A(3,2).A(3,3) = 114, and the rank of 114 with respect to 3 is floor(116/3) = 38.
Square array A(m,n) begins:
  1,  1, 11, 1111, 11111111, 1111111111111111,  ...
  1,  1,  6,   58,     5829,         58292915,  ...
  1,  1,  4,   38,     3813,         38127938,  ...
  1,  1,  3,   29,     2833,         28323209,  ...
  1,  1,  3,   23,     2265,         22646453,  ...
  1,  1,  2,   19,     1870,         18698645,  ...

Alois P. Heinz, Antidiagonals n = 1..12, flattened

Row m=2 gives: A156147.
Main diagonal gives: A159862.
Cf. A156146, A010783.

R:= (S,m)-> iquo(S+m-1, m):
A:= proc(m, n) option remember; `if`(n=1, 1,
      R(parse(cat(seq(A(m, j), j=1..n-1))), m))
    end:
seq(seq(A(m, d-m), m=1..d-1), d=1..10);

R[S_, m_] := Quotient[S + m - 1, m];
A[m_, n_] := If[n == 1, 1, R[ToExpression@StringJoin[ToString /@ Table[A[m, j], {j, 1, n - 1}]], m]];
Table[Table[A[m, d - m], {m, 1, d - 1}], {d, 1, 10}] // Flatten (* Jean-François Alcover, Feb 13 2023, after Maple code *)

A159862 Main diagonal of A159861.

Original entry on oeis.org

1, 1, 4, 29, 2265, 18698645, 1602308616574727, 14017675267522095175220940844027, 1245902734717669791621141496863001384336371908521990690157218737

Offset: 1

Eric Angelini and Alois P. Heinz, Apr 24 2009

The length (number of decimal digits) of a(n) may be a power of 2 and often simply doubles, when n is increased by 1. But there are many exceptions: n = 11, 12, 13 give lengths 2^8, 3*2^7, 2^9, respectively. A factor of 3 is found in the lengths of a(n) for n = 12, 112..123, 1113..1234, 11123..12345, and so on. A factor of 7 is found for n = 1112, 11112..11122, and so on. 15 is factor of the length of a(11111112).

Alois P. Heinz, Table of n, a(n) for n = 1..12

Cf. A156146, A156147, A000040, A010783.

R:= (S,m)-> iquo(S+m-1, m):
A:= proc(m, n) option remember; `if`(n=1, 1,
      R(parse(cat(seq(A(m, j), j=1..n-1))), m))
    end:
a:= n-> A(n,n):
seq(a(n), n=1..10);

R[S_, m_] := Quotient[S + m - 1, m];
A[m_, n_] := If[n == 1, 1, R[ToExpression@StringJoin[ToString /@ Table[A[m, j], {j, 1, n - 1}]], m]];
a[n_] := A[n, n];
Table[a[n], {n, 1, 10}] (* Jean-François Alcover, Feb 13 2023, after Maple code *)

cp's OEIS Frontend

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.

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A159861 Square array A(m,n), m>=1, n>=1, read by antidiagonals: A(m,1)=1, A(m,n) is the rank with respect to m of the concatenation of all preceding terms in row m, and the rank of S with respect to m is floor ((S+m-1)/m).

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A159862 Main diagonal of A159861.

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