A156146 -id:A156146 - cp's OEIS frontend

A156147 a(n+1) = round( c(n)/2 ), where c(n) is the concatenation of all preceding terms a(1)...a(n) and a(1)=1.

1, 1, 6, 58, 5829, 58292915, 5829291479146458, 58292914791464577914645739573229, 5829291479146457791464573957322929146457395732288957322869786615

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".
The sequence has some characteristics reminiscent of Thue-Morse type sequences. It "converges" to a non-periodic sequence of digits: all but the last digit of a given term will remain the initial digits of all subsequent terms. - M. F. Hasler
It's interesting that the number of digits of a(k) for k>2 equals to 2^(k-3). - Farideh Firoozbakht

M. F. Hasler et al., Table of n, a(n) for n = 1..12
E. Angelini, Rang dans les Pairs/Impairs
E. Angelini, Rang dans les Pairs/Impairs [Cached copy, with permission]
E. Angelini et al., Rank of n in the Odd/Even sequence and follow-up messages on the SeqFan list, Feb 03 2009

Cf. A156146 (other starting values).

rank:= n-> `if`(irem(n,2)=0, n/2, (n+1)/2): a:= proc(n) option remember; if n=1 then 1 else rank(parse(cat(seq(a(j), j=1..n-1)))) fi end: seq(a(n), n=1..10);  # Alois P. Heinz

a[1]=1; a[n_]:=a[n]=(v={};Do[v= Join[v,IntegerDigits[a[k]]],{k,n-1}]; Floor[(1+FromDigits[v])/2]) (* Farideh Firoozbakht *)

A156147(n)={local(a=1,t=1); while(n-->1,t=round(1/2*a=eval(Str(a,t))));t} /* 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

A156147 a(n+1) = round( c(n)/2 ), where c(n) is the concatenation of all preceding terms a(1)...a(n) and a(1)=1.

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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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