A087468 -id:A087468 - cp's OEIS frontend

A087469 a(n) = number of the row (counting from initial row 0) of the array R in A087468 that contains n.

0, 1, 0, 2, 1, 3, 0, 2, 4, 1, 3, 0, 5, 2, 4, 1, 6, 3, 0, 5, 2, 7, 4, 1, 6, 3, 0, 8, 5, 2, 7, 4, 1, 9, 6, 3, 0, 8, 5, 2, 10, 7, 4, 1, 9, 6, 3, 0, 11, 8, 5, 2, 10, 7, 4, 1, 12, 9, 6, 3, 0, 11, 8, 5, 2, 13, 10, 7, 4, 1, 12, 9, 6, 3, 0, 14, 11, 8, 5, 2, 13, 10, 7, 4, 1, 15, 12, 9, 6, 3, 0, 14, 11, 8, 5, 2, 16

Offset: 1

Clark Kimberling, Sep 09 2003

nonn

A fractal sequence.

			Northwest corner of R:
.1 3 7 12 19
.2 5 10 16 24
.4 8 14 21 30
.6 11 18 26 36
.9 15 23 32 43
a(11)=3 because 11 is in row 3.

Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Cf. A087468, A087470.

A087470 a(n) = number of the row (counting from initial row 1) of the array R in A087468 that contains n.

Original entry on oeis.org

1, 2, 1, 3, 2, 4, 1, 3, 5, 2, 4, 1, 6, 3, 5, 2, 7, 4, 1, 6, 3, 8, 5, 2, 7, 4, 1, 9, 6, 3, 8, 5, 2, 10, 7, 4, 1, 9, 6, 3, 11, 8, 5, 2, 10, 7, 4, 1, 12, 9, 6, 3, 11, 8, 5, 2, 13, 10, 7, 4, 1, 12, 9, 6, 3, 14, 11, 8, 5, 2, 13, 10, 7, 4, 1, 15, 12, 9, 6, 3, 14, 11, 8, 5, 2, 16, 13, 10, 7, 4, 1, 15, 12, 9, 6

Offset: 1

Clark Kimberling, Sep 09 2003

nonn

A fractal sequence.

			Northwest corner of R:
1 3 7 12 19
2 5 10 16 24
4 8 14 21 30
6 11 18 26 36
9 15 23 32 43
a(11)=4 because 11 is in row 4.

Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Cf. A087468, A087469.

A087465 Rank array R of 3/2 read by antidiagonals; this array is the dispersion of the complement of the sequence given by r(n) = r(n-1) + 1 + floor(3n/2) for n>=1, with r(0) = 1; that is, A077043(n+1).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 8, 10, 12, 9, 11, 14, 16, 19, 13, 15, 18, 21, 24, 27, 17, 20, 23, 26, 30, 33, 37, 22, 25, 29, 32, 36, 40, 44, 48, 28, 31, 35, 39, 43, 47, 52, 56, 61, 34, 38, 42, 46, 51, 55, 60, 65, 70, 75, 41, 45, 50, 54, 59, 64, 69, 74, 80, 85, 91, 49, 53, 58, 63, 68, 73

Offset: 0

Clark Kimberling, Sep 09 2003

The sequence is a permutation of the positive integers and the array is a transposable dispersion.
Let T(n,k) be the rectangular version of the array at A036561, with northwest corner as shown here:
   1     2     4     8    16    32
   3     6    12    24    48    96
   9    18    36    72   144   288
  27    54   108   216   432   864
Then R(n,k) is the rank of T(n,k) when all the numbers in {T(n,k)} are jointly ranked. - Clark Kimberling, Jan 25 2018

			Northwest corner of R:
   1    2    4    6    9   13   17   22
   3    5    8   11   15   20   25   31
   7   10   14   18   23   29   35   42
  12   16   21   26   32   39   46   54
  19   24   30   36   43   51   59   68
  27   33   40   47   55   64   73   83
  37   44   52   60   69   79   89  100
Let t=3/2; then R(i,j) = rank of (j,i) when all nonnegative integer pairs (a,b) are ranked by the relation << defined as follows: (a,b) << (c,d) if a + b*t < c + d*t, and also (a,b) << (c,d) if a + b*t = c + d*t and b < d.  Thus R(2,1) = 10 is the rank of (1,2) in the list (0,0) << (1,0) << (0,1) << (2,0) << (1,1) << (3,0) << (0,2) << (2,1) << (4,0) << (1,2).

Clark Kimberling, Antidiagonals n = 1..60, flattened
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Cf. A087466, A087468, A087483, A007780 (row 1), A077043 (column 1).

r = 20; r1 = 12;(*r=# rows of T,r1=# rows to show*);
c = 20; c1 = 12;(*c=# cols of T,c1=# cols to show*);
s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[3 n/2]; u = Table[s[n], {n, 0, 100}]
v = Complement[Range[Max[u]], u]; f[n_] := v[[n]]; Table[f[n], {n, 1, 30}]
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,   Length[Union[list]]]; rows = {NestList[f, 1, c]}; Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}]; w[i_, j_] := rows[[i, j]];
TableForm[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]]   (* A087465 array *)
Flatten[Table[w[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A087465 sequence *)
TableForm[Table[w[i, 1] + w[1, j] + (i - 1)*(j - 1) - 1, {i, 1, 10}, {j, 1, 10}]] (* A087465 array, by formula *)

R(i,j) = R(i,0) + R(0,j) + i*j - 1, for i>=1, j>=1.

Updated by Clark Kimberling, Sep 23 2014

A087483 Row 0 of the order array of 3/2, i.e., row 0 of the transposable dispersion in A087465.

Original entry on oeis.org

1, 2, 4, 6, 9, 13, 17, 22, 28, 34, 41, 49, 57, 66, 76, 86, 97, 109, 121, 134, 148, 162, 177, 193, 209, 226, 244, 262, 281, 301, 321, 342, 364, 386, 409, 433, 457, 482, 508, 534, 561, 589, 617, 646, 676, 706, 737, 769, 801, 834, 868, 902, 937, 973, 1009, 1046, 1084

Offset: 0

Clark Kimberling, Sep 09 2003

Also, column 0 of the transposable dispersion in A087468.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs. 7 (2004), article 04.1.6.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A053799, A087465, A087468.

A087483 := proc(n)
    1+floor((n+1)^2/3) ;
end proc:
seq(A087483(n),n=0..10) ; # R. J. Mathar, Aug 10 2017

LinearRecurrence[{2, -1, 1, -2, 1}, {1, 2, 4, 6, 9}, 100] (* Jean-François Alcover, Mar 29 2020 *)

a(n) = n + 1 - floor(n/3) + Sum_{i=1..n} floor(2i/3).
a(n) = 1 + floor((n+1)^2/3) = 1 + A000212(n+1).
a(n) = A192735(n+2) / (n+2). - Reinhard Zumkeller, Jul 08 2011
G.f.: -(x^4-x^3+x^2+1) / ((x-1)^3*(x^2+x+1)). - Colin Barker, Mar 31 2013

Edited by Max Alekseyev, Dec 05 2013

cp's OEIS Frontend

A087469 a(n) = number of the row (counting from initial row 0) of the array R in A087468 that contains n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A087470 a(n) = number of the row (counting from initial row 1) of the array R in A087468 that contains n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A087465 Rank array R of 3/2 read by antidiagonals; this array is the dispersion of the complement of the sequence given by r(n) = r(n-1) + 1 + floor(3n/2) for n>=1, with r(0) = 1; that is, A077043(n+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A087483 Row 0 of the order array of 3/2, i.e., row 0 of the transposable dispersion in A087465.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions