A087465 -id:A087465 - cp's OEIS frontend

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

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

A087468 Dispersion, read by antidiagonals, of the complement of row 0 of the array R in A087465.

Original entry on oeis.org

1, 3, 2, 7, 5, 4, 12, 10, 8, 6, 19, 16, 14, 11, 9, 27, 24, 21, 18, 15, 13, 37, 33, 30, 26, 23, 20, 17, 48, 44, 40, 36, 32, 29, 25, 22, 61, 56, 52, 47, 43, 39, 35, 31, 28, 75, 70, 65, 60, 55, 51, 46, 42, 38, 34, 91, 85, 80, 74, 69, 64, 59, 54, 50, 45, 41, 108, 102, 96, 90, 84, 79

Offset: 0

Clark Kimberling, Sep 09 2003

The sequence is a permutation of the natural numbers and the array is a transposable dispersion.

			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
(See example at A087465.)

Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004, Article 04.1.6, 22 pp.

Cf. A087465.

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[j, i], {i, 1, 10}, {j, 1, 10}]]   (*A087468 array*)
Flatten[Table[w[n - k + 1, k], {n, 1, c1}, {k, 1, n}]] (*A087468 sequence*)
TableForm[Table[w[j, 1] + w[1, i] + (i - 1)*(j - 1) - 1, {i, 1, 10}, {j, 1, 10}]] (*A087468 array,by formula*)

Transpose of the array R in A087465.

Updated by Clark Kimberling, Sep 23 2014

A087466 a(n) = number of the row (counting from initial row 0) of the array R in A087465 that contains n.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 2, 1, 0, 2, 1, 3, 0, 2, 1, 3, 0, 2, 4, 1, 3, 0, 2, 4, 1, 3, 5, 0, 2, 4, 1, 3, 5, 0, 2, 4, 6, 1, 3, 5, 0, 2, 4, 6, 1, 3, 5, 7, 0, 2, 4, 6, 1, 3, 5, 7, 0, 2, 4, 6, 8, 1, 3, 5, 7, 0, 2, 4, 6, 8, 1, 3, 5, 7, 9, 0, 2, 4, 6, 8, 1, 3, 5, 7, 9, 0, 2, 4, 6, 8, 10, 1, 3, 5, 7, 9, 0, 2, 4, 6, 8, 10, 1, 3

Offset: 1

Clark Kimberling, Sep 09 2003

nonn

A sequence that contains itself as a proper subsequence (infinitely many times); that is, a fractal sequence.

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

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

Cf. A087465, A087467.

A087467 a(n) = number of the row (counting from initial row 1) of the array R in A087465 that contains n.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 4, 1, 3, 2, 4, 1, 3, 5, 2, 4, 1, 3, 5, 2, 4, 6, 1, 3, 5, 2, 4, 6, 1, 3, 5, 7, 2, 4, 6, 1, 3, 5, 7, 2, 4, 6, 8, 1, 3, 5, 7, 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, 8, 10, 1, 3, 5, 7, 9, 2, 4, 6, 8, 10, 1, 3, 5, 7, 9, 11, 2, 4, 6, 8, 10, 1, 3, 5, 7, 9, 11, 2

Offset: 1

Clark Kimberling, Sep 09 2003

nonn

A sequence that contains itself as a proper subsequence (infinitely many times); that is, a fractal sequence.

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

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

Cf. A087465, A087466.

A087466(n)+1

A322026 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(i) = A007814(j) and A007949(i) = A007949(j), for all i, j, where A007814 and A007949 give the 2- and 3-adic valuations of n.

Original entry on oeis.org

1, 2, 3, 4, 1, 5, 1, 6, 7, 2, 1, 8, 1, 2, 3, 9, 1, 10, 1, 4, 3, 2, 1, 11, 1, 2, 12, 4, 1, 5, 1, 13, 3, 2, 1, 14, 1, 2, 3, 6, 1, 5, 1, 4, 7, 2, 1, 15, 1, 2, 3, 4, 1, 16, 1, 6, 3, 2, 1, 8, 1, 2, 7, 17, 1, 5, 1, 4, 3, 2, 1, 18, 1, 2, 3, 4, 1, 5, 1, 9, 19, 2, 1, 8, 1, 2, 3, 6, 1, 10, 1, 4, 3, 2, 1, 20, 1, 2, 7, 4, 1, 5, 1, 6, 3

Offset: 1

Antti Karttunen, Dec 03 2018

nonn

Restricted growth sequence transform of the ordered pair [A007814(n), A007949(n)].
For all i, j:
  A305900(i) = A305900(j) => a(i) = a(j),
  a(i) = a(j) => A122841(i) = A122841(j),
  a(i) = a(j) => A244417(i) = A244417(j),
  a(i) = a(j) => A322316(i) = A322316(j) => A072078(i) = A072078(j).
If and only if a(k) > a(i) for all k > i then k is in A003586, - David A. Corneth, Dec 03 2018
That is, A003586 gives the positions of records (1, 2, 3, 4, 5, ...) in this sequence.
Sequence A126760 (without its initial zero) and this sequence are ordinal transforms of each other.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Jon Maiga, Computer-generated formulas for A322026, Sequence Machine.

Cf. A003586 (positions of records, the first occurrence of n), A007814, A007949, A065331, A071521, A072078, A087465, A122841, A126760 (ordinal transform), A322316, A323883, A323884.

Cf. also A247714 and A255975.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007814(n) = valuation(n,2);
A007949(n) = valuation(n,3);
v322026 = rgs_transform(vector(up_to, n, [A007814(n), A007949(n)]));
A322026(n) = v322026[n];

A065331(n) = (3^valuation(n, 3)<A065331
A071521(n) = { my(t=1/3); sum(k=0, logint(n, 3), t*=3; logint(n\t, 2)+1); }; \\ From A071521.
A322026(n) = A071521(A065331(n)); \\ Antti Karttunen, Sep 08 2024

For s = A003586(n), a(s) = n = a((6k+1)*s) = a((6k-1)*s), where s is the n-th 3-smooth number and k > 0. - David A. Corneth, Dec 03 2018
A065331(n) = A003586(a(n)). - David A. Corneth, Dec 04 2018
From Antti Karttunen, Sep 08 2024: (Start)
a(n) = Sum{k=1..n} [A126760(k)==A126760(n)], where [ ] is the Iverson bracket.
a(n) = A071521(A065331(n)). [Found by Sequence Machine and also by LODA miner]
a(n) = A323884(25*n). [Conjectured by Sequence Machine]
(End)

A283734 Rank array, R, of the golden ratio, read by antidiagonals downwards.

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, 28, 17, 20, 23, 26, 30, 34, 38, 22, 25, 29, 32, 36, 41, 45, 50, 27, 31, 35, 39, 43, 48, 53, 58, 63, 33, 37, 42, 46, 51, 56, 61, 67, 72, 78, 40, 44, 49, 54, 59, 65, 70, 76, 82, 88, 95, 47

Offset: 1

Clark Kimberling, Mar 16 2017

Every row intersperses all other rows, and every column intersperses all other columns. The array is the dispersion of the complement of column 1; column 1 is given by r(n) = r(n-1) + 1 + L(n), where L = lower Wythoff sequence (A000201).

			The corner of R begins:
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
28   34   41   48   56   65    74    84
38   45   53   61   70   80    90    101
50   58   67   76   86   97    108   120
Let t = golden ratio = (1 + sqrt(5))/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. A001622, A255977 (row 1), A283733 (column 1), A000201, A087465.

r = 40; r1 = 12;(*r=# rows of T,r1=# rows to show*);
c = 40; c1 = 12;(*c=# cols of T,c1=# cols to show*);
s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*GoldenRatio];
u = Table[s[n], {n, 0, 400}] (* A283733 *)
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, r1}, {j, 1, c1}]]   (* A283734, array *)
Flatten[Table[w[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A283734, sequence *)
TableForm[Table[w[i, 1] + w[1, j] + (i - 1)*(j - 1) - 1, {i, 1, r1}, {j, 1, c1}]] (* A283734, array, by formula *)

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

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

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A087468 Dispersion, read by antidiagonals, of the complement of row 0 of the array R in A087465.

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A087466 a(n) = number of the row (counting from initial row 0) of the array R in A087465 that contains n.

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A087467 a(n) = number of the row (counting from initial row 1) of the array R in A087465 that contains n.

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A322026 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(i) = A007814(j) and A007949(i) = A007949(j), for all i, j, where A007814 and A007949 give the 2- and 3-adic valuations of n.

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A283734 Rank array, R, of the golden ratio, read by antidiagonals downwards.

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