A163255 -id:A163255 - cp's OEIS frontend

A167430 Fractal sequence of the interspersion A163255.

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

Offset: 1

Clark Kimberling, Nov 03 2009

nonn

Both the upper trim and the lower trim of A167430 are identical to A167430.
(The upper trim of a fractal sequence s is what remains after the first
occurrence of each term is deleted; the lower trim of s is what remains
after all 0's are deleted from the sequence s-1.)

Cf. A002260, A163255.

The sequence is formed by concatenating rows:
Row 1: ... 1 2
Row 2: ... 1 3 2 4
Row 3: ... 1 3 5 2 4 6
Row n comes from row n-1 by putting 2n-1 just before 2 and 2*n
just after 2n-2.

A163253 An interspersion: the order array of the odd-numbered columns of the double interspersion at A161179.

Original entry on oeis.org

1, 4, 2, 9, 5, 3, 16, 10, 7, 6, 25, 17, 13, 11, 8, 36, 26, 21, 18, 14, 12, 49, 37, 31, 27, 22, 19, 15, 64, 50, 43, 38, 32, 28, 23, 20, 81, 65, 57, 51, 44, 39, 33, 29, 24, 100, 82, 73, 66, 58, 52, 45, 40, 34, 30, 121, 101, 91, 83, 74, 67, 59, 53, 46, 41, 35

Offset: 1

Clark Kimberling, Jul 23 2009

A permutation of the natural numbers.
Row 1 consists of the squares.
Beginning with row 5, the columns obey a 3rd-order recurrence:
c(n)=c(n-1)+c(n-2)-c(n-3)+1; thus disregarding row 1, the nonsquares are partitioned by this recurrence.
Except for initial terms, the first ten rows match A000290, A002522, A002061, A059100, A014209, A117950, A027688, A087475, A027689, A117951, and the first column, A035106.

			Corner:
1....4....9...16...25
2....5...10...17...26
3....7...13...21...31
6...11...18...27...38
The double interspersion A161179 begins thus:
1....4....7...12...17...24
2....3....8...11...18...23
5....6...13...16...25...30
9...10...19...22...33...38
Expel the even-numbered columns, leaving
1....7...17...
2....8...18...
5...13...25...
9...19...33...
Then replace each of those numbers by its rank when all the numbers are jointly ranked.

Clark Kimberling, Doubly interspersed sequences, double interspersions and fractal sequences, The Fibonacci Quarterly 48 (2010) 13-20.

Cf. A000037, A000290, A161179, A163254, A163255, A163256, A163257, A163258.

Let S(n,k) denote the k-th term in the n-th row. Three cases:
S(1,k)=k^2;
if n is even, then S(n,k)=k^2+(n-2)k+(n^2-2*n+4)/4;
if n>=3 is odd, then S(n,k)=k^2+(n-2)k+(n^2-2*n+1)/4.

Edited and augmented by Clark Kimberling, Jul 24 2009

A163254 Array of the nonsquares; the columns satisfy c(n)=c(n-1)+c(n-2)-c(n-3)+1.

Original entry on oeis.org

2, 5, 3, 10, 7, 6, 17, 13, 11, 8, 26, 21, 18, 14, 12, 37, 31, 27, 22, 19, 15, 50, 43, 38, 32, 28, 23, 20, 65, 57, 51, 44, 39, 33, 29, 24, 82, 73, 66, 58, 52, 45, 40, 34, 30, 101, 91, 83, 74, 67, 59, 53, 46, 41, 35, 122, 111, 102, 92, 84, 75, 68, 60, 54

Offset: 1

Clark Kimberling, Jul 23 2009

This is the array remaining after row 1 is expelled from the array in A163253.

			Corner:
2....5...10...17...26
3....7...13...21...31
6...11...18...27...38

Clark Kimberling, Doubly interspersed sequences, double interspersions and fractal sequences, The Fibonacci Quarterly 48 (2010) 13-20.

Cf. A000290, A163253, A163255, A163256, A163257, A163258.

Let S(n,k) for the k-th term in the n-th row.
if n is odd, then S(n,k)=k^2+(n-1)k+(n^2+3)/4;
if n is even, then S(n,k)=k^2+(n-1)k+(n^2)/4.

A163256 Fractal sequence of the interspersion A163253.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 24 2009

nonn

As a fractal sequence, A163256 contains every positive integer; indeed, A163256 properly contains itself (infinitely many times).

			Append the following segments:
  1 2 3
  1 2 4 3 5
  1 2 4 6 3 5 7
  1 2 4 6 8 3 5 7 9
For n>1, the n-th segment arises from the (n-1)st by inserting 2*n at position n+1 and appending 2*n+1 at position 2*n+1.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Clark Kimberling, Doubly interspersed sequences, double interspersions and fractal sequences, The Fibonacci Quarterly 48 (2010) 13-20.

Cf. A163253, A163254, A163255, A163257, A163258.

Flatten[FoldList[Append[Insert[#1, 2 #2, #2 + 1], 2 #2 + 1] &, {1}, Range[10]]] (* Birkas Gyorgy, Jul 09 2012 *)

A163258 Fractal sequence of the interspersion A163257.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 24 2009

nonn

As a fractal sequence, A163258 contains every positive integer; indeed, A163258 properly contains itself (infinitely many times).

			Append the following segments:
1 2 3 4
1 2 5 3 6 4
1 2 7 5 3 8 6 4
1 2 9 7 5 3 10 8 6 4
For n>1, the n-th segment arises from the (n-1)st by inserting 2*n+1 at position 3 and 2*n+2 at position n+3.

Clark Kimberling, Doubly interspersed sequences, double interspersions and fractal sequences, The Fibonacci Quarterly 48 (2010) 13-20.

Cf. A163253, A163254, A163255, A163256, A163258.

A163257 An interspersion: the order array of the even-numbered columns (after swapping the first two rows) of the double interspersion at A161179.

Original entry on oeis.org

1, 5, 2, 11, 6, 3, 19, 12, 8, 4, 29, 20, 15, 10, 7, 41, 30, 24, 18, 14, 9, 55, 42, 35, 28, 23, 17, 13, 71, 56, 48, 40, 34, 27, 22, 16, 89, 72, 63, 54, 47, 39, 33, 26, 21, 109, 90, 80, 70, 62, 53, 46, 38, 32, 25, 131, 110, 99, 88, 79, 69, 61, 52, 45, 37, 31, 155, 132, 120, 108

Offset: 1

Clark Kimberling, Jul 24 2009

A permutation of the natural numbers.
Beginning at row 6, the columns obey a 3rd-order recurrence:
c(n)=c(n-1)+c(n-2)-c(n-3)+1.
Except for initial terms, the first seven rows are A028387, A002378, A005563, A028552, A008865, A014209, A028873, and the first column, A004652.

			Corner:
1....5...11...19
2....6...12...20
3....8...15...24
4...10...18...28
The double interspersion A161179 begins thus:
1....4....7...12...17...24
2....3....8...11...18...23
5....6...13...16...25...30
9...10...19...22...33...38
Expel the odd-numbered columns and then swap rows 1 and 2, leaving
3....11...23...39
4....12...24...40
6....16...30...48
10...22...38...58
Then replace each of those numbers by its rank when all the numbers are jointly ranked.

Clark Kimberling, Doubly interspersed sequences, double interspersions and fractal sequences, The Fibonacci Quarterly 48 (2010) 13-20.

Cf. A163253, A163254, A163255, A163256, A163258.

Let S(n,k) denote the k-th term in the n-th row. Four cases:
S(1,k)=k^2+k-1
S(2,k)=k^2+k
if n>1 is odd, then S(n,k)=k^2+(n-1)k+(n-1)(n-3)/4
if n>2 is even, then S(n,k)= k^2+(n-1)k+n(n-4)/4.

A361974 (1,2)-block array, B(1,2), of the natural number array (A000027), read by descending antidiagonals.

Original entry on oeis.org

3, 11, 8, 27, 20, 15, 51, 40, 31, 24, 83, 68, 55, 44, 35, 123, 104, 87, 72, 59, 48, 171, 148, 127, 108, 91, 76, 63, 227, 200, 175, 152, 131, 112, 95, 80, 291, 260, 231, 204, 179, 156, 135, 116, 99, 363, 328, 295, 264, 235, 208, 183, 160, 139, 120, 443, 404

Offset: 1

Clark Kimberling, Apr 01 2023

We begin with a definition. Suppose that W = (w(i,j)), where i >= 1 and j >= 1, is an array of numbers such that if m and n satisfy 1 <= m < n, then there exists k such that w(m,k+h) < w(n,h+1) < w(m,k+h+1) for every h >= 0. Then W is a row-splitting array. The array B(1,2) is a row-splitting array. The rows and columns of B(1,2) are linearly recurrent with signature (3,-3,1). It appears that the order array (as defined in A333029) of B(1,2) is given by A163255.

			Corner of B(1,2):
   3   11   27   51   83  123   171   227
   8   20   40   68  104  148   200   260
  15   31   55   87  127  175   231   295
  24   44   72  108  152  204   264   332
  35   59   91  131  179  235   299   371
  48   76  112  156  298  268   336   412
(row 1 of A000027) = (1,2,4,7,11,16,22,29,...), so (row 1 of B(1,2)) = (3,11,27,58,...);
(row 2 of A000027) = (3,5,8,12,17,23,30,38,...), so (row 2 of B(1,2)) = (8,20,40,68,...).

Cf. A000027, A163255, A333029, A361975 (array B(2,1)), A361976 (array B(2,2)).

zz = 10; z = 13;
w[n_, k_] := n + (n + k - 2) (n + k - 1)/2;
t[h_, k_] := w[h, 2 k - 1] + w[h, 2 k];
Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (*A361974 sequence*)
TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (*A361974 array*)

B(1,2) = (b(i,j)), where b(i,j) = w(i, 2j-1) + w(i, 2j) for i >= 1, j >= 1, where (w(i,j)) is the natural number array (A000027).

b(i,j) = 2i + (i + 2j - 2)^2.

A361996 Order array of A361994, read by descending antidiagonals.

Original entry on oeis.org

1, 2, 3, 6, 7, 4, 15, 17, 11, 5, 39, 43, 28, 14, 8, 102, 112, 73, 38, 20, 9, 268, 292, 191, 100, 51, 23, 10, 568, 592, 491, 263, 132, 61, 27, 12, 868, 892, 791, 563, 345, 159, 72, 32, 13, 1168, 1192, 1091, 863, 645, 416, 189, 83, 35, 16, 1468, 1492, 1391

Offset: 1

Clark Kimberling, Apr 05 2023

This array is an interspersion (hence a dispersion, as in A114537 and A163255), so every positive integer occurs exactly once. See A333029 for the definition of order array.

			Corner:
  1    2    6   15   39  102  268 ...
  3    7   17   43  112  292  592 ...
  4   11   28   73  191  491  791 ...
  5   14   38  100  263  563  863 ...
  8   20   51  132  345  645  945 ...
  9   23   61  159  416  716 1016 ...
  ...

Cf. A114537, A163255, A333029, A361993, A361996.

zz = 300; z = 30;
w[n_, k_] := w[n, k] = Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
b[h_, k_] := b[h, k] = w[2 h - 1, 2 k - 1] + w[2 h - 1, 2 k] + w[2 h, 2 k - 1] + w[2 h, 2 k];
s = Flatten[Table[b[h, k], {h, 1, zz}, {k, 1, z}]];
r[h_, k_] := Length[Select[s, # <= b[h, k] &]]
TableForm[Table[r[h, k], {h, 1, 50}, {k, 1, 12}]](*A351996, array*)
v = Table[r[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (*A351996, sequence *)

A234305 Irregular triangle read by rows. Theoretical distribution of electrons based on the Janet's sequence A167268.

Original entry on oeis.org

1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 4, 2, 2, 5, 2, 2, 6, 2, 2, 6, 1, 2, 2, 6, 2, 2, 2, 6, 2, 1, 2, 2, 6, 2, 2, 2, 2, 6, 2, 3, 2, 2, 6, 2, 4, 2, 2, 6, 2, 5, 2, 2, 6, 2, 6, 2, 2, 6, 2, 6, 1, 2, 2, 6, 2, 6, 2, 2, 2, 6, 2, 6, 2, 1, 2, 2, 6, 2, 6, 2, 2, 2, 2, 6, 2, 6, 2, 3, 2, 2, 6, 2, 6, 2, 4

Offset: 1

Paul Curtz, Jan 02 2014

a(n) is not A173642, a compact Bohr-Stoner model (1924), modified by Charles Janet in 1930. The good distribution is A168208.
Only sequences N16(n) in A234398 are used:
N16(1)= 1 followed by 2's              = A040000,
N16(2)= 1, 2, 3, 4, 5, followed by 6's = A101272,
N16(3)= 1 to  9, followed by 10's,
N16(4)= 1 to 13, followed by 14's, etc.
The distribution by rows are in the example.
The N16(n)'s are respectively on columns (hence triangle T)
1, 2, 4,  6,  9, 12, 16, 20, 25, 30, 36,   A002620(n+2)
   3, 5,  8, 11, 15, 19, 24, 29, 35,       A024206(n+2)
      7, 10, 14, 18, 23, 28, 34,           A014616(n+3)
         13, 17, 22, 27, 33,               A004116(n+4)
             21, 26, 32,
                 31, etc.
See A163255.
Antidiagonals give the natural numbers A000027, like rows sums in the example.
A033638=1, 1, 2, 3, 5, 7,... is upon the triangle T.

			1,      H
2,       He
2, 1,    Li
2, 2,    Be
2, 2, 1,
2, 2, 2,
2, 2, 3,
2, 2, 4,
2, 2, 5,
2, 2, 6,
2, 2, 6, 1,
2, 2, 6, 2,
2, 2, 6, 2, 1,
2, 2, 6, 2, 2,
2, 2, 6, 2, 3,
2, 2, 6, 2, 4,
2, 2, 6, 2, 5,
2, 2, 6, 2, 6,
2, 2, 6, 2, 6, 1,
2, 2, 6, 2, 6, 2,
2, 2, 6, 2, 6, 2, 1,
2, 2, 6, 2, 6, 2, 2,
2, 2, 6, 2, 6, 2, 3, etc.

Cf. A002061, A002522 (or A160457), A014206, A059100, diagonals of the triangle T. A004526.

A361995 Order array of A361993, read by descending antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 6, 7, 5, 10, 11, 8, 9, 16, 18, 14, 12, 15, 26, 29, 23, 19, 13, 24, 42, 46, 38, 31, 22, 17, 39, 68, 74, 62, 50, 36, 28, 20, 63, 110, 119, 100, 81, 59, 45, 32, 21, 102, 111, 192, 101, 131, 97, 73, 52, 35, 25, 165, 179, 310, 162, 212, 158, 118, 84

Offset: 1

Clark Kimberling, Apr 05 2023

This array is an interspersion (hence a dispersion, as in A114537 and A163255), so every positive integer occurs exactly once. See A333029 for the definition of order array.

			Corner:
   1    2    3    5    9   15   24 ...
   4    6   10   16   26   42   68 ...
   7   11   18   29   46   74  119 ...
   8   14   23   38   62  100  162 ...
  12   19   31   50   81  131  212 ...
  13   22   36   59   97  158  191 ...
  ...

Cf. A114537, A163255, A333029, A361993, A361996.

zz = 300; z = 40;
w[n_, k_] := w[n, k] = Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
b[h_, k_] := b[h, k] = w[2 h - 1, k] + w[2 h, k];
s = Flatten[Table[b[h, k], {h, 1, zz}, {k, 1, z}]];
r[h_, k_] := Length[Select[s, # <= b[h, k] &]]
TableForm[Table[r[h, k], {h, 1, 50}, {k, 1, 12}]]  (*A351995, array*)
v = Table[r[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (*A351995, sequence*)

cp's OEIS Frontend

A167430 Fractal sequence of the interspersion A163255.

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A163253 An interspersion: the order array of the odd-numbered columns of the double interspersion at A161179.

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A163254 Array of the nonsquares; the columns satisfy c(n)=c(n-1)+c(n-2)-c(n-3)+1.

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A163256 Fractal sequence of the interspersion A163253.

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A163258 Fractal sequence of the interspersion A163257.

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A163257 An interspersion: the order array of the even-numbered columns (after swapping the first two rows) of the double interspersion at A161179.

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A361974 (1,2)-block array, B(1,2), of the natural number array (A000027), read by descending antidiagonals.

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A361996 Order array of A361994, read by descending antidiagonals.

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A234305 Irregular triangle read by rows. Theoretical distribution of electrons based on the Janet's sequence A167268.

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