A049474 -id:A049474 - cp's OEIS frontend

A194959 Fractalization of (1 + floor(n/2)).

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

Offset: 1

Clark Kimberling, Sep 06 2011

Suppose that p(1), p(2), p(3), ... is an integer sequence satisfying 1 <= p(n) <= n for n >= 1. Define g(1)=(1) and for n > 1, form g(n) from g(n-1) by inserting n so that its position in the resulting n-tuple is p(n). The sequence f obtained by concatenating g(1), g(2), g(3), ... is clearly a fractal sequence, here introduced as the fractalization of p. The interspersion associated with f is here introduced as the interspersion fractally induced by p, denoted by I(p); thus, the k-th term in the n-th row of I(p) is the position of the k-th n in f. Regarded as a sequence, I(p) is a permutation of the positive integers; its inverse permutation is denoted by Q(p).
...
Example: Let p=(1,2,2,3,3,4,4,5,5,6,6,7,7,...)=A008619. Then g(1)=(1), g(2)=(1,2), g(3)=(1,3,2), so that
f=(1,1,2,1,3,2,1,3,4,2,1,3,5,4,2,1,3,5,6,4,2,1,...)=A194959; and I(p)=A057027, Q(p)=A064578.
The interspersion I(P) has the following northwest corner, easily read from f:
  1  2  4  7 11 16 22
  3  6 10 15 21 28 36
  5  8 12 17 23 30 38
  9 14 20 27 35 44 54
  ...
Following is a chart of selected p, f, I(p), and Q(p):
   p         f        I(p)      Q(p)
A000027   A002260   A000027   A000027
A008619   A194959   A057027   A064578
A194960   A194961   A194962   A194963
A053824   A194965   A194966   A194967
A053737   A194973   A194974   A194975
A019446   A194968   A194969   A194970
A049474   A194976   A194977   A194978
A194979   A194980   A194981   A194982
A194964   A194983   A194984   A194985
A194986   A194987   A194988   A194989
Count odd numbers up to n, then even numbers down from n. - Franklin T. Adams-Watters, Jan 21 2012
This sequence defines the square array A(n,k), n > 0 and k > 0, read by antidiagonals and the triangle T(n,k) = A(n+1-k,k) for 1 <= k <= n read by rows (see Formula and Example). - Werner Schulte, May 27 2018

			The sequence p=A008619 begins with 1,2,2,3,3,4,4,5,5,..., so that g(1)=(1). To form g(2), write g(1) and append 2 so that in g(2) this 2 has position p(2)=2: g(2)=(1,2). Then form g(3) by inserting 3 at position p(3)=2: g(3)=(1,3,2), and so on. The fractal sequence A194959 is formed as the concatenation g(1)g(2)g(3)g(4)g(5)...=(1,1,2,1,3,2,1,3,4,2,1,3,5,4,2,...).
From _Werner Schulte_, May 27 2018: (Start)
This sequence seen as a square array read by antidiagonals:
  n\k: 1  2  3  4  5   6   7   8   9  10  11  12 ...
  ===================================================
   1   1  2  2  2  2   2   2   2   2   2   2   2 ... (see A040000)
   2   1  3  4  4  4   4   4   4   4   4   4   4 ... (see A113311)
   3   1  3  5  6  6   6   6   6   6   6   6   6 ...
   4   1  3  5  7  8   8   8   8   8   8   8   8 ...
   5   1  3  5  7  9  10  10  10  10  10  10  10 ...
   6   1  3  5  7  9  11  12  12  12  12  12  12 ...
   7   1  3  5  7  9  11  13  14  14  14  14  14 ...
   8   1  3  5  7  9  11  13  15  16  16  16  16 ...
   9   1  3  5  7  9  11  13  15  17  18  18  18 ...
  10   1  3  5  7  9  11  13  15  17  19  20  20 ...
  etc.
This sequence seen as a triangle read by rows:
  n\k:  1  2  3  4  5   6   7   8   9  10  11  12  ...
  ======================================================
   1    1
   2    1  2
   3    1  3  2
   4    1  3  4  2
   5    1  3  5  4  2
   6    1  3  5  6  4   2
   7    1  3  5  7  6   4   2
   8    1  3  5  7  8   6   4   2
   9    1  3  5  7  9   8   6   4   2
  10    1  3  5  7  9  10   8   6   4   2
  11    1  3  5  7  9  11  10   8   6   4   2
  12    1  3  5  7  9  11  12  10   8   6   4   2
  etc.
(End)

Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168.

Paul Lévy, Sur quelques classes de permutations, Compositio Mathematica, Volume 8, 1951, pages 1-48. P_n = g(n).
Eric Weisstein's World of Mathematics, Fractal sequence
Eric Weisstein's World of Mathematics, Interspersion
Wikipedia, Fractal sequence

Cf. A000142, A000217, A005408, A005843, A008619, A057027, A064578, A209229, A210535, A219977; A000012 (col 1), A157532 (col 2), A040000 (row 1), A113311 (row 2); A194029 (introduces the natural fractal sequence and natural interspersion of a sequence - different from those introduced at A194959).

Cf. A003558 (g permutation order), A102417 (index), A330081 (on bits), A057058 (inverse).

r = 2; p[n_] := 1 + Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A008619 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A194959 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]]  (* A057027 *)
q[n_] := Position[w, n]; Flatten[
Table[q[n], {n, 1, 80}]]  (* A064578 *)
Flatten[FoldList[Insert[#1, #2, Floor[#2/2] + 1] &, {}, Range[10]]] (* Birkas Gyorgy, Jun 30 2012 *)

T(n,k) = min(k<<1-1,(n-k+1)<<1); \\ Kevin Ryde, Oct 09 2020

From Werner Schulte, May 27 2018 and Jul 10 2018: (Start)
Seen as a triangle: It seems that the triangle T(n,k) for 1 <= k <= n (see Example) is the mirror image of A210535.
Seen as a square array A(n,k) and as a triangle T(n,k):
A(n,k) = 2*k-1 for 1 <= k <= n, and A(n,k) = 2*n for 1 <= n < k.
A(n+1,k+1) = A(n,k+1) + A(n,k) - A(n-1,k) for k > 0 and n > 1.
A(n,k) = A(k,n) - 1 for n > k >= 1.
P(n,x) = Sum_{k>0} A(n,k)*x^(k-1) = (1-x^n)*(1-x^2)/(1-x)^3 for n >= 1.
Q(y,k) = Sum_{n>0} A(n,k)*y^(n-1) = 1/(1-y) for k = 1 and Q(y,k) = Q(y,1) + P(k-1,y) for k > 1.
G.f.: Sum_{n>0, k>0} A(n,k)*x^(k-1)*y^(n-1) = (1+x)/((1-x)*(1-y)*(1-x*y)).
Sum_{k=1..n} A(n+1-k,k) = Sum_{k=1..n} T(n,k) = A000217(n) for n > 0.
Sum_{k=1..n} (-1)^(k-1) * A(n+1-k,k) = Sum_{k=1..n} (-1)^(k-1) * T(n,k) = A219977(n-1) for n > 0.
Product_{k=1..n} A(n+1-k,k) = Product_{k=1..n} T(n,k) = A000142(n) for n > 0.
A(n+m,n) = A005408(n-1) for n > 0 and some fixed m >= 0.
A(n,n+m) = A005843(n) for n > 0 and some fixed m > 0.
Let A_m be the upper left part of the square array A(n,k) with m rows and m columns. Then det(A_m) = 1 for some fixed m > 0.
The P(n,x) satisfy the recurrence equation P(n+1,x) = P(n,x) + x^n*P(1,x) for n > 0 and initial value P(1,x) = (1+x)/(1-x).
Let B(n,k) be multiplicative with B(n,p^e) = A(n,e+1) for e >= 0 and some fixed n > 0. That yields the Dirichlet g.f.: Sum_{k>0} B(n,k)/k^s = (zeta(s))^3/(zeta(2*s)*zeta(n*s)).
Sum_{k=1..n} A(k,n+1-k)*A209229(k) = 2*n-1. (conjectured)
(End)
From Kevin Ryde, Oct 09 2020: (Start)
T(n,k) = 2*k-1 if 2*k-1 <= n, or 2*(n+1-k) if 2*k-1 > n. [Lévy, chapter 1 section 1 equations (a),(b)]
Fixed points T(n,k)=k for k=1 and k = (2/3)*(n+1) when an integer. [Lévy, chapter 1 section 2 equation (3)]
(End)

Name corrected by Franklin T. Adams-Watters, Jan 21 2012

A049472 a(n) = floor(n/sqrt(2)).

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 19, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 31, 31, 32, 33, 33, 34, 35, 36, 36, 37, 38, 38, 39, 40, 41, 41, 42, 43, 43, 44, 45, 45, 46, 47

Offset: 0

Thomas Kellar

For n > 0: A006337(n) = number of repeating n's. - Reinhard Zumkeller, Jul 04 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Robbert Fokkink, The Pell Tower and Ostronometry, arXiv:2309.01644 [math.CO], 2023.
Index entries for sequences related to Beatty sequences

First differences give A080764.

Cf. A006337, A049474.

a049472 = floor . (/ sqrt 2) . fromIntegral
-- Reinhard Zumkeller, Jul 04 2015

[Floor(n/Sqrt(2)): n in [0..70] ]; // Vincenzo Librandi, Aug 23 2011

A049472:=n->floor(n/sqrt(2)): seq(A049472(n), n=0..100); # Wesley Ivan Hurt, Jun 25 2016

Floor[Range[0,70]/Sqrt[2]] (* Harvey P. Dale, Aug 22 2011 *)

a(n)=sqrtint(n^2\2) \\ Charles R Greathouse IV, Sep 02 2015

A022776 Place where n-th 1 occurs in A023115.

Original entry on oeis.org

1, 2, 4, 7, 10, 14, 19, 24, 30, 37, 45, 53, 62, 72, 82, 93, 105, 118, 131, 145, 160, 175, 191, 208, 225, 243, 262, 282, 302, 323, 345, 367, 390, 414, 439, 464, 490, 517, 544, 572, 601, 630, 660, 691, 723, 755, 788, 822, 856, 891, 927, 964, 1001

Offset: 1

Clark Kimberling

nonn

Positions of the integers when the numbers a + b*sqrt(2) are arranged in increasing order. - Clark Kimberling, Mar 16 2015

It seems the name of this sequence could also be "Indices where records occur in A007336". - Ivan N. Ianakiev, Sep 09 2019

			The ordering of numbers a+b*r, where r = sqrt(2) as in Comments, begins with 0, 1, r, 2, 1+r, 2r, 3, 2+r, 1+2r, 4, ... in which the positions of integers are 1, 2, 4, 7, 10.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A023115, A049474.

t = Table[n + 1 + Sum[Floor[(n - k)/Sqrt[2]], {k, 0, n}], {n, 0, 200}] (* A022776 *)
Differences[t] (* A049474 *) (* Clark Kimberling, Mar 14 2015 *)

a(n)=1+sum(k=1,n-1,ceil(k/sqrt(2))) \\ Benoit Cloitre, Jan 24 2009

a(n) = 1 + Sum_{k=1..n-1} ceiling(r*k) where r=1/sqrt(2). - Benoit Cloitre, Jan 24 2009

A194977 Interspersion fractally induced by A194976, a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 9, 11, 12, 15, 13, 14, 16, 17, 21, 18, 19, 20, 22, 23, 28, 24, 25, 27, 26, 29, 30, 36, 31, 32, 35, 33, 34, 37, 38, 45, 39, 40, 44, 41, 42, 43, 46, 47, 55, 48, 49, 54, 50, 51, 52, 53, 56, 57, 66, 58, 59, 65, 60, 61, 62, 64, 63, 67, 68, 78, 69, 70, 77, 71, 72, 73, 76, 74, 75

Offset: 1

Clark Kimberling, Sep 07 2011

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. Every pair of rows eventually intersperse. As a sequence, A194977 is a permutation of the positive integers, with inverse A194978.

			Northwest corner:
   1  2  4  7 11 16 22
   3  5  8 12 17 23 30
   6 10 15 21 28 36 45
   9 13 18 24 31 39 48
  14 19 25 32 40 49 59

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A194959, A194976, A194978.

r = Sqrt[2]; p[n_] := 1 + Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A049474 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A194976 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]] (* A194977 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]] (* A194978 *)

Terms a(70) and beyond from G. C. Greubel, Mar 28 2018

A194976 Fractalization of (1+[n/sqrt(2)]), where [ ]=floor.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 07 2011

nonn

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. The sequence (1+[n/sqrt(2)]) is A049474.

G. C. Greubel, Table of n, a(n) for n = 1..11325

Cf. A194959, A049474, A194977, A194978.

r = Sqrt[2]; p[n_] := 1 + Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A049474 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A194976 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]] (* A194977 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]] (* A194978 *)

A099188 a(n) = 2*ceiling(n/sqrt(2)).

Original entry on oeis.org

0, 2, 4, 6, 6, 8, 10, 10, 12, 14, 16, 16, 18, 20, 20, 22, 24, 26, 26, 28, 30, 30, 32, 34, 34, 36, 38, 40, 40, 42, 44, 44, 46, 48, 50, 50, 52, 54, 54, 56, 58, 58, 60, 62, 64, 64, 66, 68, 68, 70, 72, 74, 74, 76, 78, 78, 80, 82, 84, 84, 86, 88, 88, 90, 92, 92, 94, 96, 98, 98, 100

Offset: 0

David W. Wilson, Mar 26 2005

Conjecturally, length of shortest polygonal path from (0,0) to (n,n) with integer vertices and edges. This is true for n <= 10000.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

a099188 = (* 2) . a049474  -- Reinhard Zumkeller, Jul 03 2015

[2*Ceiling(n/Sqrt(2)): n in [0..100]]; // G. C. Greubel, Aug 17 2018

2Ceiling[Range[0,80]/Sqrt[2]] (* Harvey P. Dale, Dec 21 2011 *)

a(n)=if(n,2+2*sqrtint((n^2-1)\2)) \\ Charles R Greathouse IV, Jan 10 2013

a(n) = 2*A049474(n).

cp's OEIS Frontend

A194959 Fractalization of (1 + floor(n/2)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A049472 a(n) = floor(n/sqrt(2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

A022776 Place where n-th 1 occurs in A023115.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A194977 Interspersion fractally induced by A194976, a rectangular array, by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A194976 Fractalization of (1+[n/sqrt(2)]), where [ ]=floor.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A099188 a(n) = 2*ceiling(n/sqrt(2)).

Views

Author

Keywords

Comments

Links

Programs

Haskell

Magma

Mathematica

PARI

Formula