A022775 -id:A022775 - cp's OEIS frontend

A283962 Interspersion of the signature sequence of sqrt(1/2).

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10, 12, 13, 15, 17, 14, 16, 18, 20, 22, 25, 19, 21, 23, 26, 28, 31, 34, 24, 27, 29, 32, 35, 38, 41, 44, 30, 33, 36, 39, 42, 46, 49, 52, 56, 37, 40, 43, 47, 50, 54, 58, 61, 65, 69, 45, 48, 51, 55, 59, 63, 67, 71, 75, 79, 84, 53

Offset: 1

Clark Kimberling, Mar 19 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 = A022776).

R(n,m) = position of n*r + m when all the numbers k*r + h, where r = sqrt(2), k >= 1, h >= 0, are jointly ranked. - Clark Kimberling, Oct 06 2017

			Northwest corner of R:
   1   2   4   7  10  14  19  24  30
   3   5   8  12  16  21  27  33  40
   6   9  13  18  23  29  36  43  51
  11  15  20  26  32  39  47  44  64
  17  22  28  35  42  50  59  68  78
  25  31  38  46  54  63  73  83  94

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. A010503, A022775, A022776, A283939.

r = Sqrt[1/2]; z = 100;
s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*r];
u = Table[n + 1 + Sum[Floor[(n - k)/r], {k, 0, n}], {n, 0, z}] (* A022775, col 1 of A283962 *)
v = Table[s[n], {n, 0, z}] (* A022776, row 1 of A283962*)
w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1;
Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283962, array *)
Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283962, sequence *)

r = sqrt(1/2);
z = 100;
s(n) = if(n<1, 1, s(n - 1) + 1 + floor(n*r));
p(n) = n + 1 + sum(k=0, n, floor((n - k)/r));
u = v = vector(z + 1);
for(n=1, 101, (v[n] = s(n - 1)));
for(n=1, 101, (u[n] = p(n - 1)));
w(i, j) = u[i] + v[j] + (i - 1) * (j - 1) - 1;
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(w(k, n - k + 1), ", "); );print(); ); };
tabl(10) \\ Indranil Ghosh, Mar 21 2017

r = 0.5 ** 0.5
def s(n): return 1 if n<1 else s(n - 1) + 1 + int(n*r)
def p(n): return n + 1 + sum([int((n - k)/r) for k in range(0, n+1)])
v=[s(n) for n in range(0, 101)]
u=[p(n) for n in range(0, 101)]
def w(i,j): return u[i - 1] + v[j - 1] + (i - 1) * (j - 1) - 1
for n in range(1, 11):
    print([w(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 21 2017

import numpy as np
r = np.sqrt(1/2)
x = np.arange(11)
u = np.cumsum(np.ceil(x / r)).astype(int)
v = np.cumsum(np.ceil(x * r)).astype(int)
print(*[1 + u[k] + v[n-k] + k*(n-k) for n in range(11) for k in range(n+1)], sep=', ')
# David Radcliffe, May 10 2025

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

A007336 Signature sequence of sqrt 2 (arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Clark Kimberling

Clark Kimberling, "Fractal Sequences and Interspersions", Ars Combinatoria, vol. 45 p 157 1997.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..1000
Dale Gerdemann, Plot of Adjacent Terms, YouTube Video, 2015.
Clark Kimberling, Interspersions
Index entries for sequences related to signature sequences

Cf. A007337, A022775.

Take[ Transpose[ Sort[ Flatten[ Table[{i + j*Sqrt[2], i}, {i, 17}, {j, 15}], 1], #1[[1]] < #2[[1]] &]][[2]], 96] (* Robert G. Wilson v, Jul 24 2004 *)
Quiet[Block[{$ContextPath}, Needs["Combinatorica`"]], {General::compat}]
memos = <||>;
zeroBasedC[theta_, i_] := zeroBasedC[theta, i] = Module[{memo, depth},
  memo = Lookup[memos, theta, {-1, 0}];
  While[memo[[-1]] <= i, AppendTo[memo, memo[[-1]] + Ceiling[theta * (Length[memo] - 1)]]];
  memos[i] = memo;
  depth = Combinatorica`BinarySearch[memo, i] - 3/2;
  If[IntegerQ[depth] && depth <= i, 1 + zeroBasedC[theta, i - depth], 0]
];
A007336[i_] := zeroBasedC[2^(1/2), i - 1] + 1;
Table[A007336[i], {i, 1, 100}] (* Brady J. Garvin, Aug 19 2024 *)

from bisect import bisect
from collections import defaultdict
from functools import cache
from math import ceil
memos = defaultdict(lambda: [-1, 0])
@cache
def zero_based_c(theta, i):
    memo = memos[theta]
    while memo[-1] <= i:
        memo.append(memo[-1] + ceil(theta * (len(memo) - 1)))
    depth = bisect(memo, i) - 1
    return 0 if depth > i or memo[depth] == i else 1 + zero_based_c(theta, i - depth)
def A007336(i):
    return zero_based_c(2 ** 0.5, i - 1) + 1
print([A007336(i) for i in range(1, 1001)])  # Brady J. Garvin, Aug 18 2024

If delete first occurrence of 1, 2, 3, ... the sequence is unchanged.

More terms from Robert G. Wilson v, Jul 24 2004

A283939 Interspersion of the signature sequence of sqrt(2).

Original entry on oeis.org

1, 3, 2, 6, 5, 4, 11, 9, 8, 7, 17, 15, 13, 12, 10, 25, 22, 20, 18, 16, 14, 34, 31, 28, 26, 23, 21, 19, 44, 41, 38, 35, 32, 29, 27, 24, 56, 52, 49, 46, 42, 39, 36, 33, 30, 69, 65, 61, 58, 54, 50, 47, 43, 40, 37, 84, 79, 75, 71, 67, 63, 59, 55, 51, 48, 45, 100

Offset: 1

Clark Kimberling, Mar 19 2017

Row n is the ordered sequence of numbers k such that A007336(k)=n. As a sequence, A283939 is a permutation of the positive integers. As an array, A283939 is the joint-rank array (defined at A182801) of the numbers {i+j*r}, for i>=1, j>=1, where r = sqrt(2). This is a transposable interspersion; i.e., every row intersperses all other rows, and every column intersperses all other columns.

			Northwest corner:
  1   3   6   11   17   25   34   44   56
  2   5   9   15   22   31   41   52   65
  4   8   13  20   28   38   49   61   75
  7   12  18  26   35   46   58   71   86
  10  16  23  32   42   54   67   81   97
  14  21  29  39   50   63   77   91   109

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. A007336, A002193, A022776, A283962.

r = Sqrt[2]; z = 100;
s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*r];
u = Table[n + 1 + Sum[Floor[(n - k)/r], {k, 0, n}], {n, 0, z}] (* A022776, col 1 of A283939 *)
v = Table[s[n], {n, 0, z}] (* A022775, row 1 of A283939*)
w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1;
Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283939, array *)
p = Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283939, sequence *)

r = sqrt(2);
z = 100;
s(n) = if(n<1, 1, s(n - 1) + 1 + floor(n*r));
p(n) = n + 1 + sum(k=0, n, floor((n - k)/r));
u = v = vector(z + 1);
for(n=1, 101, (v[n] = s(n - 1)));
for(n=1, 101, (u[n] = p(n - 1)));
w(i, j) = u[i] + v[j] + (i - 1) * (j - 1) - 1;
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(w(k, n - k + 1), ", "); );print(); ); };
tabl(10) \\ Indranil Ghosh, Mar 21 2017

sqrt2 = 2 ** 0.5
def s(n): return 1 if n<1 else s(n - 1) + 1 + int(n*sqrt2)
def p(n): return n + 1 + sum([int((n - k)/sqrt2) for k in range(0, n+1)])
v=[s(n) for n in range(0, 101)]
u=[p(n) for n in range(0, 101)]
def w(i,j): return u[i - 1] + v[j - 1] + (i - 1) * (j - 1) - 1
for n in range(1, 11):
    print ([w(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 21 2017

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A283962 Interspersion of the signature sequence of sqrt(1/2).

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A007336 Signature sequence of sqrt 2 (arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x).

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A283939 Interspersion of the signature sequence of sqrt(2).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python