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A283939 Interspersion of the signature sequence of sqrt(2). 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

LINKS

Clark Kimberling, Antidiagonals n = 1..60, flattened

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

EXAMPLE

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

MATHEMATICA

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 *)

PROG

(PARI)

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

(Python)

from sympy import sqrt

import math

def s(n): return 1 if n<1 else s(n - 1) + 1 +

int(math.floor(n*sqrt(2)))

def p(n): return n + 1 + sum([int(math.floor((n - k)/sqrt(2))) 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

CROSSREFS

Cf. A007336, A002193, A022776, A283962.

Sequence in context: A038722 A277881 A145522 * A293056 A131968 A191740

Adjacent sequences:  A283936 A283937 A283938 * A283940 A283941 A283942

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Mar 19 2017

STATUS

approved

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Last modified July 24 06:56 EDT 2021. Contains 346273 sequences. (Running on oeis4.)