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A283944 Interspersion of the signature sequence of Pi (rectangular array by antidiagonals). 2
1, 5, 2, 12, 7, 3, 22, 15, 9, 4, 35, 26, 18, 11, 6, 51, 40, 30, 21, 14, 8, 70, 57, 45, 34, 25, 17, 10, 92, 77, 63, 50, 39, 29, 20, 13, 118, 100, 84, 69, 56, 44, 33, 24, 16, 147, 127, 108, 91, 76, 62, 49, 38, 28, 19, 179, 157, 136, 116, 99, 83, 68, 55, 43, 32 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row n is the ordered sequence of numbers k such that A023133(k) = n. As a sequence, it is a permutation of the positive integers. 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  5   12  22   35   51   70   92    118

2  7   15  26   40   57   77   100   127

3  9   18  30   45   63   84   108   136

4  11  21  34   50   69   91   115   145

6  14  25  39   56   76   99   125   155

8  17  29  44   62   83   107  134   165

MATHEMATICA

r = Pi; 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}] (* A022796, col 1 of A283944 *)

v = Table[s[n], {n, 0, z}] (* A022795, row 1 of A283944 *)

w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1;

Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283944, array*)

Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283944, sequence *)

PROG

(PARI)

\\ Produces the triangle when the array is read by antidiagonals

r = Pi;

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 26 2017

(Python)

# Produces the triangle when the array is read by antidiagonals

import math

from mpmath import *

mp.dps = 100

def s(n): return 1 if n<1 else s(n - 1) + 1 + int(math.floor(n*pi))

def p(n): return n + 1 + sum([int(math.floor((n - k)/pi)) 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 26 2017

CROSSREFS

Cf. A000796, A023133, A022796, A022795.

Sequence in context: A130298 A128116 A082153 * A294684 A013946 A261327

Adjacent sequences:  A283941 A283942 A283943 * A283945 A283946 A283947

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Mar 26 2017

STATUS

approved

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Last modified June 30 06:00 EDT 2022. Contains 354914 sequences. (Running on oeis4.)