A023133 -id:A023133 - cp's OEIS frontend

A022795 Place where n-th 1 occurs in A023133.

1, 5, 12, 22, 35, 51, 70, 92, 118, 147, 179, 214, 252, 293, 337, 385, 436, 490, 547, 607, 670, 736, 806, 879, 955, 1034, 1116, 1201, 1289, 1381, 1476, 1574, 1675, 1779, 1886, 1996, 2110, 2227, 2347, 2470, 2596, 2725, 2857, 2993, 3132, 3274

Offset: 1

Clark Kimberling

nonn

			a(n) = 1 + Sum_{k=1..n-1} ceiling(r*k) where r=Pi. [_Benoit Cloitre_, Jan 24 2009]

Cf. A023133.

a(n)=1+sum(k=1,n-1,ceil(Pi*k)) \\ Benoit Cloitre, Jan 24 2009

A283944 Interspersion of the signature sequence of Pi (rectangular array by antidiagonals).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Mar 26 2017

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.

			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

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. A000796, A023133, A022796, A022795.

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 *)

\\ 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

# 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

A179547 Transcendental signature sequence of Pi.

Original entry on oeis.org

1, 2, 3, 4, 1, 5, 2, 6, 3, 7, 4, 1, 8, 5, 2, 9, 6, 3, 10, 7, 4, 1, 1, 11, 8, 5, 2, 2, 12, 9, 6, 3, 3, 13, 10, 7, 4, 1, 4, 14, 1, 11, 8, 5, 2, 5, 15, 2, 12, 9, 6, 3, 6, 16, 3, 13, 10, 7, 4, 1, 7, 17, 4, 14, 1, 11, 8, 5, 2, 8, 18, 5, 15, 2, 12, 9, 6, 3, 9, 19, 6, 16, 3, 13, 10, 7, 4, 1, 10, 20, 7, 17

Offset: 1

Kerry Mitchell, Jul 19 2010

Let x be a transcendental number greater than 1. S1(x) is the standard signature sequence for an irrational number - sort y = a0 + a1x (for positive integers a0 and a1) and S1 is the sequence of a0 values. S2(x) is the sequence of a0's for sorted y = a0 + a1x + a2x^2, S3(x) is the a0 sequence for y = a0 + a1x + a2x^2 + a3x^3, etc. The transcendental signature sequence is the limit of Sn(x) as n approaches infinity.

L. K. Mitchell, Table of n, a(n) for n=1..10000
Franklin T. Adams-Watters and Eric W. Weisstein, Signature Sequence
Eric Weisstein's World of Mathematics, Transcendental Number

Cf. A023133 (signature sequence of Pi).

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A022795 Place where n-th 1 occurs in A023133.

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A283944 Interspersion of the signature sequence of Pi (rectangular array by antidiagonals).

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A179547 Transcendental signature sequence of Pi.

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