Zacariaz Martinez - cp's OEIS frontend

User: Zacariaz Martinez

Zacariaz Martinez has authored 5 sequences.

A219330 Number of random selections (with replacement) needed from a normal population to assure a greater than one-half chance that the selected group contains the top 10th percentile individual, top 1st percentile individual, the 0.1 percentile, 0.01 percentile etc...

7, 69, 693, 6932, 69315, 693147, 6931472, 69314718, 693147181, 6931471806, 69314718056, 693147180560, 6931471805600, 69314718055995, 693147180559945, 6931471805599453, 69314718055994531, 693147180559945310, 6931471805599453094, 69314718055994530942

Offset: 1

Zacariaz Martinez, Apr 11 2013

nonn

a(n) is the least number of random selections (with replacement) required that assures a group having at least a 0.5 probability of containing the top 100/(10^n)th percentile individual. Converges toward log(2)*10^n with increasing n.

			To assure a greater than 1/2 chance of finding an individual with, say, an IQ in the top one out of a thousand or 1/(10^3), you must select at least (with replacement) a(3) = 693 individuals.

F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill, 1965

Cf. A002162, A014088, A033810, A050255.

a(n)=ceil((log(2))/(log(10^n/(10^n-1)))); \\ Joerg Arndt, Apr 15 2013

a(n) = ceiling((log(2))/(log(10^n/(10^n-1)))).

A127314 Decimal expansion of number x satisfying log(x) = x/10.

Original entry on oeis.org

3, 5, 7, 7, 1, 5, 2, 0, 6, 3, 9, 5, 7, 2, 9, 7, 2, 1, 8, 4, 0, 9, 3, 9, 1, 9, 6, 3, 5, 1, 1, 9, 9, 4, 8, 8, 0, 4, 0, 1, 7, 9, 6, 2, 5, 7, 7, 9, 3, 0, 7, 5, 9, 2, 3, 6, 8, 3, 5, 2, 7, 7, 5, 5, 7, 9, 1, 6, 8, 7, 2, 3, 6, 3, 5, 0, 5, 7, 5, 4, 6, 2, 8, 6, 1, 4, 6, 3, 6, 5, 5, 6, 2, 0, 8, 4, 6, 8, 0, 8, 0, 1, 7, 7, 3

Offset: 2

Zacariaz Martinez, Apr 08 2007

			log(35.771520639...) = 3.5771520639.

Cf. A128804.

Corrected and extended by Klaus Brockhaus, Apr 11 2007

A128804 Decimal expansion of number x satisfying log_10(x) = x/100.

Original entry on oeis.org

2, 3, 7, 5, 8, 1, 2, 0, 8, 7, 5, 9, 3, 4, 2, 6, 3, 2, 2, 2, 7, 0, 3, 4, 1, 4, 9, 6, 5, 9, 1, 7, 4, 0, 3, 9, 8, 9, 2, 6, 1, 4, 4, 7, 5, 9, 2, 8, 8, 6, 2, 3, 8, 8, 3, 6, 1, 3, 2, 9, 9, 5, 2, 7, 4, 5, 8, 5, 9, 9, 7, 9, 5, 2, 0, 7, 5, 4, 2, 4, 7, 7, 1, 7, 5, 7, 4, 9, 2, 6, 7, 4, 2, 8, 4, 8, 9, 9, 2, 1, 2, 7, 7, 0, 2

Offset: 3

Zacariaz Martinez, Apr 08 2007

			log_10(237.5812087...) = 2.375812087.

Cf. A127314.

Extended by Klaus Brockhaus, Apr 11 2007

A124507 a(n) = floor(exp(n*Pi/2)).

Original entry on oeis.org

1, 4, 23, 111, 535, 2575, 12391, 59609, 286751, 1379410, 6635623, 31920519, 153552935, 738662922, 3553321280, 17093171648, 82226315585, 395547831244, 1902773895292, 9153250784394, 44031505860632, 211812562992413, 1018919543279304, 4901489415968642

Offset: 0

Zacariaz Martinez, Dec 27 2006

nonn

Roger Penrose, The Road to Reality, (2005), p. 88 (figure 5.3).

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A062360 (even bisection), A042972.

R:= RealField(10); [Floor(Exp(n*Pi(R)/2)): n in [0..30]]; // G. C. Greubel, Nov 25 2018

Digits:= 2000:
a:= n-> floor(exp(n*Pi/2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 25 2018

Table[ Floor@ Exp[n*Pi/2], {n, 0, 21}] (* Robert G. Wilson v, Dec 31 2006 *)

vector(30, n, n--; floor(exp(n*Pi/2))) \\ G. C. Greubel, Nov 25 2018

[floor(exp(n*pi/2)) for n in range(30)] # G. C. Greubel, Nov 25 2018

Edited and extended by Robert G. Wilson v, Dec 31 2006

Comments edited by Jon E. Schoenfield, Nov 25 2018

A123663 Number of shared edges in a spiral of n unit squares.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 10, 12, 13, 15, 17, 18, 20, 22, 24, 25, 27, 29, 31, 32, 34, 36, 38, 40, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 60, 61, 63, 65, 67, 69, 71, 72, 74, 76, 78, 80, 82, 84, 85, 87, 89, 91, 93, 95, 97, 98, 100, 102, 104, 106, 108, 110, 112, 113, 115, 117, 119

Offset: 1

Zacariaz Martinez, Nov 15 2006

If one constructs a square (square 1) and then draws another square of identical size beside it (square 2), the squares share 1 edge. If one then places an identical square above square 2 (instead of continuing in a straight path), there are now 2 shared edges. Continuing this pattern in an outward spiral, one finds that the number of shared edges is 4, 5, 7, ...
Numbers a(n) such that a(n+1) = a(n) + 1 are (except for the leading zero) A074148. Otherwise a(n+1) = a(n) + 2. - Franklin T. Adams-Watters, Oct 17 2014.
This sequence is also the maximal number of shared edges among all polyominoes with n square cells.  This is the result of Harary and Harborth cited in the references. Once this is known the formula 2n - ceiling(2*sqrt(n)) comes from geometrical considerations and A027709. Namely, the 4n sides of the n squares making up the polyomino form the perimeter and come together in pairs along shared edges. Hence, 4n = perimeter + 2*shared edges. Maximizing shared edges minimizes perimeter and so maximum shared edges = (4n - minimum perimeter)/2 = (4n - 2ceiling(2*sqrt(n)))/2 = 2n - ceiling(2*sqrt(n)). This interpretation is important to landscape ecologists and is called the aggregation index in the GIS program FRAGSTATS. - Julian F. Fleron, Nov 29 2016
a(n) is also the maximum degree of the cover graphs of lattice quotients of lattice congruences of the weak order on the symmetric group S_n. See Table 1 in the Hoang/Mütze reference in the Links section. - Torsten Muetze, Nov 28 2019
a(n) is also the number of pixels in H_{n-1}, where H_n (a pixelated piece of hyperbola x*y = n) is the set of the (x, y), ordered pairs of positive integers, such that x*y = n or (x*y < n and ((x+1)*y > n or x*(y+1) > n)). - Luc Rousseau, Dec 28 2019

F. Harary and H. Harborth, Extremal Animals, Journal of Combinatorics, Information & System Sciences, Vol. 1, No 1, 1-8 (1976).

Peter Kagey, Table of n, a(n) for n = 1..10000
Richard A. Brualdi and Geir Dahl, Frobenius-König theorem for classes of (0,+/-1)-matrices, Disc. Math. (2024) Vol. 347, Issue 6, 113951.
Hung Phuc Hoang and Torsten Mütze, Combinatorial generation via permutation languages. II. Lattice congruences, arXiv:1911.12078 [math.CO], 2019.

Cf. A002620.

A[1]:= 0:
for n from 2 to 100 do
  if issqr(2*A[n-1]+1) or issqr(2*A[n-1]+2) then A[n]:= A[n-1]+1
  else A[n]:= A[n-1]+2
  fi
od:
seq(A[n],n=1..100); # Robert Israel, Oct 21 2014

FoldList[Plus, 0, t = Table[2, {72}]; t[[ Table[ Ceiling[n/2] Floor[n/2], {n, 2, 16}] ]]--; t] (* Robert G. Wilson v, Jan 19 2007 *)

a(n)=2*n - sqrtint(4*n-1) - 1 \\ Charles R Greathouse IV, Nov 29 2016

from math import isqrt
def A123663(n): return (m:=n<<1)-1-isqrt((m<<1)-1) # Chai Wah Wu, Jul 28 2022

a123663 = [0]; k = 0; a_n = 0; (1..N).to_a.each{ |i| 2.times{ k.times{ a_n += 2; a123663 << a_n }; a_n += 1; a123663 << a_n; }; k += 1}

a(n) = 2n - ceiling(2*sqrt(n)). - Julian F. Fleron, Nov 29 2016

a(n) = a(n-1) + 2 - [n-1 is a square or a pronic number], where [] stands for the Iverson bracket. - Luc Rousseau, Dec 28 2019

Extended by Robert G. Wilson v, Jan 19 2007

cp's OEIS Frontend

User: Zacariaz Martinez

A219330 Number of random selections (with replacement) needed from a normal population to assure a greater than one-half chance that the selected group contains the top 10th percentile individual, top 1st percentile individual, the 0.1 percentile, 0.01 percentile etc...

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A127314 Decimal expansion of number x satisfying log(x) = x/10.

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A128804 Decimal expansion of number x satisfying log_10(x) = x/100.

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A124507 a(n) = floor(exp(n*Pi/2)).

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A123663 Number of shared edges in a spiral of n unit squares.

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