Sharvil Kesarwani - cp's OEIS frontend

User: Sharvil Kesarwani

Sharvil Kesarwani has authored 2 sequences.

A355884 Number of circles in an n X n grid passing through at least three points.

0, 0, 1, 34, 223, 997, 3402, 9141, 21665, 46390, 90874, 167539, 293443, 487082, 781537, 1209469, 1816528, 2661113, 3822203, 5369662, 7420495, 10086360, 13494376

Offset: 0

Sharvil Kesarwani, Jul 20 2022

C. Lin, Number of circles in configuration, Mathematics Stack Exchange, 2013.

Cf. A192493, A192494, A357301.

\\ after user joriki's Java code at Mathematics Stack Exchange link
circles(n) = {
  my(C = List());
  for (x1 = 1, n,
    for (y1 = 1, n,
      for (x2 = 1, x1,
        for (y2 = 1, n,
          for (x3 = 1, x2,
            for (y3 = 1, n,
                my( ax2 = 2 * (x2 - x1),
                  ay2 = 2 * (y2 - y1),
                  ax3 = 2 * (x3 - x1),
                  ay3 = 2 * (y3 - y1),
                  den = ax2 * ay3 - ax3 * ay2
                );
              if (den == 0, next);
                my( b2 = x2^2 + y2^2 - x1^2 - y1^2,
                  b3 = x3^2 + y3^2 - x1^2 - y1^2,
                  x = b2 * ay3 - b3 * ay2,
                  y = ax2 * b3 - ax3 * b2,
                  gc = gcd(gcd(x, y), den)
                );
              if (den < 0, gc = -gc);
                x /= gc; y /= gc; den /= gc;
                my( dx = x - den * x1,
                  dy = y - den * y1,
                  s = dx^2 + dy^2
                );
              listput(C, [x, y, s, den])
  ))))));
  Set(C)
};
for (k = 0, 10, print1(#circles(k), ", "))  \\ Hugo Pfoertner, Sep 22 2022

A282473 Multiples of 9 which cannot be expressed as the difference between a natural number k and its digit sum s(k).

Original entry on oeis.org

90, 189, 288, 387, 486, 585, 684, 783, 882, 981, 990, 1089, 1188, 1287, 1386, 1485, 1584, 1683, 1782, 1881, 1980, 1989, 2088, 2187, 2286, 2385, 2484, 2583, 2682, 2781, 2880, 2979, 2988, 3087, 3186, 3285, 3384, 3483, 3582, 3681, 3780, 3879, 3978, 3987, 4086, 4185, 4284, 4383, 4482, 4581, 4680, 4779, 4878, 4977, 4986

Offset: 1

Sharvil Kesarwani, Feb 16 2017

Based on empirical observations, it can be noted that the difference between consecutive terms is usually 99. However, this breaks down when the hundreds digit is 9, in which case the difference between consecutive terms is 9. This changes back, however.

			If k=90, then k-s(k)=81. If k=100, then k-s(k)=99. This is an increasing function, so k-s(k)=90 is unachievable.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Brilliant.org, Those are some nifty numbers
Australian Mathematics Olympiad, Question 2, 2015, p. 84.

Cf. A003052, A176995.

okQ[n_] := Catch[ Do[ If[ x- Total@ IntegerDigits@ x == n, Throw@ False], {x, n, n+ 9 IntegerLength[n]}]; True]; Select[9 Range[1000], okQ[#] &] (* Giovanni Resta, Feb 27 2017 *)

is(n)=for(k=n,n+9*#Str(n)+9, if(k-sumdigits(k)==n, return(0))); n%9==0 \\ Charles R Greathouse IV, Feb 27 2017

from math import ceil
def a(n): #Outputs all numbers less than n which are in the sequence
    def s(n):
        r = 0
        while n:
            r, n = r + n% 10, n//10
        return r
    mult9=[]
    if n%9==0:
        for x in range(1, ceil(n/9)+1):
            mult9.append(9*x)
    else:
        for x in range(1, ceil(n/9)):
            mult9.append(9*x)
    for y in range(1, ceil(n/10)+1):
        mult9.remove(10*y-s(10*y))
    return mult9

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User: Sharvil Kesarwani

A355884 Number of circles in an n X n grid passing through at least three points.

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A282473 Multiples of 9 which cannot be expressed as the difference between a natural number k and its digit sum s(k).

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