A055979 -id:A055979 - cp's OEIS frontend

A056161 Solutions (value of x) of Diophantine equation 2x^2 + 3x + 2 = r^2.

2, 7, 94, 263, 3218, 8959, 109342, 304367, 3714434, 10339543, 126181438, 351240119, 4286454482, 11931824527, 145613270974, 405330793823, 4946564758658, 13769315165479, 168037588523422, 467751384832487, 5708331445037714, 15889777769139103, 193915231542758878

Offset: 0

Helge Robitzsch (hrobi(AT)math.uni-goettingen.de), Aug 01 2000

The same equation also has negative solutions x=-c(n), where c would be the sequence {1,2,17,46,553,1538,18761,52222,...} with the corresponding values of r being {1,2,23,64,781,2174,26531,73852,...}. Moreover, replacing x with x+K, one obtains the Diophantine equation 2*x^2+(4*K+3)*x+(2*K^2+3*K+2)=r^2. Since K can be any integer (for example K=-1, giving 2*x^2-x+1=r^2), this amounts to an infinite family of Diophantine equations with closely related solutions. For example, if the present equation has a solution pair {a(n), A055979(n)}, the one with x replaced by x+K will have a solution {a(n)-K, A055979(n)}.

Colin Barker, Table of n, a(n) for n = 0..1000
Seon-Hong Kim and Kenneth B. Stolarsky, Translations and extensions of the Nicomachus identity, arXiv:2306.17402 [math.NT], 2023. See also J. Int. Seq. (2024), Vol. 27, Issue 6, Art. No. 24.6.3, p. 12.
Index entries for linear recurrences with constant coefficients, signature (1,34,-34,-1,1).

Cf. A055979.

I:=[2,7,94,263,3218]; [n le 5 select I[n] else Self(n-1)+34*Self(n-2)-34*Self(n-3)-Self(n-4)+Self(n-5): n in [1..30]]; // Vincenzo Librandi, Jan 10 2016

a:= n-> (Matrix([94,7,2,-1,-2]). Matrix([[1,1,0,0,0], [34,0,1,0,0], [ -34,0,0,1,0], [ -1,0,0,0,1], [1,0,0,0,0]])^n)[1,3]: seq(a(n), n=0..25); # Alois P. Heinz, Jun 03 2009

CoefficientList[Series[(x^4 + x^3 - 19 x^2 - 5 x - 2)/(x^5 - x^4 - 34 x^3 + 34 x^2 + x - 1), {x, 0, 22}], x] (* Michael De Vlieger, Jan 09 2016 *)
LinearRecurrence[{1, 34, -34, -1, 1}, {2, 7, 94, 263, 3218}, 30] (* Vincenzo Librandi, Jan 10 2016 *)

Vec((x^4+x^3-19*x^2-5*x-2)/((x-1)*(x^2-6*x+1)*(x^2+6*x+1)) + O(x^100)) \\ Colin Barker, May 17 2015

a(n) = floor(A055979(n)/sqrt(2)).
G.f.: (x^4 + x^3 - 19*x^2 - 5*x - 2) / (x^5 - x^4 - 34*x^3 + 34*x^2 + x - 1). - Alois P. Heinz, Jun 03 2009
a(n) = a(n-1) + 34*a(n-2) - 34*a(n-3) - a(n-4) + a(n-5). - Colin Barker, May 17 2015

More terms from Alois P. Heinz, Jun 03 2009

A341198 Number of points on or inside the circle of radius n, as rasterized by the midpoint circle algorithm.

Original entry on oeis.org

1, 5, 21, 37, 61, 97, 129, 177, 221, 277, 349, 413, 489, 569, 657, 749, 845, 957, 1073, 1193, 1313, 1441, 1581, 1733, 1877, 2025, 2209, 2369, 2553, 2725, 2909, 3117, 3305, 3513, 3721, 3941, 4181, 4405, 4645, 4889, 5145, 5401, 5653, 5941, 6213, 6493, 6769, 7065

Offset: 0

Pontus von Brömssen, Feb 06 2021

nonn

The number of points on the rasterized circle itself (of radius n) is given by 4*A022846(n) for n > 0.

			In the figure below, the points on the rasterized circle of radius n are labeled with the number n. (Points without a label do not lie on any such circle.)
                9 9 9 9 9
            9 9 8 8 8 8 8 9 9
        9 9 8 8 7 7 7 7 7 8 8 9 9
      9 . 8 7 7 6 6 6 6 6 7 7 8 . 9
      9 8 7 . 6 5 5 5 5 5 6 . 7 8 9
    9 8 7 . 6 5 . 4 4 4 . 5 6 . 7 8 9
    9 8 7 6 5 4 4 3 3 3 4 4 5 6 7 8 9
  9 8 7 6 5 . 4 3 2 2 2 3 4 . 5 6 7 8 9
  9 8 7 6 5 4 3 2 . 1 . 2 3 4 5 6 7 8 9
  9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9
  9 8 7 6 5 4 3 2 . 1 . 2 3 4 5 6 7 8 9
  9 8 7 6 5 . 4 3 2 2 2 3 4 . 5 6 7 8 9
    9 8 7 6 5 4 4 3 3 3 4 4 5 6 7 8 9
    9 8 7 . 6 5 . 4 4 4 . 5 6 . 7 8 9
      9 8 7 . 6 5 5 5 5 5 6 . 7 8 9
      9 . 8 7 7 6 6 6 6 6 7 7 8 . 9
        9 9 8 8 7 7 7 7 7 8 8 9 9
            9 9 8 8 8 8 8 9 9
                9 9 9 9 9
Counting the points on or inside a circle of given radius, one obtains a(0)=1, a(1)=5, a(2)=21, a(3)=37, a(4)=61, a(5)=97, ...

Eric Weisstein's World of Mathematics, Gauss's Circle Problem
Wikipedia, Midpoint circle algorithm

First differences: A341199.

Cf. A022846, A055979, A057655, A373193.

def A341198(n):
  n2=n**2
  x=n
  y=A=0
  while y<=x:
    dx=x**2+(y+1)**2-n2-x>=0
    A+=x+(y!=0 and y!=x)*(x-2*y)+(dx and y==x-1)*(x-1)
    x-=dx
    y+=1
  return 4*A+1

a(n) == 1 (mod 4).

a(n) ~ Pi*n^2. More precisely, it is reasonable to expect that a(n) = Pi*n^2 + sqrt(8)*n + o(n), because there are Pi*n^2 + o(n) points in the disk x^2 + y^2 <= n^2 (Gauss's circle problem), all of which are inside the rasterized circle, and we can expect about half of the 4*sqrt(2)*n + O(1) points on the rasterized circle itself to be outside this disk (and there are no points between the disk and the rasterized circle).

A133217 Indices of decagonal numbers (A001107) that are also triangular (A000217).

Original entry on oeis.org

0, 1, 2, 20, 55, 667, 1856, 22646, 63037, 769285, 2141390, 26133032, 72744211, 887753791, 2471161772, 30157495850, 83946756025, 1024467105097, 2851718543066, 34801724077436, 96874483708207, 1182234151527715, 3290880727535960, 40161159427864862

Offset: 1

Richard Choulet, Oct 11 2007; Ant King, Nov 04 2011

nonn

For n>0, a(n) = (A055979(n) - A056161(n))/2, with those two sequences related through the Diophantine equation 2x^2 + 3x + 2 = r^2. - Richard R. Forberg, Nov 24 2013

			The third number which is both decagonal (A001107) and triangular (A000217) is A133216(3)=10. As this is the second decagonal number, we have a(3) = 2.

Index entries for linear recurrences with constant coefficients, signature (1, 34, -34, -1, 1).

Cf. A000217, A001107, A077443, A077442, A133216, A133218.

LinearRecurrence[{1, 34, -34, -1, 1} , {0, 1, 2, 20, 55, 667}, 24] (* first term 0 corrected by Georg Fischer, Apr 02 2019 *)

For n>5, a(n) = 34*a(n-2) - a(n-4) - 12.
For n>6, a(n) = a(n-1) + 34*a(n-2) - 34*a(n-3) - a(n-4) + a(n-5).
For n>1, a(n) = 1/16 * ((2*sqrt(2) + (-1)^n)*(1 + sqrt(2))^(2*n - 3) - (2*sqrt(2) - (-1)^n)*(1 - sqrt(2))^(2*n - 3) + 6).
For n>1, a(n) = ceiling (1/16*(2*sqrt(2) + (-1)^n)*(1 + sqrt(2))^(2*n - 3)).
G.f.: ( 1 - 33*x^2 + 18*x^3 + 2*x^4 ) / ((1 - x ) * (1 - 6*x + x^2 ) * (1 + 6*x + x^2)).
lim (n -> Infinity, a(2n+1)/a(2n)) = 1/7*(43 + 30*sqrt(2)).
lim (n -> Infinity, a(2n)/a(2n-1)) = 1/7*(11 + 6*sqrt(2)).

Entry revised by Max Alekseyev, Nov 06 2011

cp's OEIS Frontend

A056161 Solutions (value of x) of Diophantine equation 2x^2 + 3x + 2 = r^2.

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A341198 Number of points on or inside the circle of radius n, as rasterized by the midpoint circle algorithm.

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A133217 Indices of decagonal numbers (A001107) that are also triangular (A000217).

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cp's OEIS Frontend

A056161 Solutions (value of x) of Diophantine equation 2*x^2 + 3*x + 2 = r^2.

Views

Author

Keywords

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Programs

Magma

Maple

Mathematica

PARI

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Extensions

A341198 Number of points on or inside the circle of radius n, as rasterized by the midpoint circle algorithm.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula

A133217 Indices of decagonal numbers (A001107) that are also triangular (A000217).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A056161 Solutions (value of x) of Diophantine equation 2x^2 + 3x + 2 = r^2.