A055979 Solutions (value of r) of the Diophantine equation 2*x^2 + 3*x + 2 = r^2.

4, 11, 134, 373, 4552, 12671, 154634, 430441, 5253004, 14622323, 178447502, 496728541, 6061962064, 16874148071, 205928262674, 573224305873, 6995498968852, 19472752251611, 237641036678294, 661500352248901, 8072799748093144, 22471539224211023

Offset: 0

Author

Helge Robitzsch (hrobi(AT)math.uni-goettingen.de), Jul 24 2000

Keywords

• nonn
• easy
• nice

Comments

A necessary condition on any solution of the equation is x = [r/sqrt(2)] where [] denotes the floor function. The sequence lists the radii of circles for which a "best" digital approximation, as drawn by Bresenham-like algorithms, contains a point sequence [(x-1,x), (x,x), (x,x-1)] that is multiply connected by king moves. - clarified by M. Douglas McIlroy, May 18 2015

Corresponding values of x for above equation are given by A056161(n). The numbers a(n) are also solutions (value of r) to the Diophantine equation: 2x^2 - x + 1 = r^2, (excluding r = 1 at x = 0). - Richard R. Forberg, Nov 24 2013

This sequence lists the degrees n of those Chebyshev polynomials T(n,x) of the first kind which have the following exceptional property: There are exactly two coefficients in the power form of T(n,x) whose absolute values are identical and coincide with the height of T(n,x). This property is exceptional because for all remaining degrees n there is only one coefficient in the power form of T(n,x) whose absolute value coincides with the height of T(n,x). Recall that the height of a polynomial in power form is the maximum of the absolute value of its coefficients. Example: T(4,x) = 1 - 8x^2 + 8x^4; T(11,x) = - 11x + 220x^3 - 1232x^5 + 2816x^7 - 2816x^9 + 1024x^11. - Heinz-Joachim Rack, Nov 14 2015

References

• H.-J. Rack, On the length and height of Chebyshev polynomials in one and two variables, East Journal on Approximations, 16 (2010), 35 - 91. See Theorem 5.2.1, Remark (k), and Table 5.

Links

• Harvey P. Dale, 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.
• Zenon Kulpa, On the properties of discrete circles, rings, and disks, Computer Graphics and Image Processing, 10(1979), 348-365.
• M. D. McIlroy, Best approximate circles on integer grids, ACM Transactions on Graphics 2(1983), 237-263.
• Heinz-Joachim Rack, A comment on the Integer Sequence A055979
• Ville Salo, Subshifts with sparse traces, University of Turki, Finland (2019).
• Index entries for linear recurrences with constant coefficients, signature (0,34,0,-1).

Crossrefs

Cf. A006452.

Programs

Magma

I:=[4,11,134,373]; [n le 4 select I[n] else 34*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, May 19 2015

Maple

a:= n-> (Matrix([11,4,1,2]). Matrix([[0,1,0,0], [34,0,1,0], [0,0,0,1], [ -1,0,0,0]])^n)[1,2]: seq(a(n), n=0..25); # Alois P. Heinz, Jun 03 2009

Mathematica

LinearRecurrence[{0,34,0,-1},{4,11,134,373},20] (* Harvey P. Dale, Feb 21 2012 *)

PARI

Vec((4+11*x-2*x^2-x^3)/(1+x^4-34*x^2) + O(x^50)) \\ Altug Alkan, Nov 15 2015

Formula

a(n) = A006452(2n+3) if n=0, 2, 4, ... a(n) = A006452(2n+2) if n=1, 3, 5, ...

G.f.: (4+11*x-2*x^2-x^3) / (1+x^4-34*x^2). - Alois P. Heinz, Jun 03 2009

a(n) = 34*a(n-2) - a(n-4); a(0)=4, a(1)=11, a(2)=134, a(3)=373. - Harvey P. Dale, Feb 21 2012

Extensions

More terms from Alois P. Heinz, Jun 03 2009