OFFSET
0,1
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).
FORMULA
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
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 *)
PROG
(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
(PARI) Vec((4+11*x-2*x^2-x^3)/(1+x^4-34*x^2) + O(x^50)) \\ Altug Alkan, Nov 15 2015
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
Helge Robitzsch (hrobi(AT)math.uni-goettingen.de), Jul 24 2000
EXTENSIONS
More terms from Alois P. Heinz, Jun 03 2009
STATUS
approved