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


%S 4,11,134,373,4552,12671,154634,430441,5253004,14622323,178447502,

%T 496728541,6061962064,16874148071,205928262674,573224305873,

%U 6995498968852,19472752251611,237641036678294,661500352248901,8072799748093144,22471539224211023

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

%C 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

%C 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

%C 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

%D 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.

%H Harvey P. Dale, <a href="/A055979/b055979.txt">Table of n, a(n) for n = 0..1000</a>

%H Zenon Kulpa, <a href="http://dx.doi.org/10.1016/S0146-664X(79)80043-X">On the properties of discrete circles, rings, and disks</a>, Computer Graphics and Image Processing, 10(1979), 348-365.

%H M. D. McIlroy, <a href="http://dx.doi.org/10.1145/245.246">Best approximate circles on integer grids</a>, ACM Transactions on Graphics 2(1983), 237-263.

%H Heinz-Joachim Rack, <a href="/A055979/a055979.rtf">A comment on the Integer Sequence A055979</a>

%H Ville Salo, <a href="http://www.villesalo.com/article/SwST.pdf">Subshifts with sparse traces</a>, University of Turki, Finland (2019).

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0, 34, 0, -1).

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

%F G.f.: (4+11*x-2*x^2-x^3) / (1+x^4-34*x^2). - _Alois P. Heinz_, Jun 03 2009

%F a(0)=4, a(1)=11, a(2)=134, a(3)=373, a(n)=34*a(n-2)-a(n-4). - _Harvey P. Dale_, Feb 21 2012

%p 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

%t LinearRecurrence[{0,34,0,-1},{4,11,134,373},20] (* _Harvey P. Dale_, Feb 21 2012 *)

%o (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

%o (PARI) Vec((4+11*x-2*x^2-x^3)/(1+x^4-34*x^2) + O(x^50)) \\ _Altug Alkan_, Nov 15 2015

%Y Cf. A006452.

%K nonn,easy,nice

%O 0,1

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

%E More terms from _Alois P. Heinz_, Jun 03 2009

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Last modified January 27 14:39 EST 2020. Contains 331295 sequences. (Running on oeis4.)