

A055979


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


3



4, 11, 134, 373, 4552, 12671, 154634, 430441, 5253004, 14622323, 178447502, 496728541, 6061962064, 16874148071, 205928262674, 573224305873, 6995498968852, 19472752251611, 237641036678294, 661500352248901, 8072799748093144, 22471539224211023
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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 Bresenhamlike algorithms, contains a point sequence [(x1,x), (x,x), (x,x1)] 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.  HeinzJoachim 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
Zenon Kulpa, On the properties of discrete circles, rings, and disks, Computer Graphics and Image Processing, 10(1979), 348365.
M. D. McIlroy, Best approximate circles on integer grids, ACM Transactions on Graphics 2(1983), 237263.
HeinzJoachim 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

a(n) = A006452(2n+3) if n=0, 2, 4, ... a(n) = A006452(2n+2) if n=1, 3, 5, ...
G.f.: (4+11*x2*x^2x^3) / (1+x^434*x^2).  Alois P. Heinz, Jun 03 2009
a(0)=4, a(1)=11, a(2)=134, a(3)=373, a(n)=34*a(n2)a(n4).  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(n2)Self(n4): n in [1..30]]; // Vincenzo Librandi, May 19 2015
(PARI) Vec((4+11*x2*x^2x^3)/(1+x^434*x^2) + O(x^50)) \\ Altug Alkan, Nov 15 2015


CROSSREFS

Cf. A006452.
Sequence in context: A320501 A214113 A167418 * A018242 A006248 A119571
Adjacent sequences: A055976 A055977 A055978 * A055980 A055981 A055982


KEYWORD

nonn,easy,nice


AUTHOR

Helge Robitzsch (hrobi(AT)math.unigoettingen.de), Jul 24 2000


EXTENSIONS

More terms from Alois P. Heinz, Jun 03 2009


STATUS

approved



