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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 (list; graph; refs; listen; history; text; internal format)
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

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

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(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

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

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.uni-goettingen.de), Jul 24 2000

EXTENSIONS

More terms from Alois P. Heinz, Jun 03 2009

STATUS

approved

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Last modified December 10 18:10 EST 2019. Contains 329901 sequences. (Running on oeis4.)