This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A281065 #32 Feb 25 2025 04:03:57
%S A281065 4,2,1,2,7,9,5,4,3,9,8,3,9,0,3,4,3,2,7,6,8,8,2,1,7,6,0,6,5,0,2,9,8,0,
%T A281065 9,1,6,1,0,3,6,7,2,1,4,0,7,2,6,1,2,2,3,2,1,6,5,4,3,7,5,4,5,4,0,6,5,1,
%U A281065 7,2,9,3,9,2,2,4,3,7,7,9,1,5,3,6,3,2,9,0,6,8,8,4,7,1,9,2,4,6,2,4,3,9
%N A281065 Decimal expansion of the greatest minimum separation between ten points in a unit square.
%C A281065 The corresponding values for two to nine points have simple expressions:
%C A281065 N ... d_min
%C A281065 2 ... sqrt(2) (A002193)
%C A281065 3 ... sqrt(6) - sqrt(2) (A120683)
%C A281065 4 ... 1 (A000007)
%C A281065 5 ... sqrt(2) / 2 (A010503)
%C A281065 6 ... sqrt(13) / 6 (A381485)
%C A281065 7 ... 4 - 2*sqrt(3) (A379338)
%C A281065 8 ... sqrt(2 - sqrt(3)) (A101263)
%C A281065 9 ... 1 / 2 (A020761)
%C A281065 In contrast, the value for ten points has a minimal polynomial of degree 18.
%C A281065 The smallest square ten unit circles will fit into has side length s = 2 + 2/d = 6.74744152... and the maximum radius of ten non-overlapping circles in the unit square is 1 / s = 0.14820432...
%D A281065 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.
%H A281065 C. de Groot, R. Peikert, and D. Würtz, <a href="https://www.researchgate.net/publication/228327432_The_Optimal_Packing_of_Ten_Equal_Circles_in_a_Square">The Optimal Packing of Ten Equal Circles in a Square</a>, IPS Research Report, ETH Zürich, No. 90-12, August 1990.
%H A281065 Eckard Specht, <a href="http://hydra.nat.uni-magdeburg.de/packing/csq/">The best known packings of equal circles in a square</a>.
%H A281065 Jeremy Tan, <a href="https://gist.github.com/Parclytaxel/fe51678ea07cc448c56c3927afc44ac1">Sympy (Python) program</a>.
%H A281065 <a href="/index/Al#algebraic_18">Index entries for algebraic numbers, degree 18</a>.
%F A281065 d is the smallest real root of 1180129*d^18 - 11436428*d^17 + 98015844*d^16 - 462103584*d^15 + 1145811528*d^14 - 1398966480*d^13 + 227573920*d^12 + 1526909568*d^11 - 1038261808*d^10 - 2960321792*d^9 + 7803109440*d^8 - 9722063488*d^7 + 7918461504*d^6 - 4564076288*d^5 + 1899131648*d^4 - 563649536*d^3 + 114038784*d^2 - 14172160*d + 819200.
%e A281065 0.421279543983903432768821760650298...
%t A281065 RealDigits[x /. FindRoot[x^Range[18, 0, -1].{1180129, -11436428, 98015844, -462103584, 1145811528, -1398966480, 227573920, 1526909568, -1038261808, -2960321792, 7803109440, -9722063488, 7918461504, -4564076288, 1899131648, -563649536, 114038784, -14172160, 819200}, {x, 2/5}, WorkingPrecision -> 120]][[1]] (* _Amiram Eldar_, Feb 24 2025 *)
%o A281065 (PARI) my(p = Pol([1180129, -11436428, 98015844, -462103584, 1145811528, -1398966480, 227573920, 1526909568, -1038261808, -2960321792, 7803109440, -9722063488, 7918461504, -4564076288, 1899131648, -563649536, 114038784, -14172160, 819200])); polrootsreal(p)[1]
%Y A281065 Cf. A281115 (10 points in unit circle), A000007, A002193, A010503, A020761, A101263, A120683, A379338, A381485.
%K A281065 nonn,cons
%O A281065 0,1
%A A281065 _Jeremy Tan_, Jan 14 2017