%I #71 Sep 14 2024 13:31:04

%S 1,7,25,55,97,151,217,295,385,487,601,727,865,1015,1177,1351,1537,

%T 1735,1945,2167,2401,2647,2905,3175,3457,3751,4057,4375,4705,5047,

%U 5401,5767,6145,6535,6937,7351,7777,8215,8665,9127,9601,10087,10585,11095,11617,12151

%N a(n) = 6*n^2 + 1.

%C Least splitter is defined for x < y at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y. Conjecture: a(n) is the least splitter of s(n) and s(n+1), where s(n) = n*sin(1/n).

%H Clark Kimberling, <a href="/A227776/b227776.txt">Table of n, a(n) for n = 0..1000</a>

%H Leo Tavares, <a href="/A227776/a227776.jpg">Illustration: Hexagonal Star Rays</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

%F G.f.: (1 + 4*x + 7*x^2)/(1 - x)^3.

%F a(n) = A287326(2n, n). - _Kolosov Petro_, Nov 06 2017

%F From _Amiram Eldar_, Jul 15 2020: (Start)

%F Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(6))*coth(Pi/sqrt(6)))/2.

%F Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(6))*csch(Pi/sqrt(6)))/2. (End)

%F From _Amiram Eldar_, Feb 05 2021: (Start)

%F Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(6))*sinh(Pi/sqrt(3)).

%F Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(6))*csch(Pi/sqrt(6)).(End)

%F From _Leo Tavares_, Nov 20 2021: (Start)

%F a(n) = A003154(n+1) - A008458(n). See Hexagonal Star Rays illustration.

%F a(n) = A003215(n) + A028896(n-1).

%F a(n) = A054554(n+1) + A046092(n).

%F a(n) = A080855(n) + A045943(n).

%F a(n) = A172043(n) + A002378(n).

%F a(n) = A033581(n) + 1. (End)

%F E.g.f.: exp(x)*(1 + 6*x + 6*x^2). - _Stefano Spezia_, Sep 14 2024

%e The first eight least splitting rationals for {n*sin(1/n), n >=1 } are these fractions: 6/7, 24/25, 54/55, 96/97, 150/151, 216/217, 294/295, 384/385.

%t z = 40; r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; s[n_] := s[n] = n*Sin[1/n]; t = Table[r[s[n], s[n + 1]], {n, 1, z}] (* least splitting rationals *); fd = Denominator[t] (* _Peter J. C. Moses_, Jul 15 2013 *)

%t Array[6 #^2 + 1 &, 45] (* _Michael De Vlieger_, Nov 08 2017 *)

%t LinearRecurrence[{3,-3,1},{7,25,55},50] (* _Harvey P. Dale_, Dec 16 2017 *)

%o (PARI) a(n)=6*n^2+1 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A227631, A287326.

%Y Cf. A003154, A008458, A003215, A000217, A028896, A054554, A046092, A080855, A045943, A172043, A002378, A033581.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Jul 30 2013

%E a(0) = 1 prepended by _Robert P. P. McKone_, Oct 09 2023