A227776 a(n) = 6*n^2 + 1.

1, 7, 25, 55, 97, 151, 217, 295, 385, 487, 601, 727, 865, 1015, 1177, 1351, 1537, 1735, 1945, 2167, 2401, 2647, 2905, 3175, 3457, 3751, 4057, 4375, 4705, 5047, 5401, 5767, 6145, 6535, 6937, 7351, 7777, 8215, 8665, 9127, 9601, 10087, 10585, 11095, 11617, 12151

Offset: 0

Author

Clark Kimberling, Jul 30 2013

Keywords

• nonn
• easy

Comments

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).

Examples

			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.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000
• Leo Tavares, Illustration: Hexagonal Star Rays.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A227631, A287326.

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

Programs

Mathematica

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 *)
Array[6 #^2 + 1 &, 45] (* Michael De Vlieger, Nov 08 2017 *)
LinearRecurrence[{3,-3,1},{7,25,55},50] (* Harvey P. Dale, Dec 16 2017 *)

PARI

a(n)=6*n^2+1 \\ Charles R Greathouse IV, Jun 17 2017

Formula

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

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

a(n) = A287326(2n, n). - Kolosov Petro, Nov 06 2017

From Amiram Eldar, Jul 15 2020: (Start)

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

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

From Amiram Eldar, Feb 05 2021: (Start)

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

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

From Leo Tavares, Nov 20 2021: (Start)

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

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

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

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

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

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

E.g.f.: exp(x)*(1 + 6*x + 6*x^2). - Stefano Spezia, Sep 14 2024

Extensions

a(0) = 1 prepended by Robert P. P. McKone, Oct 09 2023