cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 30 2013

Keywords

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

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

A247643 a(n) = ( 10*n*(n+1)+(2*n+1)*(-1)^n+7 )/8.

Original entry on oeis.org

1, 3, 9, 15, 27, 37, 55, 69, 93, 111, 141, 163, 199, 225, 267, 297, 345, 379, 433, 471, 531, 573, 639, 685, 757, 807, 885, 939, 1023, 1081, 1171, 1233, 1329, 1395, 1497, 1567, 1675, 1749, 1863, 1941, 2061, 2143, 2269, 2355, 2487, 2577, 2715, 2809, 2953, 3051
Offset: 0

Views

Author

N. J. A. Sloane, Sep 23 2014

Keywords

Comments

From Paul Curtz, Jan 01 2020: (Start)
In the following pentagonal spiral of odd numbers
101
99 61 63
97 59 31 33 65
95 57 29 11 13 35 67
93 55 27 9 1 3 15 37 69
91 53 25 7 5 17 39 71
89 51 23 21 19 41 73
87 49 47 45 43 75
85 83 81 79 77
the terms of this sequence appear on the x axis. A062786 and A172043 are in the spiral as well. (End)

Links

Crossrefs

A diagonal of triangle in A247646.
Cf. A005408, A062786, A085787, A090772, A172043.

Programs

  • Maple
    f:=n->(10*n*(n+1)+(2*n+1)*(-1)^n+7)/8;
  • Mathematica
    Table[(10 n (n + 1) + (2 n + 1) (-1)^n + 7)/8, {n, 0, 60}] (* Vincenzo Librandi, Sep 26 2014 *)
  • PARI
    Vec(-(x^4+2*x^3+4*x^2+2*x+1) / ((x-1)^3*(x+1)^2) + O(x^100)) \\ Colin Barker, Sep 25 2014

Formula

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Colin Barker, Sep 25 2014
G.f.: -(x^4+2*x^3+4*x^2+2*x+1) / ((x-1)^3*(x+1)^2). - Colin Barker, Sep 25 2014
From Paul Curtz, Jan 01 2020: (Start)
a(n) = 1 + 2*A085787(n).
a(n+1) = a(n-1) + A090772(n+1). (End)
E.g.f.: (1/4)*((1 + x)*(4 + 5*x)*cosh(x) + (3 + x*(11 + 5*x))*sinh(x)). - Stefano Spezia, Jan 01 2020

Extensions

More terms from Colin Barker, Sep 25 2014

A172193 a(n) = 5*n^2 + 31*n + 1.

Original entry on oeis.org

1, 37, 83, 139, 205, 281, 367, 463, 569, 685, 811, 947, 1093, 1249, 1415, 1591, 1777, 1973, 2179, 2395, 2621, 2857, 3103, 3359, 3625, 3901, 4187, 4483, 4789, 5105, 5431, 5767, 6113, 6469, 6835, 7211, 7597, 7993, 8399, 8815, 9241, 9677, 10123, 10579
Offset: 0

Views

Author

Vincenzo Librandi, Jan 29 2010

Keywords

Links

Crossrefs

Cf. A172043 (5*n^2-n+1).

Programs

  • Magma
    [ 5*n^2+31*n+1: n in [0..50] ];
  • Mathematica
    CoefficientList[Series[(1 +34x -25x^2)/(1-x)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Apr 07 2013 *)
LinearRecurrence[{3,-3,1},{1,37,83},50] (* Harvey P. Dale, May 16 2025 *)
  • PARI
    a(n)=5*n^2+31*n+1 \\ Charles R Greathouse IV, Jun 17 2017
  • SageMath
    [((10*n+31)^2 -941)/20 for n in (0..50)] # G. C. Greubel, Apr 28 2022

Formula

G.f.: (1+34*x-25*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Apr 07 2013
E.g.f.: (1 + 36*x + 5*x^2)*exp(x). - G. C. Greubel, Apr 28 2022

Extensions

Replaced definition with formula. - N. J. A. Sloane, Mar 03 2010
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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