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 1-4 of 4 results.

A166138 Trisection A022998(3n+1).

Original entry on oeis.org

1, 8, 7, 20, 13, 32, 19, 44, 25, 56, 31, 68, 37, 80, 43, 92, 49, 104, 55, 116, 61, 128, 67, 140, 73, 152, 79, 164, 85, 176, 91, 188, 97, 200, 103, 212, 109, 224, 115, 236, 121, 248, 127, 260, 133, 272, 139, 284, 145, 296, 151, 308, 157, 320, 163, 332, 169, 344, 175, 356, 181, 368, 187, 380, 193, 392
Offset: 0

Views

Author

Paul Curtz, Oct 08 2009

Keywords

Links

Crossrefs

Cf. A165988, A166304.

Programs

  • Mathematica
    LinearRecurrence[{0,2,0,-1},{1,8,7,20},70] (* Harvey P. Dale, Aug 15 2012 *)
Table[If[OddQ@ #, #, 2 #] &[3 n + 1], {n, 0, 65}] (* or *)
CoefficientList[Series[(1 + 8 x + 5 x^2 + 4 x^3)/((1 - x)^2 (1 + x)^2), {x, 0, 65}], x] (* Michael De Vlieger, Apr 27 2016 *)

Formula

a(2n) = 6n+1 = A016921(n).
a(2n+1) = 12n+8 = A017617(n).
a(n) = 2*a(n-2)-a(n-4) = (3n+1)*(3-(-1)^n)/2.
From G. C. Greubel, Apr 26 2016: (Start)
O.g.f.: (1 + 8*x + 5*x^2 + 4*x^3)/((1 - x)^2*(1 + x)^2).
E.g.f.: (1/2)*(-1 + 3*x + (3+9*x)*exp(2*x))*exp(-x). (End)

A166577 Inverse binomial transform of A166517.

Original entry on oeis.org

1, 4, -5, 10, -20, 40, -80, 160, -320, 640, -1280, 2560, -5120, 10240, -20480, 40960, -81920, 163840, -327680, 655360, -1310720, 2621440, -5242880, 10485760, -20971520, 41943040, -83886080, 167772160, -335544320, 671088640, -1342177280, 2684354560, -5368709120
Offset: 0

Views

Author

Paul Curtz, Oct 17 2009

Keywords

Comments

The definition assumes that the offset of A166517 is changed to 0.
A166517 mod 9 yields a periodic sequence with period 1, 5, 4, 8, 7, 2.
This set of numbers in the period appears also in A153130, A146501, and A166304.

Links

Crossrefs

Cf. A010716, A010692, A010859.

Programs

  • Mathematica
    Join[{1,4},NestList[-2#&,-5,40]] (* Harvey P. Dale, Aug 02 2012 *)
Join[{1, 4}, LinearRecurrence[{-2}, {-5}, 48]] (* G. C. Greubel, May 17 2016 *)

Formula

a(n) = -2*a(n-1), n>2.
a(n) = (-1)^(n+1)*A020714(n-2), n>1.
From Colin Barker, Jan 07 2013: (Start)
a(n) = -5*(-1)^n*2^(n-2) for n>1.
G.f.: (3*x^2+6*x+1)/(2*x+1). (End)
E.g.f.: (9/4) + (3/2)*x - (5/4)*exp(-2*x). - Alejandro J. Becerra Jr., Feb 15 2021

Extensions

Edited, comments not concerning this sequence removed, and extended by R. J. Mathar, Oct 21 2009

A177883 Period 6: repeat [4, 5, 7, 2, 1, 8].

Original entry on oeis.org

4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5
Offset: 0

Views

Author

Paul Curtz, Dec 14 2010

Keywords

Comments

Represents also the decimal expansion of 16934/37037 and the continued fractions of 0.23839... = (sqrt(496555)-667)/158 or of 4.194699... = (667+sqrt(496555))/327. - R. J. Mathar, Dec 20 2010

Links

Crossrefs

Cf. A173598, A141425, A153130 (permutations).

Programs

Formula

a(n) = A166304(n) mod 9 = A022998(3n+2) mod 9.
a(2n) + a(2n+1) = 9.
G.f.: (4+5*x+7*x^2+2*x^3+x^4+8*x^5) / ( (1-x)*(1+x)*(1+x+x^2)*(x^2-x+1) ). - R. J. Mathar, Dec 20 2010
From Wesley Ivan Hurt, Jun 18 2016: (Start)
a(n) = a(n-6) for n>5.
a(n) = (9 -cos(n*Pi) + 3*cos(n*Pi/3) - 3*cos(2*n*Pi/3) + sqrt(3)*sin(n*Pi/3) - 3*sqrt(3)*sin(2*n*Pi/3))/2. (End)

A281098 a(n) is the GCD of the sequence d(n) = A261327(k+n) - A261327(k) for all k.

Original entry on oeis.org

0, 1, 1, 3, 4, 1, 3, 1, 8, 3, 5, 1, 12, 1, 7, 3, 16, 1, 9, 1, 20, 3, 11, 1, 24, 1, 13, 3, 28, 1, 15, 1, 32, 3, 17, 1, 36, 1, 19, 3, 40, 1, 21, 1, 44, 3, 23, 1, 48, 1, 25, 3, 52, 1, 27, 1, 56, 3, 29, 1, 60, 1, 31, 3, 64, 1, 33, 1, 68, 3, 35, 1, 72, 1, 37, 3, 76, 1, 39, 1
Offset: 0

Views

Author

Paul Curtz, Jan 14 2017

Keywords

Comments

Successive sequences:
0: 0, 0, 0, 0, ... = 0 * ( )
1: 4, -3, 11, -8, ... = 1 * ( )
2: 1, 8, 3, 16, ... = 1 * ( ) A195161
3: 12, 0, 27, -3, ... = 3 * (4, 0, 9, -1, ...)
4: 4, 24, 8, 40, ... = 4 * (1, 6, 2, 10, ...) A064680
5; 28, 5, 51, 4, ... = 1 * ( )
6: 9, 48, 15, 72, ... = 3 * (3, 16, 5, 24, ...) A195161
7: 52, 12, 83, 13, ... = 1 * ( )
8: 16, 80, 24, 112, ... = 8 * (2, 10, 3, 14, ...) A064080
9: 84 21, 123, 24, ... = 3 * (28, 7, 41, 8, ...)
10: 25, 120, 35, 160, ... = 5 * (5, 24, 7, 32, ...) A195161

Links

Crossrefs

Cf. A064680, A144433 or A195161.
Cf. A000012, A005408, A008586, A010701, A109007 (bisection), A016825, A165988 (via A022998), A166138, A166304, A280579.

Programs

  • Mathematica
    CoefficientList[Series[(-x (-1 - x - 4 x^2 - 5 x^3 - 3 x^4 - 6 x^5 + 3 x^6 - 5 x^7 + 4 x^8 - x^9 + x^10))/((x^2 - x + 1) (1 + x + x^2) (x - 1)^2*(1 + x)^2*(1 + x^2)^2), {x, 0, 79}], x] (* Michael De Vlieger, Feb 02 2017 *)
  • PARI
    f(n) = numerator((4 + n^2)/4);
a(n) = gcd(vector(1000, k, f(k+n) - f(k))); \\ Michel Marcus, Jan 15 2017
  • PARI
    A281098(n) = if(n%2, gcd((n\2)-1,3), n>>(bitand(n,2)/2)); \\ Antti Karttunen, Feb 15 2023

Formula

G.f.: -x*( -1 - x - 4*x^2 - 5*x^3 - 3*x^4 - 6*x^5 + 3*x^6 - 5*x^7 + 4*x^8 - x^9 + x^10 )/( (x^2 - x + 1)*(1 + x + x^2)*(x - 1)^2*(1 + x)^2*(1 + x^2)^2 ). - R. J. Mathar, Jan 31 2017
a(2*k) = A022998(k).
a(2*k+1) = A109007(k-1).
a(3*k) = interleave 3*k*(3 +(-1)^k)/2, 3.
a(3*k+1) = interleave 1, A166304(k).
a(3*k+2) = interleave A166138(k), 1.
a(4*k) = 4*k.
a(4*k+1) = period 3: repeat [1, 1, 3].
a(4*k+2) = 1 + 2*k.
a(4*k+3) = period 3: repeat [3, 1, 1].
a(n+12) - a(n) = 6*A131743(n+3).
a(n) = (18*n + 40 - 16*cos(n*Pi/3) + 9*n*cos(n*Pi/2) + 32*cos(2*n*Pi/3) + (18*n - 40)*cos(n*Pi) + 3*n*cos(3*n*Pi/2) - 16*cos(5*n*Pi/3))/48. - Wesley Ivan Hurt, Oct 04 2018

Extensions

Corrected and extended by Michel Marcus, Jan 15 2017
Showing 1-4 of 4 results.

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