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.

A188134 a(4*n) = n, a(1+2*n) = 4+8*n, a(2+4*n) = 2+4*n.

Original entry on oeis.org

0, 4, 2, 12, 1, 20, 6, 28, 2, 36, 10, 44, 3, 52, 14, 60, 4, 68, 18, 76, 5, 84, 22, 92, 6, 100, 26, 108, 7, 116, 30, 124, 8, 132, 34, 140, 9, 148, 38, 156, 10, 164, 42, 172, 11, 180, 46, 188, 12, 196, 50, 204, 13, 212, 54, 220, 14, 228, 58, 236, 15, 244, 62
Offset: 0

Views

Author

Paul Curtz, Mar 21 2011

Keywords

Links

Crossrefs

Cf. A016825, A017113, A060819, A061037, A064680, A146160, A176895, A177499.

Programs

  • Magma
    [(64-3*(1+(-1)^n)*(9+(-1)^(n div 2)))*n/16 : n in [0..80]]; // Wesley Ivan Hurt, Jul 06 2016
  • Maple
    A188134:=n->8*n/(11 + 9*cos(Pi*n) + 12*cos(n*Pi/2)): seq(A188134(n), n=0..100); # Wesley Ivan Hurt, Jul 06 2016
  • Mathematica
    Table[8 n/(11 + 9 Cos[Pi*n] + 12 Cos[n*Pi/2]), {n, 0, 80}] (* Wesley Ivan Hurt, Jul 06 2016 *)
CoefficientList[Series[x*(4+2*x+12*x^2+x^3+12*x^4+2*x^5+4*x^6)/(1-x^4)^2, {x, 0, 50}], x] (* G. C. Greubel, Sep 20 2018 *)
LinearRecurrence[{0,0,0,2,0,0,0,-1},{0,4,2,12,1,20,6,28},70] (* Harvey P. Dale, Aug 14 2019 *)
  • PARI
    x='x+O('x^50); concat([0], Vec(x*(4+2*x+12*x^2+x^3+12*x^4+ 2*x^5 +4*x^6)/(1-x^4)^2)) \\ G. C. Greubel, Sep 20 2018

Formula

a(n) = 2*a(n-4) - a(n-8) for n>7.
a(n) = A176895(n) * A060819(n).
a(n) = (4*A061037(n+2))/(n+4).
a(n) = 4*n / A146160(n).
a(2*n) = A064680(n).
a(1+2*n) = A017113(n).
a(4*n) = a(-4+4*n) + 1.
a(1+4*n) = a(-3+4*n) + 16.
a(2+4*n) = a(-2+4*n) + 4.
a(3+4*n) = a(-1+4*n) + 16. See A177499.
From Bruno Berselli, Mar 22 2011: (Start)
G.f.: x*(4+2*x+12*x^2+x^3+12*x^4+2*x^5+4*x^6)/(1-x^4)^2.
a(n) = (64-3*(1+(-1)^n)*(9+i^n))*n/16 with i=sqrt(-1).
a(n)/a(n-4) = n/(n-4) for n>4. (End)
a(n) = 8*n/(11 + 9*cos(Pi*n) + 12*cos(n*Pi/2)). - Wesley Ivan Hurt, Jul 06 2016
a(n) = lcm(4,n)/gcd(4,n). - R. J. Mathar, Feb 12 2019
Sum_{k=1..n} a(k) ~ (37/32)*n^2. - Amiram Eldar, Oct 07 2023

A177838 Decimal expansion of (44+sqrt(2442))/88.

Original entry on oeis.org

1, 0, 6, 1, 5, 5, 2, 2, 3, 1, 8, 4, 5, 7, 1, 9, 8, 4, 2, 9, 2, 7, 5, 3, 0, 6, 8, 6, 2, 5, 0, 2, 1, 8, 0, 2, 7, 5, 3, 5, 0, 0, 7, 5, 2, 2, 5, 0, 6, 1, 2, 3, 0, 9, 8, 7, 9, 9, 4, 0, 3, 8, 1, 9, 6, 2, 3, 0, 6, 3, 1, 1, 1, 0, 1, 4, 7, 2, 3, 1, 8, 3, 8, 9, 9, 6, 4, 9, 7, 6, 6, 2, 3, 2, 2, 6, 8, 9, 9, 0, 8, 6, 4, 3, 6
Offset: 1

Views

Author

Klaus Brockhaus, May 14 2010

Keywords

Comments

Continued fraction expansion of (44+sqrt(2442))/88 is A177499.

Examples

			(44+sqrt(2442))/88 = 1.06155223184571984292...

Links

Crossrefs

Cf. A177839 (decimal expansion of sqrt(2442)), A177499 (repeat 1, 16, 4, 16).

Programs

  • Mathematica
    RealDigits[(44+Sqrt[2442])/88, 10, 100][[1]] (* Georg Fischer, Jun 23 2020 *)

A226044 Period of length 8: 1, 64, 16, 64, 4, 64, 16, 64.

Original entry on oeis.org

1, 64, 16, 64, 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, 64
Offset: 0

Views

Author

Paul Curtz, May 24 2013

Keywords

Comments

A002378(n)/A016754(n) gives 0/1, 2/9, 6/25, 12/49, 20/81, 30/121, 42/169, 56/225,..., where A016754(n) = 4*A002378(n) + 1;
A142705(n)/A154615(n+1) gives 0/1, 3/16, 2/9, 15/64, 6/25, 35/144, 12/49, 63/256,..., where A142705(n) = 4*A154615(n+1) + A010685(n);
A061037(n)/A061038(n) gives 0/1, 5/36, 3/16, 21/100, 2/9, 45/196, 15/64, 77/324,..., where A061038(n) = 4*A061037(n) + A177499(n);
A225948(n)/A226008(n) gives 0/1, 9/100, 5/36, 33/196, 3/16, 65/324, 21/100, 105/484,..., where A226008(n) = 4*A225948(n) + a(n).
See also the triangle in Example lines.

Examples

			Triangle in which the terms of each line are repeated:
A000012: 1,   ...
A010685: 1,   4,  ...
A177499: 1,  16,  4,  16,  ...
A226044: 1,  64, 16,  64,  4,  64, 16,  64, ...
         1, 256, 64, 256, 16, 256, 64, 256, 4, 256, 64, 256, 16, 256, 64, 256, ...

Links

Crossrefs

Cf. A010685, A177499, A205383, A225948.

Programs

Formula

a(n) = A205383(n+7)^2.
G.f.: (1+64*x+16*x^2+64*x^3+4*x^4+64*x^5+16*x^6+64*x^7)/((1-x)*(1+x)*(1+x^2)*(1+x^4)). [Bruno Berselli, May 25 2013]

A177842 Period 27: repeat 1, 81, 81, 3, 81, 81, 9, 81, 81, 3, 81, 81, 3, 81, 81, 9, 81, 81, 3, 81, 81, 1, 81, 81, 9, 81, 81.

Original entry on oeis.org

1, 81, 81, 3, 81, 81, 9, 81, 81, 3, 81, 81, 3, 81, 81, 9, 81, 81, 3, 81, 81, 1, 81, 81, 9, 81, 81, 1, 81, 81, 3, 81, 81, 9, 81, 81, 3, 81, 81, 3, 81, 81, 9, 81, 81, 3, 81, 81, 1, 81, 81, 9, 81, 81, 1, 81, 81, 3, 81, 81, 9, 81, 81, 3, 81, 81, 3, 81, 81, 9, 81, 81, 3, 81, 81, 1, 81, 81
Offset: 0

Views

Author

Paul Curtz, May 14 2010

Keywords

Comments

The generating formula is a(n) = A061040(n+3) - 9*A061039(n+3). This is a member of the family of sequences with A000012(n) = A000290(n+1) -A005563(n+1), with period length 1, and A177499(n) = A061038(n+2) -4*A061037(n+2), with period length 4.
a(n) here has period length 3^3 and the general series of this family has period length k^k.

Links

Programs

  • PARI
    a(n)=3^[0, 4, 4, 1, 4, 4, 2, 4, 4, 1, 4, 4, 1, 4, 4, 2, 4, 4, 1, 4, 4, 0, 4, 4, 2, 4, 4][n%27+1] \\ Charles R Greathouse IV, Jul 17 2016

Formula

G.f.: ( -1 -81*x -3*x^9 -3*x^3 -81*x^4 -81*x^5 -9*x^6 -81*x^7 -81*x^8 -81*x^10 -3*x^12 -81*x^13 -81*x^14 -9*x^15 -81*x^16 -81*x^17 -3*x^18 -81*x^19 -81*x^20 -x^21 -81*x^22 -81*x^23 -9*x^24 -81*x^25 -81*x^26 -81*x^11 -81*x^2 ) / ( (x-1) *(1+x+x^2) *(1+x^3+x^6) *(1+x^9+x^18) ). - R. J. Mathar, Dec 09 2010
a(n) = a(n+27).
Showing 1-4 of 4 results.

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