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 1-5 of 5 results.

A102905 a(n) = A113655(Fibonacci(n+1)).

Original entry on oeis.org

3, 3, 2, 1, 5, 8, 15, 19, 36, 57, 89, 142, 233, 377, 612, 985, 1599, 2586, 4181, 6763, 10946, 17711, 28659, 46366, 75027, 121395, 196418, 317809, 514229, 832040, 1346271, 2178307, 3524580, 5702889, 9227465, 14930350, 24157817, 39088169
Offset: 0

Views

Author

Roger L. Bagula, Mar 16 2005

Keywords

Links

Crossrefs

Cf. A000045, A014017, A113655.

Programs

Formula

a(n) = f(Fibonacci(n+1)), where f(n) = n-2 if (n mod 3) = 0, f(n) = n+2 if (n mod 3) = 1, otherwise f(n) = n.
a(n) = A113655(Fibonacci(n+1)).
G.f.: (3-4*x^2-4*x^3+2*x^4+2*x^5+2*x^6-4*x^7-x^8+2*x^9) / ((1-x)*(1+x)*(1+x^2)*(1-x-x^2)*(1+x^4)). - Colin Barker, Dec 11 2012
a(n) = (1 + 3*(-1)^n)/4 + Fibonacci(n+1) + (3/2)*(-1)^floor(n/2) * (n mod 2) + A014017(n) + A014017(n-1) - A014017(n-2). - G. C. Greubel, Dec 09 2022

Extensions

Edited by G. C. Greubel, Dec 09 2022

A330396 Permutation of the nonnegative integers partitioned into triples [3*k+2, 3*k+1, 3*k] for k >= 0.

Original entry on oeis.org

2, 1, 0, 5, 4, 3, 8, 7, 6, 11, 10, 9, 14, 13, 12, 17, 16, 15, 20, 19, 18, 23, 22, 21, 26, 25, 24, 29, 28, 27, 32, 31, 30, 35, 34, 33, 38, 37, 36, 41, 40, 39, 44, 43, 42, 47, 46, 45, 50, 49, 48, 53, 52, 51, 56, 55, 54, 59, 58, 57, 62, 61, 60, 65, 64, 63, 68, 67, 66, 71, 70, 69, 74, 73, 72, 77, 76, 75, 80, 79, 78, 83, 82
Offset: 0

Views

Author

Guenther Schrack, Mar 03 2020

Keywords

Comments

Partition the nonnegative integer sequence into triples starting with (0,1,2); transpose the first and third elements of the triple, repeat for all triples.
A self-inverse sequence: a(a(n)) = n.
The sequence is an interleaving of A016789 with A016777 and with A008585, in that order.

Links

Crossrefs

Cf. A001477, A064429, A236348, A004482, A004483.
Fixed point sequence: A016777.
Relationships:
a(n) = a(n-1) - 1 + 6*A079978(n).
a(n) = 2*a(n-1) - a(n-2) + 6*A049347(n).
a(n) = A074066(n+2) - 2.
a(n) = A113655(n+1) - 1.

Programs

  • MATLAB
    a = zeros(1,10000);
w = (-1+sqrt(-3))/2;
fprintf('0 2\n');
for n = 1:10000
   a(n) = int64((3*n + 2*w^(2*n)*(w + 2) + 2*w^n*(1 - w))/3);
   fprintf('%i %i\n',n,a(n));
end

Formula

G.f.: (2 - x - x^2 + 3*x^3)/((x-1)^2*(1 + x + x^2)). [corrected by Georg Fischer, Apr 17 2020]
Linear recurrence: a(n) = a(n-1) + a(n-3) - a(n-4) for n > 4.
Simple recursion: a(n) = a(n-3) + 3 for n > 2 with a(0) = 2, a(1) = 1, a(2) = 0.
Negative domain: a(-n) = -(a(n-1) + 1).
Explicit formulas:
a(n) = n + 2 - 2*(n mod 3).
a(n) = 2 - n + 6*floor(n/3).
a(n) = n + 2*(w^(2*n)*(2 + w) + w^n*(1 - w))/3 where w = (-1 + sqrt(-3))/2.

A113778 Invert blocks of four in the sequence of natural numbers.

Original entry on oeis.org

4, 3, 2, 1, 8, 7, 6, 5, 12, 11, 10, 9, 16, 15, 14, 13, 20, 19, 18, 17, 24, 23, 22, 21, 28, 27, 26, 25, 32, 31, 30, 29, 36, 35, 34, 33, 40, 39, 38, 37, 44, 43, 42, 41, 48, 47, 46, 45, 52, 51, 50, 49, 56, 55, 54, 53, 60, 59, 58, 57, 64, 63, 62, 61, 68, 67, 66
Offset: 1

Views

Author

Zak Seidov, Jan 20 2006

Keywords

Links

Crossrefs

Cf. A113655.

Programs

  • Mathematica
    With[{k=4}, Table[k Floor[(n+k-1)/k]-Mod[n-1, k], {n, 1, 10k}]]
Reverse/@Partition[Range[100],4]//Flatten (* or *) LinearRecurrence[ {1,0,0,1,-1},{4,3,2,1,8},100] (* Harvey P. Dale, Mar 02 2020 *)

Formula

a(n) = k*floor((n+k-1)/k)-(n-1) mod k; k=4, n=1, 2, ...
a(n) = n-cos(n*pi)-2*sqrt(2)*cos((2*n+1)*pi/4). - Jaume Oliver Lafont, Dec 10 2008
G.f.: x*( 4-x-x^2-x^3+3*x^4 ) / ( (1+x)*(1+x^2)*(1-x)^2 ). - R. J. Mathar, Apr 02 2011

Extensions

More terms from Harvey P. Dale, Mar 02 2020

A165958 The digits on a number pad from lower right to upper left.

Original entry on oeis.org

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

Views

Author

Tyler Bourque (nuyashaki(AT)gmail.com), Oct 01 2009

Keywords

Comments

Similar to A113655. Used the same method as A114514.

A266084 Expansion of (5 - x - x^2 - x^3 - x^4 + 4*x^5)/( x^6 - x^5 - x + 1).

Original entry on oeis.org

5, 4, 3, 2, 1, 10, 9, 8, 7, 6, 15, 14, 13, 12, 11, 20, 19, 18, 17, 16, 25, 24, 23, 22, 21, 30, 29, 28, 27, 26, 35, 34, 33, 32, 31, 40, 39, 38, 37, 36, 45, 44, 43, 42, 41, 50, 49, 48, 47, 46, 55, 54, 53, 52, 51, 60, 59, 58, 57, 56, 65, 64, 63, 62, 61, 70
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 21 2015

Keywords

Comments

Invert blocks of five in the sequence of natural numbers.

Links

Crossrefs

Cf. A000027, A113655, A113778.

Programs

  • Magma
    [5+5*Floor(n/5)-n mod 5: n in [0..70]]; // Vincenzo Librandi, Dec 21 2015
  • Mathematica
    Table[5 + 5 Floor[n/5] - Mod[n, 5], {n, 0, 50}]
CoefficientList[Series[(5 - x - x^2 - x^3 - x^4 + 4 x^5)/(x^6 - x^5 - x + 1), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)
Reverse/@Partition[Range[80],5]//Flatten (* or *) LinearRecurrence[ {1,0,0,0,1,-1},{5,4,3,2,1,10},80] (* Harvey P. Dale, Sep 02 2016 *)
  • PARI
    a(n) = 5 + 5*(n\5) - (n % 5); \\ Michel Marcus, Dec 21 2015
  • PARI
    x='x+O('x^100); Vec((5-x-x^2-x^3-x^4+4*x^5)/(x^6-x^5-x+1)) \\ Altug Alkan, Dec 21 2015

Formula

G.f.: (5 - x - x^2 - x^3 - x^4 + 4*x^5)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6) for n>5.
a(n) = 5 + 5*floor(n/5) - n mod 5.
a(n) = n+1+2*A257145(n+3). - R. J. Mathar, Apr 12 2019
Showing 1-5 of 5 results.

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