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.

A070403 a(n) = 7^n mod 9.

Original entry on oeis.org

1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4
Offset: 0

Views

Author

N. J. A. Sloane, May 12 2002

Keywords

Comments

Also the digital root of 7^n. If we convert this to a repeating decimal 0.174174..., we get the rational number 58/333. - Cino Hilliard, Dec 31 2004
A141722 (1, 25, 121, 505, 2041, 8185) mod 9. Note A141722 = 10*A000975(2n) + A000975(2n+1). - Paul Curtz, Sep 15 2008
Digital root of the powers of any number congruent to 7 mod 9. - Alonso del Arte, Jan 26 2014

References

  • Cecil Balmond, Number 9: The Search for the Sigma Code. Munich, New York: Prestel (1998): 203.

Links

Crossrefs

Cf. Digital roots of powers of c mod 9: c = 2, A153130; c = 4, A100402; c = 5, A070366; c = 8, A010689.
Cf. A000975, A010888, A049347, A099837, A141722.

Programs

Formula

From R. J. Mathar, Feb 23 2009: (Start)
G.f.: (1+7*x+4*x^2)/((1-x)*(1+x+x^2)).
a(n+1) - a(n) = 3*A099837(n+3).
a(n) = 4 - 3*A049347(n). (End)
a(n) = a(n-3) for n>3. - G. C. Greubel, Mar 19 2016
a(n) = 4-2*sqrt(3)*sin((2*n+2)*Pi/3). - Wesley Ivan Hurt, Jun 09 2016
a(n) = A010888(7*a(n-1)). - Stefano Spezia, Mar 20 2025

A101622 A Horadam-Jacobsthal sequence.

Original entry on oeis.org

0, 1, 6, 13, 30, 61, 126, 253, 510, 1021, 2046, 4093, 8190, 16381, 32766, 65533, 131070, 262141, 524286, 1048573, 2097150, 4194301, 8388606, 16777213, 33554430, 67108861, 134217726, 268435453, 536870910, 1073741821, 2147483646, 4294967293, 8589934590
Offset: 0

Views

Author

Paul Barry, Dec 10 2004

Keywords

Comments

Companion sequence to A084639.
This is the sequence A(0,1;1,2;5) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010
Except for the initial three terms, the decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 961", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Mar 27 2017
Named after the Australian mathematician Alwyn Francis Horadam (1923-2016) and the German mathematician Ernst Jacobsthal (1882-1965). - Amiram Eldar, Jun 10 2021

References

  • Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

Crossrefs

Cf. A131953.

Programs

  • Magma
    [(2^(n+2)+(-1)^n-5)/2: n in [0..35]]; // Vincenzo Librandi, Aug 12 2011
  • Mathematica
    LinearRecurrence[{2,1,-2},{0,1,6},40] (* Harvey P. Dale, Jul 08 2014 *)
  • PARI
    concat(0, Vec(x*(1+4*x)/((1-x)*(1+x)*(1-2*x)) + O(x^30))) \\ Colin Barker, Mar 28 2017

Formula

a(n) = (2^(n+2) + (-1)^n - 5)/2.
G.f.: x*(1+4*x)/((1-x)*(1+x)*(1-2*x)).
a(n) = (A014551(n+2)-5)/2.
(1, 6, 13, 30, 61, ...) are the row sums of A131953. - Gary W. Adamson, Jul 31 2007
From Paul Curtz, Jan 01 2009: (Start)
a(n) = a(n-1) + 2*a(n-2) + 5.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
a(n) = A000079(n+1) - A010693(n).
a(n+1) = A141722(n) + 5 = A141722(n) + A010716(n).
a(2n+1) - a(2n) = 1, 7, 31, ... = A083420.
a(2n+1) - 2*a(2n) = 1.
a(2n) = A002446 = 6*A002450, a(2n+1) = A141725. (End)
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2. - Colin Barker, Mar 28 2017
a(n) = (1/2) * Sum_{k=1..n} binomial(n+1,k) * (2+(-1)^k). - Wesley Ivan Hurt, Sep 23 2017

A267777 Binary representation of the n-th iteration of the "Rule 209" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 1, 11001, 1111001, 111111001, 11111111001, 1111111111001, 111111111111001, 11111111111111001, 1111111111111111001, 111111111111111111001, 11111111111111111111001, 1111111111111111111111001, 111111111111111111111111001, 11111111111111111111111111001
Offset: 0

Views

Author

Robert Price, Jan 20 2016

Keywords

Comments

Rule 241 also generates this sequence.

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Crossrefs

Cf. A267776, A141722.

Programs

  • Mathematica
    rule=209; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]]],{k,1,rows}]   (* Binary Representation of Rows *)

Formula

Conjectures from Colin Barker, Jan 20 2016 and Apr 20 2019: (Start)
a(n) = 101*a(n-1)-100*a(n-2) for n>2.
G.f.: (1-100*x+11000*x^2) / ((1-x)*(1-100*x)).
(End)

Extensions

Removed an unjustified claim that Colin Barker's conjectures are correct. Removed a program based on a conjecture. - Michael De Vlieger, Jun 13 2022
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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