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.

A015531 Linear 2nd order recurrence: a(n) = 4*a(n-1) + 5*a(n-2).

Original entry on oeis.org

0, 1, 4, 21, 104, 521, 2604, 13021, 65104, 325521, 1627604, 8138021, 40690104, 203450521, 1017252604, 5086263021, 25431315104, 127156575521, 635782877604, 3178914388021, 15894571940104, 79472859700521, 397364298502604, 1986821492513021, 9934107462565104
Offset: 0

Views

Author

Olivier Gérard

Keywords

Comments

Number of walks of length n between any two distinct vertices of the complete graph K_6. Example: a(2)=4 because the walks of length 2 between the vertices A and B of the complete graph ABCDEF are: ACB, ADB, AEB and AFB. - Emeric Deutsch, Apr 01 2004
General form: k=5^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-4, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=charpoly(A,1). - Milan Janjic, Jan 27 2010
Pisano period lengths: 1, 2, 6, 2, 2, 6, 6, 4, 18, 2, 10, 6, 4, 6, 6, 8, 16, 18, 18, 2,... - R. J. Mathar, Aug 10 2012
The ratio a(n+1)/a(n) converges to 5 as n approaches infinity. - Felix P. Muga II, Mar 09 2014
For odd n, a(n) is congruent to 1 (mod 10). For even n > 0, a(n) is congruent to 4 (mod 10). - Iain Fox, Dec 30 2017

Links

Crossrefs

A083425 shifted right.
Cf. A033115 (partial sums), A213128.

Programs

Formula

From Paul Barry, Apr 20 2003: (Start)
a(n) = (5^n -(-1)^n)/6.
G.f.: x/((1-5*x)*(1+x)).
E.g.f.(exp(5*x)-exp(-x))/6. (End) (corrected by M. F. Hasler, Jan 29 2012)
a(n) = Sum_{k=1..n} binomial(n, k)*(-1)^(n+k)*6^(k-1). - Paul Barry, May 13 2003
a(n) = 5^(n-1) - a(n-1). - Emeric Deutsch, Apr 01 2004
a(n) = ((2+sqrt(9))^n - (2-sqrt(9))^n)/6. - Al Hakanson (hawkuu(AT)gmail.com), Jan 07 2009
a(n) = round(5^n/6). - Mircea Merca, Dec 28 2010
The logarithmic generating function 1/6*log((1+x)/(1-5*x)) = x + 4*x^2/2 + 21*x^3/3 + 104*x^4/4 + ... has compositional inverse 6/(5+exp(-6*x)) - 1, the e.g.f. for a signed version of A213128. - Peter Bala, Jun 24 2012
a(n) = (-1)^(n-1)*Sum_{k=0..(n-1)} A135278(n-1,k)*(-6)^k = (5^n - (-1)^n)/6 = (-1)^(n-1)*Sum_{k=0..(n-1)} (-5)^k. Equals (-1)^(n-1)*Phi(n,-5) when n is an odd prime, where Phi is the cyclotomic polynomial. - Tom Copeland, Apr 14 2014

A052934 Expansion of (1-x)/(1-6*x).

Original entry on oeis.org

1, 5, 30, 180, 1080, 6480, 38880, 233280, 1399680, 8398080, 50388480, 302330880, 1813985280, 10883911680, 65303470080, 391820820480, 2350924922880, 14105549537280, 84633297223680, 507799783342080
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

With formula a(n) = (5*6^n + 0^n)/6, this is the binomial transform of A083425. - Paul Barry, Apr 30 2003
For n>=1, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5,6} such that for a fixed x in {1,2,...,n} and a fixed y in {1,2,3,4,5,6} we have f(x) != y. - Aleksandar M. Janjic and Milan Janjic, Mar 27 2007
a(n) = (n+1) terms in the sequence (1, 4, 5, 5, 5, ...) dot (n+1) terms in the sequence (1, 1, 5, 30, 180, 1080, ...). Example: a(4) = (1, 4, 5, 5, 5) dot (1, 1, 5, 30, 180) = (1 + 4 + 25 + 150 + 900), where (1, 4, 25, 150, ...) = first differences of current sequence. - Gary W. Adamson, Aug 03 2010
a(n) is the number of compositions of n when there are 5 types of each natural number. - Milan Janjic, Aug 13 2010

Links

Crossrefs

Cf. A083425.

Programs

  • GAP
    Concatenation([1], List([1..30], n-> 5*6^(n-1) )); # G. C. Greubel, Oct 18 2019
  • Magma
    [1] cat [5*6^(n-1): n in [1..30]]; // G. C. Greubel, Oct 18 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 22); Coefficients(R!( (1-x)/(1-6*x))); // Marius A. Burtea, Oct 18 2019
  • Maple
    spec := [S,{S=Sequence(Prod(Sequence(Z),Union(Z,Z,Z,Z,Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(`if`(n=0,1,5*6^(n-1)), n=0..30); # G. C. Greubel, Oct 18 2019
  • Mathematica
    Join[{1},NestList[6#&,5,20]] (* Harvey P. Dale, Nov 30 2015 *)
  • PARI
    vector(31, n, if(n==1,1, 5*6^(n-2))) \\ G. C. Greubel, Oct 18 2019
  • Sage
    [1]+[5*6^(n-1) for n in (1..30)] # G. C. Greubel, Oct 18 2019

Formula

a(n) = 6*a(n-1), n>=2.
a(n) = 5*6^(n-1), n>=1. - Vincenzo Librandi, Sep 15 2011
G.f.: (1-x)/(1-6*x).
G.f.: 1/(1 - 5*Sum_{k>=1} x^k).
E.g.f.: (1/6)*(1 + 5*exp(6*x)). - Stefano Spezia, Oct 18 2019

A033115 Base-5 digits are, in order, the first n terms of the periodic sequence with initial period 1,0.

Original entry on oeis.org

1, 5, 26, 130, 651, 3255, 16276, 81380, 406901, 2034505, 10172526, 50862630, 254313151, 1271565755, 6357828776, 31789143880, 158945719401, 794728597005, 3973642985026, 19868214925130, 99341074625651, 496705373128255
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

Partial sums of A015531. - Mircea Merca, Dec 28 2010

Links

Crossrefs

Cf. A015531.

Programs

  • Magma
    [Round((5*5^n-3)/24): n in [1..30]]; // Vincenzo Librandi, Jun 25 2011
  • Maple
    seq(1/3*floor(5^(n+1)/8),n=1..32); # Mircea Merca, Dec 26 2010
  • Mathematica
    Table[FromDigits[PadRight[{},n,{1,0}],5],{n,30}] (* or *) LinearRecurrence[ {5,1,-5},{1,5,26},30] (* Harvey P. Dale, Jan 28 2017 *)

Formula

a(n) = 5*a(n-1) + a(n-2) - 5*a(n-3). - Joerg Arndt, Jan 08 2011
From Paul Barry, Nov 12 2003: (Start)
a(n) = floor(5^(n+2)/24);
a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k);
a(n) = Sum_{k=0..n} Sum_{j=0..k} (-1)^(j+k)*5^j.
Partial sums of A083425.
G.f.: 1/((1-x)*(1+x)*(1-5*x));
a(n) = 4*a(n-1) + 5*a(n-2) + 1. (End)
From Mircea Merca, Dec 28 2010: (Start)
a(n) = (1/3)*floor(5^(n+1)/8) = floor((5*5^n - 1)/24) = round((5*5^n - 3)/24) = round((5*5^n - 5)/24) = ceiling((5*5^n - 5)/24);
a(n) = a(n-2) + 5^(n-1), n > 1. (End)
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.