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.

User: Tim Honeywill

Tim Honeywill's wiki page.

Tim Honeywill has authored 3 sequences.

A098438 Numbers k such that (30^k - 1)/29 is prime.

Original entry on oeis.org

2, 5, 11, 163, 569, 1789, 8447, 72871, 78857, 82883
Offset: 1

Author

Tim Honeywill, Jon Ingram, and Paul Boddington, Oct 26 2004

Keywords

Comments

No other terms < 10^5. - Robert Price

Crossrefs

Searching in the OEIS for 'repunit' gives many similar sequences.

Programs

  • Magma
    for i in [1..500] do if i mod 50 eq 0 then print "counter equals", counter; end if; if IsPrime(i) then n := 0; for j in [0..i-1] do n +:= 30^j; end for; if IsPrime(n) then print n; print i; end if; end if; end for;
    
  • Mathematica
    Do[If[PrimeQ[(30^n - 1)/29], Print[n]], {n, 1, 10000}] (* Ryan Propper, Jun 25 2005 *)
    Select[Prime[Range[100]],PrimeQ[(30^#-1)/29]&] (* Alexander Adamchuk, Feb 11 2007 *)
  • PARI
    is(n)=n=(30^n-1)/29;denominator(n)==1&&ispseudoprime(n) \\ Charles R Greathouse IV, Jul 01 2013

Extensions

a(5)-a(7), corresponding to probable primes, from Ryan Propper, Jun 25 2005
a(7) = 8447 was found by Richard Fischer in 2004. - Alexander Adamchuk, Feb 11 2007
Edited by N. J. A. Sloane Jan 25 2008 at the suggestion of Herman Jamke (hermanjamke(AT)fastmail.fm)
Edited by T. D. Noe, Oct 30 2008
a(8)-a(10) from Robert Price, Dec 10 2011

A080254 For n>3, a(n) is the number of elements in the Coxeter complex of type D_n (although the sequence starts at n=0. See comments below for precise explanation).

Original entry on oeis.org

1, 1, 9, 75, 865, 12483, 216113, 4364979, 100757313, 2616517443, 75496735057, 2396212835283, 82968104980961, 3112139513814243, 125716310807844081, 5441108944839913587, 251195548533025953409, 12321551453507301079683
Offset: 0

Author

Keywords

Comments

The sequence makes most sense when n>3. The values for a(2) and a(3) make sense if we regard D_2=A_1 x A_1 and D_3=A_3. The values for a(0) and a(1) have to be regarded as conventions and were included to give a nice recursive description. The corresponding sequence for type B is A080253. There one can find a worked example as well as a geometric interpretation.
Also, Eulerian D-polynomials (A066094) evaluated at 2. - Ralf Stephan, Apr 23 2004

References

  • Kenneth S. Brown, Buildings, Springer-Verlag, 1988

Crossrefs

Programs

  • Mathematica
    CoefficientList[Series[(2*x-E^x)/(E^(2*x)-2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 08 2013 *)

Formula

a(0)=a(1)=1. For n>1, a(n)=1 + sum('2^r*binomial(n, r)*a(n-r)', 'r'=1..n)
E.g.f: (2*x-exp(x))/(exp(2*x)-2) - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 14 2003
a(n) ~ n! * (sqrt(2)/log(2)-1)/2 * (2/log(2))^n. - Vaclav Kotesovec, Oct 08 2013

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 14 2003

A080253 a(n) is the number of elements in the Coxeter complex of type B_n (or C_n).

Original entry on oeis.org

1, 3, 17, 147, 1697, 24483, 423857, 8560947, 197613377, 5131725123, 148070287697, 4699645934547, 162723741209057, 6103779096411363, 246564971326084337, 10671541841672056947, 492664975795819140737, 24166020791610523843203
Offset: 0

Author

Paul Boddington and Tim Honeywill, Feb 10 2003

Keywords

Comments

There is a nice geometric interpretation. Let V be a Euclidean space containing a root system of type B_n. We can decompose V into a disjoint union of 'cells', a cell being simply a maximal connected subset C of V with the property that if C has nonempty intersection with the orthogonal complement of some root a, then C lies entirely within the orthogonal complement of a. a(n) is then the number of cells.
For example, if n=2 then we can take V=R^2 and the roots to be (1,0), (0,1), (1,1), (-1, -1) and their negatives. The 17 cells are as follows: the set containing the origin O; the eight "open" halflines radiating from O and containing a root (but not O); the eight connected components of V minus the union of the nine cells already described. The corresponding sequences for types A,D are A000670, A080254 respectively.
Also number of signed orders.

Examples

			a(2)=17 as follows. Let (W,S) be a Coxeter system of type B_2. By definition the elements of the associated complex are right cosets of "special parabolic subgroups". These are simply the subgroups generated by subsets of S. In our case they have orders 1,2,2,8 and hence have 8,4,4,1 cosets respectively, giving a total of 17.
		

References

  • Kenneth S. Brown, Buildings, Springer-Verlag, 1989.

Crossrefs

Programs

  • Maple
    A080253 := proc(n) option remember; local k; if n <1 then 1 else 1 + add(2^r*binomial(n,r)*A080253(n-r),r=1..n); fi; end; seq(A080253(n),n=0..30); # Detlef Pauly
  • Mathematica
    t[n_] := Sum[StirlingS2[n, k] k!, {k, 0, n}]; c[n_] := Sum[Binomial[n, k] 2^k t[k], {k, 0, n}]; Table[c[n], {n, 0, 100}] (* Emanuele Munarini, Oct 04 2012 *)
    CoefficientList[Series[E^x/(2-E^(2*x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 07 2015 *)
    Round@Table[(-1)^(n + 1) (PolyLog[-n, Sqrt[2]] - PolyLog[-n, -Sqrt[2]])/(2 Sqrt[2]), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 31 2015 *)
  • Maxima
    t(n):=sum(stirling2(n,k)*k!,k,0,n);
    c(n):=sum(binomial(n,k)*2^k*t(k),k,0,n);
    makelist(c(n),n,0,40); // Emanuele Munarini, Oct 04 2012
    
  • Sage
    def A080253(n):
        return add(A060187(n, k) << (n-k) for k in (0..n))
    [A080253(n) for n in (0..17)]  # Peter Luschny, Apr 26 2013

Formula

a(n) = 1 + Sum_{r=1..n} 2^r *binomial(n, r) *a(n-r).
E.g.f.: exp(x)/(2-exp(2*x)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 14 2003
a(n) = Sum_{t=0..n} binomial(n, t)*2^(n-t)*A000670(n-t). Fishburn 2001, p. 57.
a(n) = Sum_{k=0..n} Stirling2(n, k)*k!*A001333(k+1). - Vladeta Jovovic, Sep 28 2003
2*a(n) = Sum_{k>=0} (2*k+1)^n/2^k = 2^n*LerchPhi(1/2,-n,1/2). - Gerson Washiski Barbosa, May 11 2009, Dec 12 2010
An approximation formula can be derived from the latter, a(n) ~ (n!/(2*sqrt(2)))*(2/log(2))^(n+1), with relative errors approaching asymptotically zero as n increases. - Gerson Washiski Barbosa, Jun 26 2009
Half the row sums of triangle A154695. - Gerson Washiski Barbosa, Jun 26 2009
G.f.: 1 + x/G(0) where G(k) = 1 - x*3*(2*k+1) + x^2*(k+1)*(k+1)*(1-3^2)/G(k+1); (continued fraction due to Stieltjes). - Sergei N. Gladkovskii, Jan 11 2013
a(n) = Sum_{k = 0..n} A060187(n, k)*2^(n-k). - Peter Luschny, Apr 26 2013
G.f.: 1/Q(0), where Q(k) = 1 - 3*x*(2*k+1) - 8*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 28 2013
a(n) = log(2) * Integral_{x = 0..oo} (2*floor(x) + 1)^n * 2^(-x) dx. - Peter Bala, Feb 06 2015
From Vladimir Reshetnikov, Oct 31 2015: (Start)
a(n) = (-1)^(n+1)*(Li_{-n}(sqrt(2)) - Li_{-n}(-sqrt(2)))/(2*sqrt(2)), where Li_n(x) is the polylogarithm.
Li_{-n}(sqrt(2)) = (-1)^(n+1)*(2*A216794(n) + a(n)*sqrt(2)).
(End)

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 14 2003