OFFSET
0,3
COMMENTS
a(n) = A125118(n, 5) for n>4. - Reinhard Zumkeller, Nov 21 2006
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=6, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Feb 21 2010
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=7, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>1, a(n-1)=(-1)^n*charpoly(A,1). - Milan Janjic, Feb 21 2010
Repunits to base 6. A repunit consisting of zero 1's (empty string) gives the empty sum, i.e., 0 (only case where leading zero is shown, for convenience). - Daniel Forgues, Jul 08 2011
3*a(n) is the total number of holes in a certain triangle fractal (start with 6 triangles, 3 holes) after n iterations. See illustration in links. - Kival Ngaokrajang, Feb 21 2015
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 375
Kival Ngaokrajang, Illustration of initial terms
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Repunit.
Index entries for linear recurrences with constant coefficients, signature (7,-6).
FORMULA
Binomial transform of A003948. If preceded by 0, then binomial transform of powers of 5, A000351 (preceded by 0). - Paul Barry, Mar 28 2003
a(n) = Sum_{k=1..n} C(n, k)*5^(k-1).
E.g.f.: (exp(6*x) - exp(x))/5. - Paul Barry, Mar 28 2003
G.f.: x/((1-x)*(1-6*x)). - Lambert Klasen (lambert.klasen(AT)gmx.net), Feb 06 2005
a(n) = 6*a(n-1) + 1 with a(1)=1. - Vincenzo Librandi, Nov 17 2010
a(n) = 7*a(n-1) - 6*a(n-2). - Vincenzo Librandi, Nov 08 2012
EXAMPLE
a(n) in base 6.................... a(n) in base 10:
0..................................0
1..................................1
11.................................7
111................................43
1111...............................259
11111..............................1555
111111.............................9331
1111111............................55987, etc. - Philippe Deléham, Mar 12 2014
MAPLE
a:=n->sum(6^(n-j), j=1..n): seq(a(n), n=1..21); # Zerinvary Lajos, Jan 04 2007
A003464:=1/(6*z-1)/(z-1); # conjectured by Simon Plouffe in his 1992 dissertation
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=5*a[n-1]+6*a[n-2]+2 od: seq(a[n], n=1..33); # Zerinvary Lajos, Dec 14 2008
MATHEMATICA
(6^Range[20]-1)/5 (* Harvey P. Dale, Dec. 14, 2010 *)
LinearRecurrence[{7, -6}, {0, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)
PROG
(PARI) for(n=1, 10, print1((6^n-1)/5, ", "));
(Sage) [lucas_number1(n, 7, 6) for n in range(1, 22)] # Zerinvary Lajos, Apr 23 2009
(Sage) [gaussian_binomial(n, 1, 6) for n in range(1, 22)] # Zerinvary Lajos, May 28 2009
(Maxima) A003464(n):=floor((6^n-1)/5)$ makelist(A003464(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */
(Magma) [n le 2 select n-1 else 7*Self(n-1) - 6*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 08 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Reinhard Zumkeller, Nov 21 2006
G.f. corrected by Philippe Deléham, Mar 11 2014
STATUS
approved