A003464 a(n) = (6^n - 1)/5. (Formerly M4425)

0, 1, 7, 43, 259, 1555, 9331, 55987, 335923, 2015539, 12093235, 72559411, 435356467, 2612138803, 15672832819, 94036996915, 564221981491, 3385331888947, 20311991333683, 121871948002099, 731231688012595, 4387390128075571

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.

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

Sequence in context: A271197 A240366 A329018 * A022036 A277670 A015451

Adjacent sequences: A003461 A003462 A003463 * A003465 A003466 A003467

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Reinhard Zumkeller, Nov 21 2006

G.f. corrected by Philippe Deléham, Mar 11 2014

Status

approved