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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Examples

			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
		

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.
• D. C. Santos, E. A. Costa, and P. M. M. C. Catarino, On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence, Axioms 14, 203, (2025). See p. 4.
• Eric Weisstein's World of Mathematics, Repunit.
• Index entries for linear recurrences with constant coefficients, signature (7,-6).

Programs

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

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 *)

Maxima

A003464(n):=floor((6^n-1)/5)$  makelist(A003464(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

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

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

Extensions

More terms from Reinhard Zumkeller, Nov 21 2006

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