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

%I A003464 M4425 #112 Jun 29 2025 21:39:58
%S A003464 0,1,7,43,259,1555,9331,55987,335923,2015539,12093235,72559411,
%T A003464 435356467,2612138803,15672832819,94036996915,564221981491,
%U A003464 3385331888947,20311991333683,121871948002099,731231688012595,4387390128075571
%N A003464 a(n) = (6^n - 1)/5.
%C A003464 a(n) = A125118(n, 5) for n>4. - _Reinhard Zumkeller_, Nov 21 2006
%C A003464 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
%C A003464 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
%C A003464 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
%C A003464 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
%D A003464 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003464 Vincenzo Librandi, <a href="/A003464/b003464.txt">Table of n, a(n) for n = 0..1000</a>
%H A003464 C. Banderier and D. Merlini, <a href="http://algo.inria.fr/banderier/Papers/infjumps.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02, Melbourne, 2002.
%H A003464 Carlos M. da Fonseca and Anthony G. Shannon, <a href="https://doi.org/10.7546/nntdm.2024.30.3.491-498">A formal operator involving Fermatian numbers</a>, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
%H A003464 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=375">Encyclopedia of Combinatorial Structures 375</a>
%H A003464 Kival Ngaokrajang, <a href="/A003464/a003464.pdf">Illustration of initial terms</a>
%H A003464 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A003464 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A003464 D. C. Santos, E. A. Costa, and P. M. M. C. Catarino, <a href="https://doi.org/10.3390/axioms14030203">On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence</a>, Axioms 14, 203, (2025). See p. 4.
%H A003464 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Repunit.html">Repunit.</a>
%H A003464 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-6).
%F A003464 Binomial transform of A003948. If preceded by 0, then binomial transform of powers of 5, A000351 (preceded by 0). - _Paul Barry_, Mar 28 2003
%F A003464 a(n) = Sum_{k=1..n} C(n, k)*5^(k-1).
%F A003464 E.g.f.: (exp(6*x) - exp(x))/5. - _Paul Barry_, Mar 28 2003
%F A003464 G.f.: x/((1-x)*(1-6*x)). - Lambert Klasen (lambert.klasen(AT)gmx.net), Feb 06 2005
%F A003464 a(n) = 6*a(n-1) + 1 with a(1)=1. - _Vincenzo Librandi_, Nov 17 2010
%F A003464 a(n) = 7*a(n-1) - 6*a(n-2). - _Vincenzo Librandi_, Nov 08 2012
%e A003464 a(n) in base 6.................... a(n) in base 10:
%e A003464 0..................................0
%e A003464 1..................................1
%e A003464 11.................................7
%e A003464 111................................43
%e A003464 1111...............................259
%e A003464 11111..............................1555
%e A003464 111111.............................9331
%e A003464 1111111............................55987, etc. - _Philippe Deléham_, Mar 12 2014
%p A003464 a:=n->sum(6^(n-j),j=1..n): seq(a(n), n=1..21); # _Zerinvary Lajos_, Jan 04 2007
%p A003464 A003464:=1/(6*z-1)/(z-1); # conjectured by _Simon Plouffe_ in his 1992 dissertation
%p A003464 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
%t A003464 (6^Range[20]-1)/5 (* _Harvey P. Dale_, Dec 14 2010 *)
%t A003464 LinearRecurrence[{7, -6}, {0, 1}, 30] (* _Vincenzo Librandi_, Nov 08 2012 *)
%o A003464 (PARI) for(n=1,10,print1((6^n-1)/5,","));
%o A003464 (Sage) [lucas_number1(n,7,6) for n in range(1, 22)] # _Zerinvary Lajos_, Apr 23 2009
%o A003464 (Sage) [gaussian_binomial(n,1,6) for n in range(1,22)] # _Zerinvary Lajos_, May 28 2009
%o A003464 (Maxima) A003464(n):=floor((6^n-1)/5)$  makelist(A003464(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */
%o A003464 (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
%K A003464 nonn,easy
%O A003464 0,3
%A A003464 _N. J. A. Sloane_
%E A003464 More terms from _Reinhard Zumkeller_, Nov 21 2006
%E A003464 G.f. corrected by _Philippe Deléham_, Mar 11 2014