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A003464
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(6^n - 1)/5.
(Formerly M4425)
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23
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0, 1, 7, 43, 259, 1555, 9331, 55987, 335923, 2015539, 12093235, 72559411, 435356467, 2612138803, 15672832819, 94036996915, 564221981491, 3385331888947, 20311991333683, 121871948002099, 731231688012595, 4387390128075571
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n) = A125118(n,5) for n>4. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 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). [From Milan R. Janjic (agnus(AT)blic.net), 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). [From Milan R. Janjic (agnus(AT)blic.net), Feb 21 2010]
Repunits to base 6. A repunit consisting of zero 1s (empty string) gives the empty sum, i.e. 0 (only case where leading zero is shown, for convenience) - Daniel Forgues, Jul 08 2011
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REFERENCES
| N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 375
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
| Binomial transform of A003948. If preceded by 0, then binomial transform of powers of 5, A000351 (preceded by 0). - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003
a(n) = Sum{k=1..n, C(n, k)5^(k-1) }. E.g.f.: (exp(6x) - exp(x))/5 (offset 0). - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003
G.f.: 1/((1-1x)(1-6x)) - Lambert Klasen (lambert.klasen(AT)gmx.net), Feb 06 2005
a(n) = 6*a(n-1)+1 with a(1)=1. [From Vincenzo Librandi, Nov 17 2010]
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MAPLE
| a:=n->sum(6^(n-j), j=1..n): seq(a(n), n=1..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 04 2007
A003464:=1/(6*z-1)/(z-1); [Conjectured by S. 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); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 14 2008]
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MATHEMATICA
| lst={}; Do[p=(6^n-1)/5; AppendTo[lst, p], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 29 2008]
Alternate: (6^Range[20]-1)/5 [from Harvey P. Dale, Dec. 14, 2010]
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PROG
| (PARI) for(n=0, 10, print1(polcoeff(1/((1-1*x)*(1-6*x)), n), ", ")) for(n=1, 10, print1((6^n-1)/5, ", "))
(Other) sage: [lucas_number1(n, 7, 6) for n in xrange(1, 22)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
(Other) sage: [gaussian_binomial(n, 1, 6) for n in xrange(1, 22)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2009]
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CROSSREFS
| Sequence in context: A043553 A049609 A161728 * A022036 A015451 A194779
Adjacent sequences: A003461 A003462 A003463 * A003465 A003466 A003467
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 21 2006
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