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A015531
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Linear 2nd order recurrence: a(n) = 4 a(n-1) + 5 a(n-2).
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28
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0, 1, 4, 21, 104, 521, 2604, 13021, 65104, 325521, 1627604, 8138021, 40690104, 203450521, 1017252604, 5086263021, 25431315104, 127156575521, 635782877604, 3178914388021, 15894571940104, 79472859700521
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of walks of length n between any two distinct vertices of the complete graph K_6. Example: a(2)=4 because the walks of length 2 between the vertices A and B of the complete graph ABCDEF are: ACB, ADB, AEB and AFB. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2004
General form: k=5^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 11 2008]
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-4, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=charpoly(A,1). [From Milan R. Janjic (agnus(AT)blic.net), Jan 27 2010]
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to sequences with linear recurrences with constant coefficients, signature (4,5).
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FORMULA
| a(n)=5^n/6-(-1)^n/6. G.f.: x/((1-5*x)*(1+x)). E.g.f. (exp(5*x)-exp(-x))/6. - Paul Barry (pbarry(AT)wit.ie), Apr 20 2003; corrected by M. F. Hasler, Jan 29 2012
a(n)=sum{k=1..n, binomial(n, k)(-1)^(n+k)*6^(k-1) }. - Paul Barry (pbarry(AT)wit.ie), May 13 2003
a(n)=5^(n-1) - a(n-1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2004
a(n) = ((2+sqrt(9))^n-(2-sqrt(9))^n)/6. [From Al Hakanson (hawkuu(AT)gmail.com), Jan 07 2009]
a(n)=round(5^n/6). [From Mircea Merca, Dec 28 2010]
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MAPLE
| seq(round(5^n/6), n=0..25); [From Mircea Merca, Dec 28 2010]
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MATHEMATICA
| k=0; lst={k}; Do[k=5^n-k; AppendTo[lst, k], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 11 2008]
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PROG
| (Sage) [lucas_number1(n, 4, -5) for n in xrange(0, 22)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
(MAGMA) [Round(5^n/6): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011
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CROSSREFS
| A083425 shifted right.
Cf. A033115 (partial sums).
Sequence in context: A080043 A113022 A014986 * A083425 A183367 A100237
Adjacent sequences: A015528 A015529 A015530 * A015532 A015533 A015534
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KEYWORD
| nonn,easy
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AUTHOR
| Olivier Gerard (olivier.gerard(AT)gmail.com)
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