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A083065
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4th row of number array A083064.
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7
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1, 4, 19, 94, 469, 2344, 11719, 58594, 292969, 1464844, 7324219, 36621094, 183105469, 915527344, 4577636719, 22888183594, 114440917969, 572204589844, 2861022949219, 14305114746094, 71525573730469, 357627868652344
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Inverse binomial transform of A090040=1,5,28,164,. With mentionned a(n)=5a(n-1)-1 also recurrence a(n)=6a(n-1)-5a(n-2). Linked to A131577=0,1,2,4,8,16, via submitted A154383,A154407 and A154410=10*A090040. [From Paul Curtz (bpcrtz(AT)free.fr), Jan 11 2009]
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-1)*charpoly(A,2). [From Milan R. Janjic (agnus(AT)blic.net), Feb 21 2010]
For an integer x, consider the sequence P(x) of polynomials p_{1}, p_{2}, p_{3}, . . . defined by p_{1} = x-1, p_{n+1} = x*p_{1} - 1. P(5) = This sequence. P(1), P(2), P(3), P(4) are A000004, A123412, A007051, A007583 respec. [From Kailasam Viswanathan Iyer (kvi(AT)nitt.edu), Jun 22 2010]
It appears that if s(n) is a first order rational sequence of the form s(0)=2, s(n)= (3*s(n-1)+2)/(2*s(n-1)+3),n>0, then s(n)=2*a(n)/(2*a(n)-1), n>0.
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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 (6,-5).
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FORMULA
| a(n)=(3*5^n+1)/4 G.f. (1-2x)/((1-5x)(1-x)) E.g.f. (3exp(5x) + exp(x))/4
a(n)=5*a(n-1)-1 (with a(0)=1) [Vincenzo Librandi, Aug 08 2010]
a(n) = 6*a(n-1)-5*a(n-2). - Vincenzo Librandi, Nov 04 2011
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EXAMPLE
| a(1)=5*1-1=4; a(2)=5*4-1=19; a(3)=5*19-1=94; a(4)5*94-1=469 [From Vincenzo Librandi, Aug 08 2010]
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MAPLE
| a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]*5-1 od: seq(a[n], n=1..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 22 2008
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MATHEMATICA
| f[n_]:=5^n; lst={}; Do[a=f[n]; Do[a-=f[m], {m, n-1, 1, -1}]; AppendTo[lst, a/5], {n, 1, 30}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 10 2010]
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PROG
| (MAGMA) [(3*5^n+1)/4: n in [0..30]]; // Vincenzo Librandi, Nov 04 2011
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CROSSREFS
| Cf. A007583, A083066.
Sequence in context: A131552 A122369 A005978 * A137636 A027618 A020060
Adjacent sequences: A083062 A083063 A083064 * A083066 A083067 A083068
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Apr 21 2003
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