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A002535
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a(n) = 2*a(n-1) + 9*a(n-2).
(Formerly M4786 N2043)
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8
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1, 1, 11, 31, 161, 601, 2651, 10711, 45281, 186961, 781451, 3245551, 13524161, 56258281, 234234011, 974792551, 4057691201, 16888515361, 70296251531, 292589141311, 1217844546401, 5068991364601, 21098583646811, 87818089575031
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Contribution from Gary W. Adamson, Sep 06 2008: (Start)
Equals right border of triangle A143970.
Starting (1, 11, 31, 161,...) = row sums of triangle A143970 and INVERT transform of (1, 10, 10, 10,...). (End)
Binomial transform of [1, 0, 10, 0, 100, 0, 1000, 0, 10000, 0, ...]=: powers of 10 (A011557) with interpolated zeros . Inverse binomial transform of A084132 . [From Philippe DELEHAM, Dec 02 2008]
a(n) is the number of compositions of n when there are 1 type of 1 and 10 types of other natural numbers. [From Milan R. Janjic (agnus(AT)blic.net), Aug 13 2010]
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REFERENCES
| N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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.
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FORMULA
| a(n)=(1+sqrt(10))^n/2+(1-sqrt(10))^n/2; G.f.: (1-x)/(1-2*x-9*x^2); E.g.f.: exp(x)*cosh(sqrt(10)*x). - Paul Barry, May 16 2003
a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*10^(n-k). - Philippe DELEHAM, Dec 26 2007
If p[1]=1, and p[i]=10,(i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A [i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. [From Milan R. Janjic (agnus(AT)blic.net), Apr 29 2010]
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MAPLE
| A002535:=(-1+z)/(-1+2*z+9*z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| Table[MatrixPower[{{1, 2}, {5, 1}}, n][[1]][[1]], {n, 0, 44}] [From Vladimir Orlovsky, Feb 20 2010]
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PROG
| (MAGMA) [Ceiling((1+Sqrt(10))^n/2+(1-Sqrt(10))^n/2): n in [0..30]]; // Vincenzo Librandi, Aug 15 2011
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CROSSREFS
| A143970 [From Gary W. Adamson, Sep 06 2008]
Sequence in context: A193645 A001604 A144727 * A128337 A093382 A098264
Adjacent sequences: A002532 A002533 A002534 * A002536 A002537 A002538
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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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