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 A002535 a(n) = 2*a(n-1) + 9*a(n-2), with a(0)=a(1)=1. (Formerly M4786 N2043) 11
 1, 1, 11, 31, 161, 601, 2651, 10711, 45281, 186961, 781451, 3245551, 13524161, 56258281, 234234011, 974792551, 4057691201, 16888515361, 70296251531, 292589141311, 1217844546401, 5068991364601, 21098583646811, 87818089575031, 365523431971361, 1521409670118001, 6332530227978251 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. - Philippe Deléham, 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. - Milan Janjic, Aug 13 2010 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. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009. Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith [Annotated scanned copy] Index entries for linear recurrences with constant coefficients, signature (2,9). FORMULA From Paul Barry, May 16 2003: (Start) a(n) = ((1+sqrt(10))^n + (1-sqrt(10))^n)/2. G.f.: (1-x)/(1-2*x-9*x^2). E.g.f.: exp(x)*cosh(sqrt(10)*x). (End) a(n) = Sum_{k=0..n} A098158(n,k)*10^(n-k). - Philippe Deléham, 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. - Milan Janjic, Apr 29 2010 MAPLE A002535:=(-1+z)/(-1+2*z+9*z**2); # conjectured by Simon Plouffe in his 1992 dissertation MATHEMATICA Table[ MatrixPower[{{1, 2}, {5, 1}}, n][[1, 1]], {n, 0, 30}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *) a[n_] := Simplify[((1 + Sqrt[10])^n + (1 - Sqrt[10])^n)/2]; Array[a, 30, 0] (* Or *) CoefficientList[Series[(1+9x)/(1-2x-9x^2), {x, 0, 30}], x] (* Or *) LinearRecurrence[{2, 9}, {1, 1}, 30] (* Robert G. Wilson v, Sep 18 2013 *) PROG (Magma) [Ceiling((1+Sqrt(10))^n/2+(1-Sqrt(10))^n/2): n in [0..30]]; // Vincenzo Librandi, Aug 15 2011 (Magma) I:=[1, 1]; [n le 2 select I[n] else 2*Self(n-1)+9*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 02 2019 (PARI) my(x='x+O('x^30)); Vec((1-x)/(1-2*x-9*x^2)) \\ G. C. Greubel, Aug 02 2019 (PARI) my(p=Mod('x, 'x^2-2*'x-9)); a(n) = vecsum(Vec(lift((p^n)))); \\ Kevin Ryde, Jan 28 2023 (Sage) ((1-x)/(1-2*x-9*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 02 2019 (GAP) a:=[1, 1];; for n in [3..30] do a[n]:=2*a[n-1]+9*a[n-2]; od; a; # G. C. Greubel, Aug 02 2019 CROSSREFS Cf. A002534 (partial sums), A111015 (primes). Cf. A011557, A084132, A098158. Sequence in context: A190781 A001604 A144727 * A128337 A093382 A098264 Adjacent sequences: A002532 A002533 A002534 * A002536 A002537 A002538 KEYWORD nonn,easy AUTHOR N. J. A. Sloane STATUS approved

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Last modified December 5 14:39 EST 2023. Contains 367592 sequences. (Running on oeis4.)