A002535 a(n) = 2*a(n-1) + 9*a(n-2), with a(0)=a(1)=1. (Formerly M4786 N2043)

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

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