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A006190 a(n) = 3*a(n-1) + a(n-2).
(Formerly M2844)
83
0, 1, 3, 10, 33, 109, 360, 1189, 3927, 12970, 42837, 141481, 467280, 1543321, 5097243, 16835050, 55602393, 183642229, 606529080, 2003229469, 6616217487, 21851881930, 72171863277, 238367471761, 787274278560, 2600190307441 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Denominators of continued fraction convergents to (3+Sqrt[13])/2. - Benoit Cloitre, Jun 14 2003

a(n) and A006497(n) occur in pairs: (a,b): (1,3), (3,11), (10,36), (33,119), (109,393), ... such that b^2 - 13a^2 = 4(-1)^n. - Gary W. Adamson, Jun 15 2003

Form the 4-node graph with matrix A=[1,1,1,1;1,1,1,0;1,1,0,1;1,0,1,1]. Then this sequence counts the walks of length n from the vertex with degree 5 to one (any) of the other vertices. - Paul Barry, Oct 02 2004

a(n+1) is the binomial transform of A006138. - Paul Barry, May 21 2006

a(n+1) is the diagonal sum of the exponential Riordan array (exp(3x),x). - Paul Barry, Jun 03 2006

Number of paths in the right half-plane from (0,0) to the line x=n-1, consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(3)=10 because we have hh, H, UD, DU, hU, Uh, UU, hD, Dh and DD. - Emeric Deutsch, Sep 03 2007

a(p) == 13^((p-1)/2)) mod p, for odd primes p. - Gary W. Adamson, Feb 22 2009

Equals INVERT transform of A000129. Example: a(5) = 109 = (29, 12, 5, 2, 1) dot (1, 1, 3, 10, 33) = (29 + 12 + 15 + 20 + 33). - Gary W. Adamson, Aug 06 2010

For n >=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 3's along the main diagonal, 1's along the superdiagonal and subdiagonal, and 0's everywhere else. - John M. Campbell, Jul 08 2011

These numbers could be named also "bronze Fibonacci numbers". Indeed, for n>=1, F(n+1)=ceil(phi*F(n)), if n is even and F(n+1)=floor(phi*F(n)), if n is odd, where phi is golden ratio; analogously, for Pell numbers (A000129), or "silver Fibonacci numbers",  P(n+1)=ceil(delta*a(n)), if n is even and P(n+1)=floor(delta*a(n)), if n is odd, where delta=delta_S=1+sqrt(2) is silver ratio. Here, for n>=1, we have a(n+1)=ceil(c*a(n)), if n is even and a(n+1)=floor(c*a(n)), if n is odd, where c=(3+sqrt(13))/2 is bronze ratio (cf. comment in A098316). - Vladimir Shevelev, Feb 23 2013

Let p(n,x) denote the Fibonacci polynomial, defined by p(1,x) = 1, p(2,x) = x, p(n,x) = x*p(n-1,x) + p(n-2,x).  Let q(n,x) be the numerator polynomial of the rational function p(n, x + 1 + 1/x).  Then q(n,1) = a(n). - Clark Kimberling, Nov 04 2013

The (1,1)-entry of the matrix A^n where A=[0,1,0;1,2,1;1,1,2]. - David Neil McGrath, Jul 18 2014

a(n+1) counts closed walks on K2, containing three loops on the other vertex. Equivalently the (1,1)-entry of A^(n+1) where the adjacency matrix of digraph is A=(0,1;1,3). - David Neil McGrath, Oct 29 2014

REFERENCES

H. L. Abbott and D. Hanson, A lattice path problem, Ars Combin., 6 (1978), 163-178.

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 128.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..200

Ayoub B. Ayoub, Fibonacci-like sequences and Pell equations, The College Mathematics Journal, Vol. 38 (2007), pp. 49-53.

A. F. Horadam, Generating identities for generalized Fibonacci and Lucas triples, Fib. Quart., 15 (1977), 289-292.

Haruo Hosoya, What Can Mathematical Chemistry Contribute to the Development of Mathematics?, HYLE--International Journal for Philosophy of Chemistry, Vol. 19, No.1 (2013), pp. 87-105.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 158

Tanya Khovanova, Recursive Sequences

W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1976), 318-328.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for sequences related to Chebyshev polynomials.

Index to sequences with linear recurrences with constant coefficients, signature (3,1).

FORMULA

G.f.: x/(1-3*x-x^2).

a(3n) = 2*A041019(5n-1), a(3n+1) = A041019(5n), a(3n+2) = A041019(5n+3).

a(2n) = 3*A004190(n-1); a(3n) = 10*A041613(n-1) for n>=1. - Benoit Cloitre, Jun 14 2003

a(n-1) + a(n+1) = A006497(n); [A006497(n)]^2 - 13[a(n)]^2 = 4(-1)^n. - Gary W. Adamson, Jun 15 2003

a(n) = U(n-1, (3/2)i)(-i)^(n-1), i^2=-1. - Paul Barry, Nov 19 2003

a(n) = sum{k=0..n-1, binomial(n-k-1,k)3^(n-2k-1)}. - Paul Barry, Oct 02 2004

a(n) = F(n, 3), the n-th Fibonacci polynomial evaluated at x=3.

Let M = {{0, 1}, {1, 3}}, v[1] = {0, 1}, v[n] = M.v[n - 1]; then a(n) = Abs[v[n][[1]]]. - Roger L. Bagula, May 29 2005. (Or a(n) = [M^(n+1)]_{1,1}. - L. Edson Jeffery, Aug 27 2013)

a(n+1) = sum{k=0..n, sum{j=0..n-k, C(k,j)C(n-j,k)*2^(k-j)}} = sum{k=0..n, sum{j=0..n-k, C(k,j)C(n-j,k)*2^(n-j-k)}}; a(n+1) = sum{k=0..floor(n/2), comb(n-k,k)3^(n-2k)}=sum{k=0..n, comb(k,n-k)3^(2k-n)}. - Paul Barry, May 21 2006

E.g.f.: exp(3x/2)*sinh(sqrt(13)x/2)/(sqrt(13)/2). - Paul Barry, Jun 03 2006

a(n) = (ap^n-am^n)/(ap-am), with ap = (3+sqrt(13))/2, am = (3-sqrt(13))/2.

Let C = (3+sqrt(13))/2 = exp arcsinh(3/2) = 3.3027756377... Then C^n, n>0 = a(n)*(1/C) + a(n+1). Example: C^3 = 36.02775637... = a(3)*(1/C) + a(4) = 10*(.302775637...) + 33. Let X = the 2 X 2 matrix [0, 1; 1, 3]. Then X^n = [a(n-1), a(n); a(n), a(n+1)]. - Gary W. Adamson, Dec 21 2007

1/3 = 3/(1*10) + 3/(3*33) + 3/(10*109) + 3/(33*360) + 3/(109*1189) + ... . - Gary W. Adamson, Mar 16 2008

a(n) = ((3 + Sqrt[3^2 + 4])/2)^(n) - ((3 - sqrt[3^2 + 4])/2)^(n))/(sqrt[3^2 + 4]). - Artur Jasinski, Oct 07 2008

a(n) = ((3+sqrt(13))^n-(3-sqrt(13))^n)/(2^n*sqrt(13)). - Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009

From Johannes W. Meijer, Jun 12 2010: (Start)

Limit(a(n+k)/a(k), k=infinity) = (A006497(n) + a(n)*sqrt(13))/2.

Limit(A006497(n)/a(n), n=infinity) = sqrt(13).

(End)

Sum_{k>=1} (-1)^(k-1)/(a(k)*a(k+1)) = (sqrt(13)-3)/2. - Vladimir Shevelev, Feb 23 2013

From Vladimir Shevelev, Feb 24 2013: (Start)

(1) Expression a(n+1) via a(n): a(n+1) = (3*a(n) + sqrt(13*a^2(n) + 4*(-1)^n)/2;

(2) a^2(n+1) - a(n)*a(n+2) = (-1)^n;

(3) sum_{k=1,...,n} (-1)^(k-1)/(a(k)*a(k+1)) = a(n)/a(n+1);

(4) a(n)/a(n+1) = (sqrt(13)-3)/2 + r(n), where |r(n)| < 1/(a(n+1)*a(n+2)). (End)

a(n) = sqrt(13*(A006497(n))^2 + (-1)^(n-1)*52)/13. - Vladimir Shevelev, Mar 13 2013

MAPLE

a[0]:=0: a[1]:=1: for n from 2 to 25 do a[n]:= 3*a[n-1]+a[n-2] end do: seq(a[n], n=0..25); # Emeric Deutsch, Sep 03 2007

A006190:=-1/(-1+3*z+z**2); # Simon Plouffe in his 1992 dissertation, without the leading 0

seq(combinat[fibonacci](n, 3), n=0..20); # R. J. Mathar, Dec 07 2011

MATHEMATICA

a[n_] := (MatrixPower[{{1, 3}, {1, 2}}, n].{{1}, {1}})[[2, 1]]; Table[ a[n], {n, -1, 24}] (* Robert G. Wilson v, Jan 13 2005 *)

m = 3; Table[Simplify[(((m + Sqrt[m^2 + 4])/2)^(n) - ((m - Sqrt[m^2 + 4])/2)^(n))/(Sqrt[m^2 + 4])], {n, 0, 20}] (* Artur Jasinski, Oct 07 2008 *)

a=0; lst={a}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*3, {n, 50}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)

LinearRecurrence[{3, 1}, {0, 1}, 40] (* or *) CoefficientList[Series[x/ (1-3x-x^2), {x, 0, 39}], x] (* Harvey P. Dale, Apr 20 2011 *)

Table[If[n==0, a1=1; a0=0, a2=a1; a1=a0; a0=3*a1+a2], {n, 0, 25}] (* Jean-François Alcover, Apr 30 2013 *)

PROG

(PARI) a(n)=if(n<1, 0, contfracpnqn(vector(n, i, 2+(i>1)))[2, 1])

(PARI) a(n)=([1, 3; 1, 2]^n)[2, 1] \\ Charles R Greathouse IV, Mar 06 2014

(Sage) [lucas_number1(n, 3, -1) for n in xrange(0, 26)] # Zerinvary Lajos, Apr 22 2009

(MAGMA)[ n eq 1 select 0 else n eq 2 select 1 else 3*Self(n-1)+Self(n-2): n in [1..40] ]; // Vincenzo Librandi, Aug 19 2011

(Haskell)

a006190 n = a006190_list !! n

a006190_list = 0 : 1 : zipWith (+) (map (* 3) $ tail a006190_list) a006190_list

-- Reinhard Zumkeller, Feb 19 2011

(PARI) concat([0], Vec(x/(1-3*x-x^2)+O(x^66))) \\ Joerg Arndt, Apr 30 2013

CROSSREFS

Row sums of Pascal's rhombus (A059317). Also row sums of triangle A054456(n, m). Cf. A000045, A000129, A001076.

Cf. A006497, A052906, A175182 (Pisano periods), A201001 (prime subsequence).

Cf. A243399.

Sequence in context: A126184 A060557 A018920 * A020704 A113299 A126931

Adjacent sequences:  A006187 A006188 A006189 * A006191 A006192 A006193

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Apr 17 2000

Second formula corrected by Johannes W. Meijer, Jun 02 2010

STATUS

approved

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Last modified November 22 03:54 EST 2014. Contains 249804 sequences.