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A052530 a(n) = 4*a(n-1) - a(n-2), with a(0) = 0, a(1) = 2. 27
0, 2, 8, 30, 112, 418, 1560, 5822, 21728, 81090, 302632, 1129438, 4215120, 15731042, 58709048, 219105150, 817711552, 3051741058, 11389252680, 42505269662, 158631825968, 592022034210, 2209456310872, 8245803209278, 30773756526240 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n-1) and a(n+1) are the solutions for c if b = a(n) in (b^2 + c^2)/(b*c + 1) = 4 and there are no other pairs of solutions apart from consecutive pairs of terms in this sequence. Cf. A061167. - Henry Bottomley, Apr 18 2001

a(n)^2 for n >= 1 gives solutions to A007913(3*x+4) = A007913(x). - Benoit Cloitre, Apr 07 2002

For all elements n of the sequence, 3*n^2 + 4 is a perfect square. Lim_{n->inf} a(n)/a(n-1) = 2 + sqrt(3). - Gregory V. Richardson, Oct 06 2002

a(n) = the number of compositions of the integer 2*n into even parts, where each part 2*i comes in 2*i colors. (Dedrickson, Theorem 3.2.6) An example is given below. Cf. A052529, A095263. - Peter Bala, Sep 17 2013

Except for an initial 1, this is the p-INVERT of (1, 1, 1, 1, 1, ...) for p(S) = 1 - 2*S - 2*S^2; see A291000.  - Clark Kimberling, Aug 24 2017

a(n+1) is the number of spanning trees of the graph P_n, where P_n is a 2 X n grid with two additional vertices, u and v, where u is adjacent to (1,1) and (2,1), and v is adjacent to (1,n) and (2,n). - Kevin Long, May 04 2018

LINKS

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

Andersen, K., Carbone, L. and Penta, D., Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.

Daniel Birmajer, Juan B. Gil, Michael D. Weiner, (an + b)-color compositions, arXiv:1707.07798 [math.CO], 2017.

C. R. Dedrickson III, Compositions, Bijections, and Enumerations Thesis, 2012, Jack N. Averitt College of Graduate Studies, Georgia Southern University.

J.-P. Ehrmann et al., Problem POLYA002, Integer pairs (x,y) for which (x^2+y^2)/(1+pxy) is an integer.

F. Goebel, A. A. Jagers, On a conjecture of Tutte concerning minimal tree numbers, J. Combin. Theory Ser. B 26 (1979), no. 3, 346-348. MR0535948 (80m:05064). [From N. J. A. Sloane, Feb 20 2012]

A. F. Horadam, Basic Properties of a Certain Generalized Sequence of Numbers, Fibonacci Quarterly, Vol. 3, No. 3, 1965, pp. 161-176.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 460

N. J. A. Sloane, Transforms

Index entries for linear recurrences with constant coefficients, signature (4,-1).

FORMULA

G.f.: 2*x/(1 - 4*x + x^2).

Invert transform of even numbers: a(n) = 2*Sum_{k=1..n} k*a(n-k). - Vladeta Jovovic, Apr 27 2001

From Gregory V. Richardson, Oct 06 2002: (Start)

a(n) = Sum(-(1/3)*(-1 + 2*alpha)*alpha^(-1 - n), alpha = root of (1 - 4*Z + Z^2),

a(n) = (((2+sqrt(3))^(n+1) -(2-sqrt(3))^(n+1)) -((2+sqrt(3))^(n) -(2 -sqrt(3))^(n)) +((2+sqrt(3))^(n-1) -(2-sqrt(3))^(n-1)))/(3*sqrt(3)). (End)

a(n) = A071954(n) - 2. - N. J. A. Sloane, Feb 20 2005

a(n) = (2*sinh(2n*arcsinh(1/sqrt(2))))/sqrt(3). - Herbert Kociemba, Apr 24 2008

a(n) = 2*A001353(n). - R. J. Mathar, Oct 26 2009

a(n) = ((3 - 2*sqrt(3))/3)*(2 - sqrt(3))^(n - 1) + ((3 + 2*sqrt(3))/3)*(2 + sqrt(3))^(n - 1). - Vincenzo Librandi, Nov 20 2010

a(n) = floor((2 + sqrt(3))^n/sqrt(3)). - Zak Seidov, Mar 31 2011

a(n) = ((2 + sqrt(3))^n - (2 - sqrt(3))^n)/sqrt(3). (See Horadam for construction.) - Johannes Boot, Jan 08 2012

a(n) = A217233(n) + A217233(n-1) with A217233(-1) = -1. - Bruno Berselli, Oct 01 2012

a(n) = A001835(n+1) - A001835(n). - Kevin Long, May 04 2018

E.g.f.: (exp((2 + sqrt(3))*x) - exp((2 - sqrt(3))*x))/sqrt(3). - Franck Maminirina Ramaharo, Nov 12 2018

EXAMPLE

Colored compositions. a(2) = 8: There are two compositions of 4 into even parts, namely 4 and 2 + 2. Using primes to indicate the coloring of parts, the 8 colored compositions are 4, 4', 4'', 4''', 2 + 2, 2 + 2', 2' + 2 and 2' + 2'. - Peter Bala, Sep 17 2013

MAPLE

spec := [S, {S=Sequence(Prod(Union(Z, Z), Sequence(Z), Sequence(Z)))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);

MATHEMATICA

p=1; c=2; a[0]=0; a[1]=c; a[n_]:=a[n]=p*c^2*a[n-1]-a[n-2]; Table[a[n], {n, 0, 20}]

NestList[2 # + Sqrt[4 + 3 #^2]&, 0, 200] (* Zak Seidov, Mar 31 2011 *)

LinearRecurrence[{4, -1}, {0, 2}, 25] (* T. D. Noe, Jan 09 2012 *)

PROG

(PARI): { polya002(p, c, m) = local(v, w, j, a); w=0; print1(w, ", "); v=c; print1(v, ", "); j=1; while(j<=m, a=p*c^2*v-w; print1(a, ", "); w=v; v=a; j++) } polya002(1, 2, 25)

(Haskell)

a052530 n = a052530_list !! n

a052530_list =

   0 : 2 : zipWith (-) (map (* 4) $ tail a052530_list) a052530_list

-- Reinhard Zumkeller, Sep 29 2011

(PARI) my(x='x+O('x^30)); concat([0], Vec(2*x/(1-4*x+x^2))) \\ G. C. Greubel, Feb 25 2019

(MAGMA) I:=[0, 2]; [n le 2 select I[n] else 4*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Feb 25 2019

(Sage) (2*x/(1-4*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 25 2019

CROSSREFS

Cf. A007913, A003699, A217233.

Cf. A052529, A095263.

Sequence in context: A230269 A278023 A077839 * A274798 A281949 A162551

Adjacent sequences:  A052527 A052528 A052529 * A052531 A052532 A052533

KEYWORD

nonn,easy,nice

AUTHOR

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

EXTENSIONS

More terms from James A. Sellers, Jun 06 2000

Edited by N. J. A. Sloane, Nov 11 2006

a(0) changed to 0 and entry revised accordingly by Max Alekseyev, Nov 15 2007

Signs in definition corrected by John W. Layman, Nov 20 2007

STATUS

approved

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Last modified May 26 02:35 EDT 2019. Contains 323579 sequences. (Running on oeis4.)