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A052530
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a(0)=0, a(1)=2; for n>=2, a(n)=4*a(n-1)-a(n-2).
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14
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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; internal format)
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OFFSET
| 0,2
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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 (se16(AT)btinternet.com), Apr 18 2001
a(n)^2 for n >= 1 gives solutions to A007913(3x+4)=A007913(x) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 07 2002
For all n we have: [[(a(n)]^2 + [(a(n+1)]^2]/[a(n)*a(n+1)+1]=4 [From Vincenzo Librandi, Nov 20 2010]
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
J.-P. Ehrmann et al., Problem POLYA002, Integer pairs (x,y) for which (x^2+y^2)/(1+pxy) is an integer.
A. F. Horadam, Basic properties of a certain generalized sequence of numbers, Fibonacci Quarterly 3 (1965), 161-176.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 460
N. J. A. Sloane, Transforms
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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 (vladeta(AT)eunet.rs), Apr 27 2001
a(n) = Sum(-(1/3)*(-1+2*_alpha)*_alpha^(-1-n), _alpha=RootOf(1-4*_Z+_Z^2)), i.e., a(n) = [ [(2+Sqrt(3)^n) - (2-Sqrt(3)^n)] - [(2+Sqrt(3)^(n-1)) - (2-Sqrt(3)^(n-1))] + [(2+Sqrt(3)^(n-2)) - (2-Sqrt(3)^(n-2))] ] / (3*Sqrt(3)) - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 06 2002
For all elements n of the sequence, 3*n^2 + 4 is a perfect square. Lim. a(n)/a(n-1) = 2 + Sqrt(3). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 06 2002
a(n) = A071954(n) - 2.
a(n) = (2*Sinh[2n*ArcSinh[1/Sqrt[2]]])/Sqrt[3] - Herbert Kociemba (kociemba(AT)t-online.de), Apr 24 2008
a(n)=2*A001353(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 26 2009]
a(n+2)=4a(n+1)-a(n); also for all n, a(n)=[(3-2*sqrt(3))/3]*[(2-sqrt(3)]^n + [(3+2*sqrt(3))/3]*[(2+sqrt(3)]^n [From 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
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MAPLE
| spec := [S, {S=Sequence(Prod(Union(Z, Z), Sequence(Z), Sequence(Z)))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
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MATHEMATICA
| a[0]=0; a[1] = c; a[n_] := a[n] = p*c^2*a[n - 1] - a[n - 2]; p = 1; c = 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 *)
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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
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CROSSREFS
| Cf. A007913, A003699.
Sequence in context: A127865 A199923 A077839 * A162551 A073663 A155116
Adjacent sequences: A052527 A052528 A052529 * A052531 A052532 A052533
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KEYWORD
| easy,nonn,nice
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AUTHOR
| encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 06 2000
Edited by N. J. A. Sloane (njas(AT)research.att.com), Nov 11 2006
Max Alekseyev (maxale(AT)gmail.com) changed a(0) to 0 and revised the entry accordingly, Nov 15 2007
Signs in definition corrected by John W. Layman, Nov 20 2007
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