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A052542
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a(0) = 1, a(1) = 2, a(2) = 4; for n>=3, a(n) = 2a(n-1) + a(n-2).
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15
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1, 2, 4, 10, 24, 58, 140, 338, 816, 1970, 4756, 11482, 27720, 66922, 161564, 390050, 941664, 2273378, 5488420, 13250218, 31988856, 77227930, 186444716, 450117362, 1086679440, 2623476242, 6333631924, 15290740090, 36915112104, 89120964298
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Apart from the initial 1, this sequence is simply twice the Pell numbers, A000129. - Antonio Alberto Olivares (tonioolivares(AT)todito.com), Dec 31 2003
Image of 1/(1-2x) under the mapping g(x)->g(x/(1+x^2)). - Paul Barry (pbarry(AT)wit.ie), Jan 16 2005
The intermediate convergents to 2^(1/2) begin with 4/3, 10/7, 24/17, 58/41; essentially, numerators=A052542 and denominators=A001333. - Clark Kimberling (ck6(AT)evansville.edu), Aug 26 2008
a(n) is the number of generalized compositions of n+1 when there are 2*i-2 different types of i, (i=1,2,...). [From Milan R. Janjic (agnus(AT)blic.net), Aug 26 2010]
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LINKS
| INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 477
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
Index to sequences with linear recurrences with constant coefficients, signature (2,1).
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FORMULA
| G.f.: (-1+x^2)/(-1+2*x+x^2)
Recurrence: {a(0)=1, a(2)=4, a(1)=2, a(n)+2*a(n+1)-a(n+2)}
Sum(-1/2*(-1+_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+2*_Z+_Z^2))
a(n)=2*A001333(n-1)+a(n-1), n>1. A001333(n)/a(n) converges to sqrt(1/2). - Mario Catalani (mario.catalani(AT)unito.it), Apr 29 2003
Binomial transform of A094024. a(n)=0^n+((1+sqrt(2))^n-(1-sqrt(2))^n)/sqrt(2). - Paul Barry (pbarry(AT)wit.ie), Apr 22 2004
a(n)=sum{k=0..floor(n/2), binomial(n-k-1, k)2^(n-2k)} - Paul Barry (pbarry(AT)wit.ie), Jan 16 2005
If p[i]=2modp(i,2) 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. [From Milan R. Janjic (agnus(AT)blic.net), May 02 2010]
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MAPLE
| spec := [S, {S=Sequence(Prod(Union(Z, Z), Sequence(Prod(Z, Z))))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
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MATHEMATICA
| a=2; b=0; c=0; lst={1}; Do[c=a+b+c; AppendTo[lst, c]; a=b; b=c, {n, 2*4!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), May 21 2010]
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PROG
| (PARI) Vec((-1+x^2)/(-1+2*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Nov 20 2011
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CROSSREFS
| Cf. A052906.
Sequence in context: A025275 A165409 A163271 * A178036 A191625 A110236
Adjacent sequences: A052539 A052540 A052541 * A052543 A052544 A052545
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KEYWORD
| easy,nonn
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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 05 2000
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