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A049684
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F(2n)^2 where F() = Fibonacci numbers A000045.
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9
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0, 1, 9, 64, 441, 3025, 20736, 142129, 974169, 6677056, 45765225, 313679521, 2149991424, 14736260449, 101003831721, 692290561600, 4745030099481, 32522920134769, 222915410843904, 1527884955772561, 10472279279564025
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| This is the r=9 member of the r-family of sequences S_r(n) defined in A092184 where more information can be found.
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REFERENCES
| A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 27.
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LINKS
| R. Stephan, Boring proof of a nonlinearity
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| G.f.: (x+x^2)/((1-x)*(1-7*x+x^2)).
a(n) = 8*a(n-1)-8*a(n-2)+a(n-3), n>2. a(0) = 0, a(1) = 1, a(2) = 9.
a(n) = 7a(n-1)-a(n-2)+2 = A001906(n)^2.
a(n) = 1/5*{-2+[(7+sqrt(45))/2]^n+[(7-sqrt(45))/2]^n}. - R. Stephan, Apr 14 2004
a(n)= 2*(T(n, 7/2)-1)/5 with twice the Chebyshev's polynomials of the first kind evaluated at x=7/2: 2*T(n, 7/2)= A056854(n). W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Oct 18 2004
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MATHEMATICA
| Join[{a=0, b=1}, Table[c=7*b-1*a+2; a=b; b=c, {n, 60}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 18 2011*)
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PROG
| (PARI) a(n)=fibonacci(2*n)^2
(Mupad) numlib::fibonacci(2*n)^2 $ n = 0..35; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 13 2008
(Other) sage: [(fibonacci(2*n))^2 for n in xrange(0, 21)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 15 2009]
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CROSSREFS
| A049684(n)=(A000032(4*n)-2)/5. First differences give A033890.
a(n) = A064170(n+2) - 1 = (1/5) A081070. Bisection of A007598 and A064841.
First differences of A103434.
Sequence in context: A143631 A083328 A000846 * A037540 A037484 A013566
Adjacent sequences: A049681 A049682 A049683 * A049685 A049686 A049687
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
| Better description and more terms from Michael Somos
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