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A064831
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Partial sums of A001654, or sum of the areas of the first n Fibonacci rectangles.
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33
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0, 1, 3, 9, 24, 64, 168, 441, 1155, 3025, 7920, 20736, 54288, 142129, 372099, 974169, 2550408, 6677056, 17480760, 45765225, 119814915, 313679521, 821223648, 2149991424, 5628750624, 14736260449, 38580030723, 101003831721
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OFFSET
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0,3
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COMMENTS
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The n-th rectangle is F(n)*F(n+1), where F(n) = n-th Fibonacci number (F(1)=1, F(2)=1, F(3)=2, etc.), A000045.
If 2*T(a_n) = the oblong number formed by substituting a(n) in the product formula x(x+1), then 2*T(a_n) = F(n-1)*F(n) * F(n)*F(n+1). Thus a(n) equals the integer part of the square root of the right hand side of the given equation. - Kenneth J Ramsey, Dec 19 2006
Contribution from Johannes W. Meijer, Sep 22 2010: (Start)
The a(n) represent several triangle sums of the Golden Triangle A180662: Kn11 (terms doubled), Kn12(n+1) (terms doubled), Kn4, Ca1 (terms tripled), Ca4, Gi1 (terms quadrupled) and Gi4. See A180662 for the definitions of these sums.
(End)
Define a 2 X (n+1) matrix with elements T(r,0)=A000032(r) and T(r,1) = Fibonacci(r), r=0,1,..,n. The matrix times its transposed is a 2 X 2 matrix with one diagonal element A001654(n+1), the other A216243(n), and A027941(n+1) on both outer diagonals. The determinant of this 2 X 2 matrix is 4*a(n). Example: For n=3 the matrix is 2 X 4 with rows 2 1 3 4; 0 1 1 2 to give as a product the 2 X 2 matrix with rows 30 12; 12 6 and determinant 180-144 = 36 =4*a(3). - J. M. Bergot, Feb 13 2013
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LINKS
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Harry J. Smith, Table of n, a(n) for n = 0..200
Index entries for sequences related to Chebyshev polynomials.
Index to sequences with linear recurrences with constant coefficients, signature (3,0,-3,1).
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FORMULA
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a(n)=F(n+1)^2 - 1 if n is even, or F(n+1)^2 if n is odd.
a(n) = A005313(n+1) - n.
G.f.: x/((1-x^2)*(1-3*x+x^2)). - N. J. A. Sloane Jul 15 2002
a(n)=sum{k=0..floor(n/2), U(n-2k-1, 3/2)} - Paul Barry, Nov 15 2003
Let M_n denote the n X n Hankel matrix M_n(i, j)=F(i+j-1) where F(n)=A000045(n) is Fibonacci numbers, then the characteristic polynomial of M_n is x^n - F(2n)x^(n-1) + a(n-1)x^(n-2) . - Michael Somos, Nov 14 2002
Contribution from Johannes W. Meijer, Sep 22 2010: (Start)
a(n)=a(n-1)+A001654(n) with a(0)=0. (Partial sums of A001654).
(End)
a(n)=floor(phi^(2*n+2)/5), where phi =(1+sqrt(5))/2. [From Gary Detlefs Mar 12 2011]
a(n) = (A027941(n) + A001654(n))/2, n>=0. [From Wolfdieter Lang, Jul 23 2012]
a(n) = A005248(n+1)/5 -1/2 -(-1)^n/10. - R. J. Mathar, Feb 21 2013
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MATHEMATICA
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Table[ Sum[ Fibonacci[k]*Fibonacci[k + 1], {k, n} ], {n, 0, 30}]
f[n_] := Floor[GoldenRatio^(2 n + 2)/5]; Array[f, 28, 0] (* Robert G. Wilson v *)
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PROG
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(PARI) a(n)=if(n<0, 0, fibonacci(n+1)^2-1+n%2)
(PARI) { for (n=0, 200, a=fibonacci(n+1)^2 - 1 + n%2; write("b064831.txt", n, " ", a) ) } [From Harry J. Smith, Sep 27 2009]
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CROSSREFS
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Cf. A000045.
Partial sums of A001654.
Odd terms of A097083
Sequence in context: A166290 A097134 A123892 * A153582 A096168 A051042
Adjacent sequences: A064828 A064829 A064830 * A064832 A064833 A064834
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KEYWORD
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nonn,easy
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AUTHOR
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Howard Stern (hsstern(AT)mindspring.com), Oct 23 2001
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EXTENSIONS
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More terms from Robert G. Wilson v, Oct 25 2001
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STATUS
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approved
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