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A064831 Partial sums of A001654, or sum of the areas of the first n Fibonacci rectangles.

%I A064831 #78 Aug 18 2024 16:43:24
%S A064831 0,1,3,9,24,64,168,441,1155,3025,7920,20736,54288,142129,372099,
%T A064831 974169,2550408,6677056,17480760,45765225,119814915,313679521,
%U A064831 821223648,2149991424,5628750624,14736260449,38580030723,101003831721
%N A064831 Partial sums of A001654, or sum of the areas of the first n Fibonacci rectangles.
%C A064831 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.
%C A064831 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
%C A064831 Contribution from _Johannes W. Meijer_, Sep 22 2010: (Start)
%C A064831 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.
%C A064831 (End)
%C A064831 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
%C A064831 a(n+1) is equal to the number of ternary strings of length n without any substring of the form 0x1, where x is in {0,1,2}. - _John M. Campbell_, Apr 03 2016
%H A064831 Harry J. Smith, <a href="/A064831/b064831.txt">Table of n, a(n) for n = 0..200</a>
%H A064831 E. Altinisik, A. Keskin, M. Yildiz, M. Demirbuken, <a href="https://doi.org/10.1016/j.laa.2015.11.023">On a conjecture of Ilmonen, Haukkanen and Merikoski concerning the smallest eigenvalues of certain GCD related matrices</a>, Linear Algebra and its Applications, Volume 493, 15 March 2016, Pages 1-13
%H A064831 S. Falcon, <a href="https://www.researchgate.net/publication/298789400_On_the_Sequences_of_Products_of_Two_k-Fibonacci_Numbers">On the Sequences of Products of Two k-Fibonacci Numbers</a>, American Review of Mathematics and Statistics, March 2014, Vol. 2, No. 1, pp. 111-120.
%H A064831 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A064831 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-3,1).
%F A064831 a(n) = F(n+1)^2 - 1 if n is even, or F(n+1)^2 if n is odd.
%F A064831 a(n) = A005313(n+1) - n.
%F A064831 G.f.: x/((1-x^2)*(1-3*x+x^2)). - _N. J. A. Sloane_ Jul 15 2002
%F A064831 a(n) = Sum_{k=0..floor(n/2)} U(n-2k-1, 3/2). - _Paul Barry_, Nov 15 2003
%F A064831 Let M_n denote the n X n Hankel matrix M_n(i, j)=F(i+j-1) where F = A000045 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
%F A064831 a(n) = a(n-1) + A001654(n) with a(0)=0. (Partial sums of A001654). - _Johannes W. Meijer_, Sep 22 2010
%F A064831 a(n) = floor(phi^(2*n+2)/5), where phi =(1+sqrt(5))/2. - _Gary Detlefs_ Mar 12 2011
%F A064831 a(n) = (A027941(n) + A001654(n))/2, n>=0. - _Wolfdieter Lang_, Jul 23 2012
%F A064831 a(n) = A005248(n+1)/5 -1/2 -(-1)^n/10. - _R. J. Mathar_, Feb 21 2013
%F A064831 Recurrence: a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 9, a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4). - _Vladimir Reshetnikov_, Oct 28 2015
%F A064831 a(n) = Sum_{i=0..n} (n+1-i)*Fibonacci(i)^2. - _Bruno Berselli_, Feb 20 2017
%t A064831 Table[ Sum[ Fibonacci[k]*Fibonacci[k + 1], {k, n} ], {n, 0, 30}]
%t A064831 f[n_] := Floor[GoldenRatio^(2 n + 2)/5]; Array[f, 28, 0] (* _Robert G. Wilson v_, Oct 25 2001 *)
%t A064831 a[0]= 0; a[1]= 1; a[2]= 3; a[3]= 9; a[n_]:= a[n]= 3a[n-1] - 3a[n-3] + a[n-4]; Table[a[n], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 28 2015 *)
%o A064831 (PARI) a(n)=if(n<0,0,fibonacci(n+1)^2-1+n%2)
%o A064831 (PARI) { for (n=0, 200, a=fibonacci(n+1)^2 - 1 + n%2; write("b064831.txt", n, " ", a) ) } \\ _Harry J. Smith_, Sep 27 2009
%o A064831 (PARI) my(x='x+O('x^30)); concat([0], Vec(x/((1-x^2)*(1-3*x+x^2)))) \\ _G. C. Greubel_, Jan 09 2019
%o A064831 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x/((1-x^2)*(1-3*x+x^2)) )); // _G. C. Greubel_, Jan 09 2019
%o A064831 (Sage) (x/((1-x^2)*(1-3*x+x^2))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 09 2019
%o A064831 (GAP) a:=[0,1,3,9];; for n in [5..30] do a[n]:=3*a[n-1]-3*a[n-3]+a[n-4]; od; a; # _G. C. Greubel_, Jan 09 2019
%Y A064831 Cf. A000045, A282464.
%Y A064831 Odd terms of A097083.
%Y A064831 Partial sums of A001654.
%K A064831 nonn,easy
%O A064831 0,3
%A A064831 Howard Stern (hsstern(AT)mindspring.com), Oct 23 2001
%E A064831 More terms from _Robert G. Wilson v_, Oct 25 2001