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

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

Offset: 0

Author

Howard Stern (hsstern(AT)mindspring.com), Oct 23 2001

Keywords

• nonn
• easy

Comments

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

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

Links

• Harry J. Smith, Table of n, a(n) for n = 0..200
• E. Altinisik, A. Keskin, M. Yildiz, M. Demirbuken, On a conjecture of Ilmonen, Haukkanen and Merikoski concerning the smallest eigenvalues of certain GCD related matrices, Linear Algebra and its Applications, Volume 493, 15 March 2016, Pages 1-13
• S. Falcon, On the Sequences of Products of Two k-Fibonacci Numbers, American Review of Mathematics and Statistics, March 2014, Vol. 2, No. 1, pp. 111-120.
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Crossrefs

Cf. A000045, A282464.

Odd terms of A097083.

Partial sums of A001654.

Programs

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

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x/((1-x^2)*(1-3*x+x^2)) )); // G. C. Greubel, Jan 09 2019

Mathematica

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, Oct 25 2001 *)
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 *)

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) ) } \\ Harry J. Smith, Sep 27 2009

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

Sage

(x/((1-x^2)*(1-3*x+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 09 2019

Formula

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 = 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

a(n) = a(n-1) + A001654(n) with a(0)=0. (Partial sums of A001654). - Johannes W. Meijer, Sep 22 2010

a(n) = floor(phi^(2*n+2)/5), where phi =(1+sqrt(5))/2. - Gary Detlefs Mar 12 2011

a(n) = (A027941(n) + A001654(n))/2, n>=0. - Wolfdieter Lang, Jul 23 2012

a(n) = A005248(n+1)/5 -1/2 -(-1)^n/10. - R. J. Mathar, Feb 21 2013

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

a(n) = Sum_{i=0..n} (n+1-i)*Fibonacci(i)^2. - Bruno Berselli, Feb 20 2017

Extensions

More terms from Robert G. Wilson v, Oct 25 2001