A005313 -id:A005313 - cp's OEIS frontend

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

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

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

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).

Cf. A000045, A282464.
Odd terms of A097083.
Partial sums of A001654.

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

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

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)
```

{ for (n=0, 200, a=fibonacci(n+1)^2 - 1 + n%2; write("b064831.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 27 2009

my(x='x+O('x^30)); concat([0], Vec(x/((1-x^2)*(1-3*x+x^2)))) \\ G. C. Greubel, Jan 09 2019

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

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

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

A005312 Maximal sum of inverse squares of the singular values of symmetric anti-Hadamard matrices of order n.

Original entry on oeis.org

1, 3, 7, 16, 46, 138, 624

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Noga Alon and Van H. Vu, Anti-Hadamard Matrices, Coin Weighing, Threshold Gates, and Indecomposable Hypergraphs, Journal of Combinatorial Theory, Series A, Volume 79, Issue 1, July 1997, Pages 133-160.
R. L. Graham and N. J. A. Sloane, Anti-Hadamard matrices, Linear Alg. Applic., 62 (1984), 113-137, Table 1.

Cf. A005313.

Definition corrected by Stefano Spezia, Jan 30 2022

a(7) confirmed and inserted by Stefano Spezia, Feb 20 2022

A351240 Numerator of the maximal sum of inverse squares of the singular values of symmetric Toeplitz anti-Hadamard matrices of order n.

Original entry on oeis.org

1, 2, 7, 10, 37, 34, 107, 262, 661, 2236, 5665, 13350, 35141, 260188, 382614, 418134, 9552743681459, 83369851038, 785598689

Offset: 1

Stefano Spezia, Feb 05 2022

R. L. Graham and N. J. A. Sloane, Anti-Hadamard matrices, Linear Alg. Applic., 62 (1984), 113-137.

Cf. A005312, A005313, A351241 (denominator).

A351241 Denominator of the maximal sum of inverse squares of the singular values of symmetric Toeplitz anti-Hadamard matrices of order n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 8546700, 9763, 49

Offset: 1

Stefano Spezia, Feb 05 2022

R. L. Graham and N. J. A. Sloane, Anti-Hadamard matrices, Linear Alg. Applic., 62 (1984), 113-137.

Cf. A005312, A005313, A351240 (numerator).

A351821 Maximal sum of inverse squares of the singular values of Toeplitz anti-Hadamard matrices of order n.

Original entry on oeis.org

1, 3, 7, 16, 46, 138, 601, 2619, 16162, 93531, 654700, 4442304

Offset: 1

Stefano Spezia, Feb 20 2022

R. L. Graham and N. J. A. Sloane, Anti-Hadamard matrices, Linear Alg. Applic., 62 (1984), 113-137, Table 1.

Cf. A005312, A005313, A351240, A351241.

a(12) confirmed and inserted by Stefano Spezia, Mar 16 2022

A351984 a(n) is the number of symmetric anti-Hadamard matrices of order n whose sum of the inverse squares of their singular values is maximal.

Original entry on oeis.org

1, 2, 6, 24, 120, 840, 5040

Offset: 1

Stefano Spezia, Feb 27 2022

The sequence of the ratio a(n+1)/a(n) begins with 2, 3, 4, 5, 7, 6, ...

Is a(n+1)/a(n) integer for all n? If that is true, excluding the number 1, is the sequence of the ratio a(n+1)/a(n) a permutation of all other natural numbers?

R. L. Graham and N. J. A. Sloane, Anti-Hadamard matrices, Linear Alg. Applic., 62 (1984), 113-137.

Cf. A005312, A005313, A119002, A351153.

A352177 a(n) is the number of symmetric Toeplitz anti-Hadamard matrices of order n whose sum of the inverse squares of their singular values is maximal.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Stefano Spezia, Mar 08 2022

R. L. Graham and N. J. A. Sloane, Anti-Hadamard matrices, Linear Alg. Applic., 62 (1984), 113-137.

Cf. A351240, A351241.

Cf. A005312, A005313, A351821, A351984.

Conjectures: (Start)
a(n) = 1 for n > 9.
G.f.: x*(x + x^3 + 2*x^5 + x^8 + 1/(1 - x)). (End)

A352478 a(n) is the number of Toeplitz anti-Hadamard matrices of order n whose sum of the inverse squares of their singular values is maximal.

Original entry on oeis.org

1, 2, 3, 6, 4, 2, 2, 2, 2, 2, 2, 12

Offset: 1

Stefano Spezia, Mar 17 2022

Conjecture: a(1) = 1 and a(3) = 3 are the only odd terms.

R. L. Graham and N. J. A. Sloane, Anti-Hadamard matrices, Linear Alg. Applic., 62 (1984), 113-137.

Cf. A351821.

Cf. A005312, A005313, A351984, A352177.

cp's OEIS Frontend

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

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A005312 Maximal sum of inverse squares of the singular values of symmetric anti-Hadamard matrices of order n.

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A351240 Numerator of the maximal sum of inverse squares of the singular values of symmetric Toeplitz anti-Hadamard matrices of order n.

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A351241 Denominator of the maximal sum of inverse squares of the singular values of symmetric Toeplitz anti-Hadamard matrices of order n.

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A351821 Maximal sum of inverse squares of the singular values of Toeplitz anti-Hadamard matrices of order n.

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A351984 a(n) is the number of symmetric anti-Hadamard matrices of order n whose sum of the inverse squares of their singular values is maximal.

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A352177 a(n) is the number of symmetric Toeplitz anti-Hadamard matrices of order n whose sum of the inverse squares of their singular values is maximal.

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A352478 a(n) is the number of Toeplitz anti-Hadamard matrices of order n whose sum of the inverse squares of their singular values is maximal.

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