A005312 -id:A005312 - cp's OEIS frontend

A005313 Maximal sum of inverse squares of the singular values of triangular anti-Hadamard matrices of order n.

1, 3, 6, 13, 29, 70, 175, 449, 1164, 3035, 7931, 20748, 54301, 142143, 372114, 974185, 2550425, 6677074, 17480779, 45765245, 119814936, 313679543, 821223671, 2149991448, 5628750649, 14736260475, 38580030750, 101003831749, 264431464469, 692290561630, 1812440220391, 4745030099513

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.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (4,-3,-3,4,-1).

Cf. A005312, A064831.

a(n) = A064831(n-1) + n.

G.f.: [x(1-x-3x^2+x^3)]/[(1-3x+x^2)(1+x)(1-x)^2]. - Conjectured by Simon Plouffe in his 1992 dissertation. [This is in fact the correct g.f. - N. J. A. Sloane, Jan 28 2022]

Definition corrected by Stefano Spezia, Jan 30 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

A005313 Maximal sum of inverse squares of the singular values of triangular anti-Hadamard matrices of order n.

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Links

Crossrefs

Extensions

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.

Views

Author

Keywords

Comments

Links

Crossrefs

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.

Views

Author

Keywords

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs