

A019442


Numbers n such that a Hadamard matrix of order n exists.


2



1, 2, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 240
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OFFSET

1,2


COMMENTS

It is conjectured that this sequence consists of 1, 2 and all multiples of 4.
Already in 1992 Hadamard matrices were known of all orders 4t up through 424.
The old entry with this sequence number was a duplicate of A007740.
Equals binomial transform of [1, 1, 1, 1, 3, 5, 7, 9,...]. [Gary W. Adamson, Oct 17 2008]
Integers n dividing the Lucas sequence u(n) defined by u(n) = 2*u(n1)  2*u(n2) with initial conditions u(0)=0, u(1)=1. [Thomas M. Bridge, Nov 08 2013]


REFERENCES

Hadamard, J. (1893). Resolution d'une question relative aux determinants. Bull. des Sciences Math. (2), 17, 240246.
Hall, Jr., M. (1961). Hadamard matrices of order 16. Research Summary No. 3610, Jet Propulsion Lab., Pasadena, CA, Vol. 1, pp. 2126.
Hall, Jr., M. (1965). Hadamard matrices of order 20. Technical Report 32761, Jet Propulsion Lab., Pasadena, CA.
Hall, Jr., M. (1986). Combinatorial Theory. 2nd edn. New York: Wiley.
S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays, SpringerVerlag, NY, 1999, Chapter 7.
Seberry, Jennifer and Yamada, Mieko; Hadamard matrices, sequences and block designs, in Dinitz and Stinson, eds., Contemporary design theory, pp. 431560, WileyIntersci. Ser. Discrete Math. Optim., Wiley, New York, 1992.
Wallis, W. D.; Street, Anne Penfold; Wallis, Jennifer Seberry; Combinatorics: Room squares, sumfree sets, Hadamard matrices. Lecture Notes in Mathematics, Vol. 292. SpringerVerlag, BerlinNew York, 1972. iv+508 pp.


LINKS

Table of n, a(n) for n=1..62.
D. Z. Djokovic, Construction of some new Hadamard matrices, Bull. Austral. Math. Soc., Volume 45, Issue 2, April 1992, pp. 327332.
D. Z. Djokovic, Five new orders for Hadamard matrices of skew type. Australas. J. Combin., Vol. 10, 1994.
D. Z. Djokovic, Two Hadamard matrices of order 956 of GoethalsSeidel type, Combinatorica, 1994, 14, 375377.
Hiroshi Kimura, Hadamard matrices of order 28 with automorphism groups of order two, J. Combin. Theory, 1986, A 43, 98102.
Hiroshi Kimura, New Hadamard matrix of order 24, Graphs Combin., 1989, 5, 235242.
Hiroshi Kimura, Classification of Hadamard matrices of order 28 with Hall sets, Discrete Math., 1994, 128, 257268.
Hiroshi Kimura, Classification of Hadamard matrices of order 28, Discrete Math., 1994, 133, 171180.
W. P. Orrick, Switching operations for Hadamard matrices, arXiv:math/0507515 [math.CO], 20052007.
R. E. A. C. Paley, On orthogonal matrices, J. Math. Phys., 12, 311320.
R. L. Plackett & J. P. Burman, The design of optimum multifactorial experiments, Biometrika, 1946, 33, 305325.
K. Sawade, A Hadamard matrix of order 268, Graphs Combin., 1985, 1, 185187.Seberry, Jennifer and Yamada, Mieko; Hadamard matrices, sequences and block designs, in Dinitz and Stinson, eds., Contemporary design theory, pp. 431560, WileyIntersci. Ser. Discrete Math. Optim., Wiley, New York, 1992.
N. J. A. Sloane, Tables of Hadamard matrices
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Edward Spence, Classification of Hadamard matrices of order 24 and 28, Discrete Math. 140 (1995), no. 13, 185243.
Eric Weisstein's World of Mathematics, Hadamard Matrix.
J. Williamson, Hadamard's determinant theorem and the sum of four squares, Duke Math. J., 1994, 11, 6581.
Index entries for sequences related to Hadamard matrices


FORMULA

Conjecture: g.f. is (2*x^3 + x^2 + 1)/(x  1)^2.  JeanFrançois Alcover, Oct 03 2016


MATHEMATICA

b[n_] := If[n<4, 1, (1)^(n1)(2n5)];
a[n_] := Sum[Binomial[n1, k] b[k], {k, 0, n1}];
Table[a[n], {n, 1, 62}] (* JeanFrançois Alcover, Aug 07 2018, after Gary W. Adamson *)


CROSSREFS

Cf. A007299, A036297, A016742.
Sequence in context: A024908 A325326 A215459 * A048166 A010066 A180490
Adjacent sequences: A019439 A019440 A019441 * A019443 A019444 A019445


KEYWORD

nonn,nice


AUTHOR

N. J. A. Sloane, Oct 16 2008


STATUS

approved



