A003485 Hurwitz-Radon function at powers of 2. (Formerly M1086)

1, 2, 4, 8, 9, 10, 12, 16, 17, 18, 20, 24, 25, 26, 28, 32, 33, 34, 36, 40, 41, 42, 44, 48, 49, 50, 52, 56, 57, 58, 60, 64, 65, 66, 68, 72, 73, 74, 76, 80, 81, 82, 84, 88, 89, 90, 92, 96, 97, 98, 100, 104, 105, 106, 108, 112, 113, 114, 116, 120, 121, 122, 124

Offset

0,2

Comments

Positive integers that are congruent to {0, 1, 2, 4} mod 8. - Michael Somos, Dec 12 2023

References

T. Y. Lam, The Algebraic Theory of Quadratic Forms. Benjamin, Reading, MA, 1973, p. 131.

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

Links

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

V. Ovsienko and Serge Tabachnikov, Affine Hopf fibration, arXiv preprint arXiv:1511.08894 [math.AT], 2015.

V. Ovsienko and Serge Tabachnikov, Hopf fibrations and Hurwitz-Radon numbers, Math. Intell. 38 (2016) 11-18

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

D. B. Shapiro, Letter to N. J. A. Sloane, 1974

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Formula

G.f.: (1 + x + 2*x^2 + 4*x^3) / ((1-x)*(1-x^4)). - Simon Plouffe in his 1992 dissertation

a(n) = ceiling((n+1)/4) + ceiling((n)/4) + 2*ceiling((n-1)/4) + 4*ceiling((n-2)/4). - Johannes W. Meijer, Jun 07 2011

a(n) = a(n-1) + a(n-4) - a(n-5); a(0)=1, a(1)=2, a(2)=4, a(3)=8, a(4)=9. - Harvey P. Dale, Jun 13 2011

a(n) = -A047507(-n) = a(n+4) - 8 for all n in Z. - Michael Somos, Dec 12 2023

Example

G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 9*x^4 + 10*x^5 + 12*x^6+ 16*x^7 + ... - Michael Somos, Dec 12 2023

Maple

A003485:= proc(n): ceil((n+1)/4) + ceil((n)/4) + 2*ceil((n-1)/4) + 4*ceil((n-2)/4) end: seq(A003485(n), n=0..62); # Johannes W. Meijer, Jun 07 2011

Mathematica

CoefficientList[Series[(1+x+2x^2+4x^3)/((1-x)(1-x^4)), {x, 0, 70}], x] (* or *) LinearRecurrence[{1, 0, 0, 1, -1}, {1, 2, 4, 8, 9}, 71] (* Harvey P. Dale, Jun 13 2011 *)
a[ n_] := 2*n + Max[0, 2-Mod[n-3, 4]]; (* Michael Somos, Dec 12 2023 *)

Prog

(Haskell)
a003485 n = a003485_list !! n
a003485_list = 1 : 2 : 4 : 8 : 9 : zipWith (+)
   (drop 4 a003485_list) (zipWith (-) (tail a003485_list) a003485_list)
-- Reinhard Zumkeller, Mar 11 2012
(PARI) {a(n) = 2*n + max(0, 2 - (n-3)%4)}; /* Michael Somos, Dec 12 2023 */

Crossrefs

Cf. A003484, A047507.

Essentially the same as A047466.

Cf. A008621. - Johannes W. Meijer, Jun 07 2011

Cf. A209675.

Sequence in context: A044952 A352831 A047466 * A072602 A374664 A049642

Adjacent sequences: A003482 A003483 A003484 * A003486 A003487 A003488

Keyword

easy,nonn,nice

Author

N. J. A. Sloane

Status

approved