A003485 - OEIS

%I M1086 #65 Dec 15 2023 19:16:57

%S 1,2,4,8,9,10,12,16,17,18,20,24,25,26,28,32,33,34,36,40,41,42,44,48,

%T 49,50,52,56,57,58,60,64,65,66,68,72,73,74,76,80,81,82,84,88,89,90,92,

%U 96,97,98,100,104,105,106,108,112,113,114,116,120,121,122,124

%N Hurwitz-Radon function at powers of 2.

%C Positive integers that are congruent to {0, 1, 2, 4} mod 8. - _Michael Somos_, Dec 12 2023

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

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

%H Nathaniel Johnston, <a href="/A003485/b003485.txt">Table of n, a(n) for n = 0..10000</a>

%H V. Ovsienko and Serge Tabachnikov, <a href="http://arxiv.org/abs/1511.08894">Affine Hopf fibration</a>, arXiv preprint arXiv:1511.08894 [math.AT], 2015.

%H V. Ovsienko and Serge Tabachnikov, <a href="https://doi.org/10.1007/s00283-015-9618-x">Hopf fibrations and Hurwitz-Radon numbers</a>, Math. Intell. 38 (2016) 11-18

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H D. B. Shapiro, <a href="/A003484/a003484.pdf">Letter to N. J. A. Sloane, 1974</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).

%F G.f.: (1 + x + 2*x^2 + 4*x^3) / ((1-x)*(1-x^4)). - _Simon Plouffe_ in his 1992 dissertation

%F 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

%F 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

%F a(n) = -A047507(-n) = a(n+4) - 8 for all n in Z. - _Michael Somos_, Dec 12 2023

%e 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

%p 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

%t 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 *)

%t a[ n_] := 2*n + Max[0, 2-Mod[n-3, 4]]; (* _Michael Somos_, Dec 12 2023 *)

%o (Haskell)

%o a003485 n = a003485_list !! n

%o a003485_list = 1 : 2 : 4 : 8 : 9 : zipWith (+)

%o    (drop 4 a003485_list) (zipWith (-) (tail a003485_list) a003485_list)

%o -- _Reinhard Zumkeller_, Mar 11 2012

%o (PARI) {a(n) = 2*n + max(0, 2 - (n-3)%4)}; /* _Michael Somos_, Dec 12 2023 */

%Y Cf. A003484, A047507.

%Y Essentially the same as A047466.

%Y Cf. A008621. - _Johannes W. Meijer_, Jun 07 2011

%Y Cf. A209675.

%K easy,nonn,nice

%O 0,2

%A _N. J. A. Sloane_