A003485 Hurwitz-Radon function at powers of 2.
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
Examples
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
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).
Crossrefs
Programs
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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
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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
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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 *) -
PARI
{a(n) = 2*n + max(0, 2 - (n-3)%4)}; /* Michael Somos, Dec 12 2023 */
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
Comments