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A093509 Multiples of 5 or 6.

%I A093509 #41 Feb 16 2025 08:32:53
%S A093509 0,5,6,10,12,15,18,20,24,25,30,35,36,40,42,45,48,50,54,55,60,65,66,70,
%T A093509 72,75,78,80,84,85,90,95,96,100,102,105,108,110,114,115,120,125,126,
%U A093509 130,132,135,138,140,144,145,150,155,156,160,162,165,168,170,174,175
%N A093509 Multiples of 5 or 6.
%C A093509 Numbers that are congruent to {0, 5, 6, 10, 12, 15, 18, 20, 24, 25} mod 30.
%C A093509 Also without 0: numbers n such that cos(Pi*x/n)+cos(Pi*y/n)=1/2 has integer solutions (x,y).
%C A093509 Numbers n such that there exists a nontrivial configuration to an n-1 X n-1 Lights Out game from the all-off state to the all-off state.
%H A093509 J. H. Conway and A. D. Jones, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa30/aa3033.pdf">Trigonometric Diophantine equations (on vanishing sums of roots of unity)</a>, Acta Arith. XXX (1976) 229-240.
%H A093509 M. Hunziker, A. Machiavelo and J. Park, <a href="http://dx.doi.org/10.1016/j.tcs.2004.03.031">Chebyshev polynomials over finite fields and reversibility of sigma-automata on square grids</a>, Theoretical Comp. Sci., 320 (2004), 465-483.
%H A093509 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LightsOutPuzzle.html">Lights Out Puzzle</a>
%H A093509 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-2,2,-2,2,-2,2,-1).
%F A093509 G.f.: x^2*(5-4*x+8*x^2-6*x^3+9*x^4-6*x^5+8*x^6-4*x^7+5*x^8) / ((x^4+x^3+x^2+x+1) * ( x^4-x^3+x^2-x+1) * (x-1)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009; corrected by _R. J. Mathar_, Sep 16 2009
%F A093509 From _Wesley Ivan Hurt_, May 01 2016: (Start)
%F A093509 a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-2*a(n-4)+2*a(n-5)-2*a(n-6)+2*a(n-7)-2*a(n-8)+2*a(n-9)-a(n-10) for n>10.
%F A093509 a(n) = 5*floor((n+9)/10)*(-1)^n/2 + 5*floor((n+9)/10)/2 - n*(-1)^n/4 - (-1)^n + 11*n/4 - 4. (End)
%e A093509 102 = 6*17 (a multiple of 6), so 102 is in the sequence.
%p A093509 A093509:=n->5*floor((n+9)/10)*(-1)^n/2+5*floor((n+9)/10)/2-n*(-1)^n/4-(-1)^n+11*n/4-4: seq(A093509(n), n=1..80); # _Wesley Ivan Hurt_, May 01 2016
%t A093509 Join[{0}, lim = 49; TakeWhile[Union@Flatten[# Range@lim & /@ {5, 6}], # < 5 lim &]] (* _Michael De Vlieger_, Mar 06 2016 *)
%t A093509 Union[Range[0,50]*6, Range[0,60]*5] (* _Giovanni Resta_, May 05 2016 *)
%t A093509 LinearRecurrence[{2,-2,2,-2,2,-2,2,-2,2,-1},{0,5,6,10,12,15,18,20,24,25},60] (* _Harvey P. Dale_, Jul 15 2023 *)
%o A093509 (PARI) isok(n) = !(n%5) || !(n%6);
%o A093509 (Magma) [5*Floor((n+9)/10)*(-1)^n/2+5*Floor((n+9)/10)/2-n*(-1)^n/4-(-1)^n+11*n/4-4 : n in [1..50]]; // _Wesley Ivan Hurt_, May 01 2016
%o A093509 (Sage)
%o A093509 def isA093509(n): return n % 5 == 0 or n % 6 == 0
%o A093509 def A093509List(upto): return [n for n in range(upto + 1) if isA093509(n)]
%o A093509 print(A093509List(175))  # _Peter Luschny_, Apr 10 2022
%Y A093509 Cf. A008587, A008588.
%K A093509 nonn,easy
%O A093509 1,2
%A A093509 _Ralf Stephan_, May 22 2004