A093509 Multiples of 5 or 6.

0, 5, 6, 10, 12, 15, 18, 20, 24, 25, 30, 35, 36, 40, 42, 45, 48, 50, 54, 55, 60, 65, 66, 70, 72, 75, 78, 80, 84, 85, 90, 95, 96, 100, 102, 105, 108, 110, 114, 115, 120, 125, 126, 130, 132, 135, 138, 140, 144, 145, 150, 155, 156, 160, 162, 165, 168, 170, 174, 175

Offset: 1

Author

Ralf Stephan, May 22 2004

Keywords

• nonn
• easy

Comments

Numbers that are congruent to {0, 5, 6, 10, 12, 15, 18, 20, 24, 25} mod 30.

Also without 0: numbers n such that cos(Pi*x/n)+cos(Pi*y/n)=1/2 has integer solutions (x,y).

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.

Examples

			102 = 6*17 (a multiple of 6), so 102 is in the sequence.
		

Links

• J. H. Conway and A. D. Jones, Trigonometric Diophantine equations (on vanishing sums of roots of unity), Acta Arith. XXX (1976) 229-240.
• M. Hunziker, A. Machiavelo and J. Park, Chebyshev polynomials over finite fields and reversibility of sigma-automata on square grids, Theoretical Comp. Sci., 320 (2004), 465-483.
• Eric Weisstein's World of Mathematics, Lights Out Puzzle
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-2,2,-2,2,-1).

Crossrefs

Cf. A008587, A008588.

Programs

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

Maple

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

Mathematica

Join[{0}, lim = 49; TakeWhile[Union@Flatten[# Range@lim & /@ {5, 6}], # < 5 lim &]] (* Michael De Vlieger, Mar 06 2016 *)
Union[Range[0,50]*6, Range[0,60]*5] (* Giovanni Resta, May 05 2016 *)
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 *)

PARI

isok(n) = !(n%5) || !(n%6);

Sage

def isA093509(n): return n % 5 == 0 or n % 6 == 0
def A093509List(upto): return [n for n in range(upto + 1) if isA093509(n)]
print(A093509List(175))  # Peter Luschny, Apr 10 2022

Formula

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

From Wesley Ivan Hurt, May 01 2016: (Start)

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.

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)