A162698 Numbers n such that the incidence matrix of the grid n X n has -1 as eigenvalue.

4, 5, 9, 11, 14, 17, 19, 23, 24, 29, 34, 35, 39, 41, 44, 47, 49, 53, 54, 59, 64, 65, 69, 71, 74, 77, 79, 83, 84, 89, 94, 95, 99, 101, 104, 107, 109, 113, 114, 119, 124, 125, 129, 131, 134, 137, 139, 143, 144, 149, 154, 155, 159, 161, 164, 167, 169, 173, 174, 179, 184, 185, 189, 191, 194, 197, 199

Offset: 1

Author

Vincent Delecroix, Jul 11 2009

Keywords

• nonn
• easy

Comments

Numbers n such that n+1 is a multiple of 5 or 6. - Tom Edgar, Dec 15 2017

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• M. Kreh, "Lights Out" and Variants, Amer. Math. Month., Vol. 124 (10), Dec. 2017, pp. 937-950.
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-2,2,-2,2,-1).

Crossrefs

Cf. A093509, A117870.

Programs

Mathematica

With[{nn=40},Select[Union[Join[5*Range[nn],6*Range[nn]]]-1,#<=5nn&]] (* Harvey P. Dale, Sep 04 2023 *)

PARI

for(n=1,100, if( matdet(matrix(n^2,n^2,i,j, (abs((i-1)\n - (j-1)\n) + abs((i-1)%n - (j-1)%n)==1) + (i==j) ))==0, print1(n,", ") ) ) \\ Max Alekseyev, Apr 23 2010

PARI

Vec(x*(x^9+4*x^8-3*x^7+7*x^6-5*x^5+8*x^4-5*x^3+7*x^2-3*x+4) / ((x-1)^2*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)) + O(x^100)) \\ Colin Barker, Dec 15 2017

Sage

[n for n in [1..200] if (n+1)%5==0 or (n+1)%6==0] # Tom Edgar, Dec 15 2017

Formula

G.f.: x*(x^9+4*x^8-3*x^7+7*x^6-5*x^5+8*x^4-5*x^3+7*x^2-3*x+4) / ((x-1)^2*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)). - Colin Barker, Dec 03 2012 ["Empirical" removed after Tom Edgar's comment by Andrey Zabolotskiy, Dec 15 2017]

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.

Extensions

Twelve more terms from Max Alekseyev, Apr 23 2010

a(33)-a(40) from Max Alekseyev, Feb 15 2013

More terms from Tom Edgar, Dec 15 2017