A093509 -id:A093509 - cp's OEIS frontend

A353280 n is a term if n = 0 or n does not divide F(n, k) for all k >= 0, where F(n, k) are the Fibonacci numbers A352744.

0, 5, 6, 10, 12, 15, 18, 20, 24, 25, 30, 35, 36, 40, 42, 45, 48, 50, 54, 55, 56, 60, 65, 66, 70, 72, 75, 78, 80, 84, 85, 90, 91, 95, 96, 100, 102, 105, 108, 110, 112, 114, 115, 120, 125, 126, 130, 132, 135, 138, 140, 144, 145, 150, 153, 155, 156, 160, 162, 165

Offset: 1

Peter Luschny, Apr 09 2022

nonn

n is a term if 0 is not a term of the sequence A352747(n, .). Since A352747(n, .) is for all n a pure periodic sequence, it is sufficient to require that 0 is not a term of period(A352747(n, .)). Since the length of the period is <= n, the condition can be checked in a finite number of steps.

The multiples of 5 and 6 (A093509) are a subsequence. The terms not of this form start 56, 91, 112, ..., and are in A353281.

			period(A352747(6, .)) = (5, 1, 3) is zero-free, therefore 6 is a term of a.
period(A352747(7, .)) = (1, 0, 6, 5, 4, 3, 2), thus 7 is not a term of a.

a = A093509 union A353281.

Cf. A352744, A352747.

f := n -> combinat:-fibonacci(n): F := (n, k) -> (n-1)*f(k) + f(k+1):
df := n -> denom(f(n)/n) - 1: period := n -> [seq(modp(F(k,n), n), k = 0..df(n))]:
isA353280 := n -> n = 0 or not member(0, period(n)):
select(isA353280, [$(0..166)]);

def F(n, k): return (n - 1)*fibonacci(k) + fibonacci(k + 1)
def df(n): return denominator(fibonacci(n) / n)
def period(n): return (Integer(n).divides(F(k, n)) for k in range(df(n)))
def isA353280(n): return n == 0 or not any([k == True for k in period(n)])
def A353280List(upto): return [n for n in range(upto + 1) if isA353280(n)]
print(A353280List(165))

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

Original entry on oeis.org

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

Vincent Delecroix, Jul 11 2009

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

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).

Cf. A093509, A117870.

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

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

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

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

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.

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

A353281 k is a term if 5 and 6 do not divide k and k does not divide F(n, j) for all j >= 0, where F(n, j) are the Fibonacci numbers.

Original entry on oeis.org

56, 91, 112, 153, 182, 224, 273, 364, 392, 406, 448, 459, 616, 637, 703, 728, 752, 784, 812, 819, 896, 952, 979, 1001, 1064, 1071, 1183, 1232, 1274, 1288, 1377, 1406, 1431, 1456, 1504, 1547, 1568, 1624, 1683, 1729, 1736, 1792, 1892, 1904, 1911, 1958, 1989

Offset: 1

Peter Luschny, Apr 10 2022

nonn

			91 is a term because period(A352747(91, .)) = [34, 60, 86, 21, 47, 73, 8] is zero-free, and 5 and 6 do not divide 91.

{a(n)} union A093509 = A353280.

Cf. A352744, A352747.

def isA353281(n): return not isA093509(n) and isA353280(n)
def A353281List(upto): return [n for n in range(upto + 1) if isA353281(n)]
print(A353281List(400))

cp's OEIS Frontend

A353280 n is a term if n = 0 or n does not divide F(n, k) for all k >= 0, where F(n, k) are the Fibonacci numbers A352744.

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

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A353281 k is a term if 5 and 6 do not divide k and k does not divide F(n, j) for all j >= 0, where F(n, j) are the Fibonacci numbers.

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Sage