A084069 Numbers k such that 7*k^2 = floor(k*sqrt(7)*ceiling(k*sqrt(7))).

1, 3, 17, 48, 271, 765, 4319, 12192, 68833, 194307, 1097009, 3096720, 17483311, 49353213, 278635967, 786554688, 4440692161, 12535521795, 70772438609, 199781794032, 1127918325583, 3183973182717, 17975920770719, 50743789129440

Offset

1,2

Comments

This is a strong divisibility sequence, that is, GCD(a(n),a(m)) = a(GCD(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Sep 01 2019

Links

Table of n, a(n) for n=1..24.

Wikipedia, Lehmer sequence

Index entries for linear recurrences with constant coefficients, signature (0,16,0,-1).

Formula

a(1)=1, a(2)=3, a(2n) = 6*a(2n-1)-a(2n-2); a(2n+1) = 3*a(2n)-a(2n-1).

a(n)*a(n+3) = -3 + a(n+1)*a(n+2).

G.f.: x*(1+3*x+x^2)/(1-16*x^2+x^4). [corrected by Harvey P. Dale, Oct 31 2011]

a(n) = 16*a(n-2) - a(n-4), n > 4. - Harvey P. Dale, Oct 31 2011

a(n) = U_n(sqrt(18),1) = (alpha^n - beta^n)/(alpha - beta) for n odd and a(n) = 3*U_n(sqrt(18),1) = (sqrt(2)/2)*(alpha^n - beta^n)/(alpha - beta) for n even, where U_n(sqrt(R),Q) denotes the Lehmer sequence with parameters R and Q and alpha = (sqrt(3) + sqrt(14))/2 and beta = (sqrt(3) - sqrt(14))/2. - Peter Bala, Sep 01 2019

Mathematica

CoefficientList[Series[(1+3x+x^2)/(1-16x^2+x^4), {x, 0, 30}], x] (* or *) LinearRecurrence[{0, 16, 0, -1}, {1, 3, 17, 48}, 31] (* Harvey P. Dale, Oct 31 2011 *)

Crossrefs

Cf. A159678, A001080, A001653, A001353, A060645, A001078, A001109, A084068, A084070, A143643.

Sequence in context: A095697 A154304 A144640 * A297514 A307862 A132124

Adjacent sequences: A084066 A084067 A084068 * A084070 A084071 A084072

Keyword

nonn

Author

Benoit Cloitre, May 10 2003

Status

approved