cp's OEIS Frontend

A306367 a(n) = numerator of (n^2 + 2)/(n + 2).

%I A306367 #22 Feb 28 2019 15:59:39
%S A306367 1,1,3,11,3,27,19,17,33,83,17,123,73,57,99,227,43,291,163,121,201,443,
%T A306367 81,531,289,209,339,731,131,843,451,321,513,1091,193,1227,649,457,723,
%U A306367 1523,267,1683,883,617,969,2027,353,2211,1153,801,1251,2603,451
%N A306367 a(n) = numerator of (n^2 + 2)/(n + 2).
%C A306367 If P(x) and Q(x) are coprime integral polynomials such that Q(n) > 0 for n >= 0 then the sequence of numerators of the rational numbers P(n)/Q(n) for n >= 0 and the sequence of denominators of P(n)/Q(n) for n >= 0 are both quasi-polynomial in n. In fact, there exists a purely periodic sequence b(n) such that numerator(P(n)/Q(n)) = P(n)/b(n) and denominator(P(n)/Q(n)) = Q(n)/b(n).
%C A306367 Here we take P(n) = n^2 + 2 and Q(n) = n + 2. Cf. A228564 (case P(n) = n^2 + 1, Q(n) = n + 1).
%H A306367 Muniru A Asiru, <a href="/A306367/b306367.txt">Table of n, a(n) for n = 0..10000</a>
%H A306367 P. Bala, <a href="/A306367/a306367.pdf">A note on the sequence of numerators of a rational function </a>
%H A306367 Wikipedia, <a href="https://en.wikipedia.org/wiki/Quasi-polynomial">Quasi-polynomial</a>
%F A306367 O.g.f.: (3*x^17 + x^16 + 11*x^15 + 9*x^14 + 9*x^13 + 19*x^12 + 42*x^11 + 8*x^10 + 50*x^9 + 24*x^8 + 14*x^7 + 16*x^6 + 27*x^5 + 3*x^4 + 11*x^3 + 3*x^2 + x + 1)/(1 - x^6)^3.
%F A306367 a(n) = (n^2 + 2)/b(n), where (b(n))n>=0 is the purely periodic sequence [2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, ...] with period 6.
%F A306367 a(n) is a quasi-polynomial in n: a(6*n) = 18*n^2 + 1; a(6*n+1) = 12*n^2 + 4*n + 1; a(6*n+2) = 18*n^2 + 12*n + 3; a(6*n+3) = 36*n^2 + 36*n + 11; a(6*n+4) = 6*n^2 + 8*n + 3; a(6*n+5) = 36*n^2 + 60*n + 27.
%F A306367 a(n) = (n^2 + 2)/( gcd(n+2,2)*gcd(n+2,3) ).
%F A306367 a(n) = (n^2 + 2)/(n + 2) * A060789(n+2).
%p A306367 seq(numer( (n^2 + 2)/(n + 2) ), n = 0..100);
%o A306367 (GAP) List([0..100],n->NumeratorRat((n^2+2)/(n+2)); # _Muniru A Asiru_, Feb 25 2019
%o A306367 (PARI) a(n) = numerator((n^2 + 2)/(n + 2)); \\ _Michel Marcus_, Feb 26 2019
%Y A306367 Cf. A060789, A228564, A306368.
%K A306367 nonn,frac,easy,look
%O A306367 0,3
%A A306367 _Peter Bala_, Feb 14 2019