Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #14 Jul 21 2021 09:33:23
%S 28,204,604,1348,2580,4468,7204,11004,16108,22780,31308,42004,55204,
%T 71268,90580,113548,140604,172204,208828,250980,299188,354004,416004,
%U 485788,563980,651228,748204,855604,974148,1104580,1247668,1404204,1575004,1760908,1962780
%N a(n) = n^4 + 14*n^3 + 63*n^2 + 98*n + 28.
%C The product of eight consecutive positive integers can always be expressed as the difference of two squares: x^2 - y^2.
%C This sequence gives the x-values for each product. The y-values are A017113(n+4).
%C a(n) is always divisible by 4. In addition, we have (a(n)+16)/4 belongs to A028387.
%C Are 4 and 8 the unique values of k such that the product of k consecutive integers is always distant to upper square by a square?
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = A239035(n)^2 - A017113(n+4)^2.
%F a(n) = 4*(A028387(A046691(n+2)) - 4).
%F G.f.: 4*(7 + 16*x - 34*x^2 + 22*x^3 - 5*x^4)/(1 - x)^5. - _Stefano Spezia_, Jul 14 2021
%Y Cf. A239035, A017113, A028387, A046691.
%K nonn,easy
%O 0,1
%A _Lamine Ngom_, Jul 14 2021