A346514 - OEIS

%I #16 Aug 23 2021 22:43:34

%S 448,1513,3264,5905,9664,14793,21568,30289,41280,54889,71488,91473,

%T 115264,143305,176064,214033,257728,307689,364480,428689,500928,

%U 581833,672064,772305,883264,1005673,1140288,1287889,1449280,1625289,1816768,2024593,2249664,2492905,2755264

%N a(n) = n^4 + 28*n^3 + 252*n^2 + 784*n + 448.

%C The product of eight positive integers shifted by 2; i.e., m * (m+2) * (m+4) * ... * (m+14) = A346515(m) 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 A152691(n+7).

%C More generally, for any k, we have n * (n+k) * (n+2*k) * ... * (n+7*k) = a(n,k) = x(n,k)^2 - y(n,k)^2, where

%C   x(n,k) = n^4 + 14*k*n^3 + 63*k^2*n^2 + 98*k^3*n + 28*k^4,

%C   y(n,k) = 4*k^3*(2*n + 7*k).

%C A239035(n) corresponds to a(n,k) in the case k = 1, with related y(n,k) = A346376(n).

%C This sequence is y(n,k) in the case k = 2, with related y(n,k) = A152691(n+7).

%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) = sqrt(A346515(n) + A152691(n+7)^2).

%F G.f.: (448 - 727*x + 179*x^2 + 235*x^3 - 111*x^4)/(1 - x)^5. - _Stefano Spezia_, Jul 22 2021

%Y Cf. A152691, A239035, A190577, A346515.

%K nonn,easy

%O 0,1

%A _Lamine Ngom_, Jul 21 2021