A346514 a(n) = n^4 + 28*n^3 + 252*n^2 + 784*n + 448.

448, 1513, 3264, 5905, 9664, 14793, 21568, 30289, 41280, 54889, 71488, 91473, 115264, 143305, 176064, 214033, 257728, 307689, 364480, 428689, 500928, 581833, 672064, 772305, 883264, 1005673, 1140288, 1287889, 1449280, 1625289, 1816768, 2024593, 2249664, 2492905, 2755264

Offset

0,1

Comments

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.

This sequence gives the x-values for each product. The y-values are A152691(n+7).

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

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

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

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

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

Links

Table of n, a(n) for n=0..34.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

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

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

Crossrefs

Cf. A152691, A239035, A190577, A346515.

Sequence in context: A093786 A251646 A235177 * A194648 A348344 A234438

Adjacent sequences: A346511 A346512 A346513 * A346515 A346516 A346517

Keyword

nonn,easy

Author

Lamine Ngom, Jul 21 2021

Status

approved