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A346514
a(n) = n^4 + 28*n^3 + 252*n^2 + 784*n + 448.
1
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).
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
KEYWORD
nonn,easy
AUTHOR
Lamine Ngom, Jul 21 2021
STATUS
approved