cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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

Original entry on oeis.org

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

Views

Author

Lamine Ngom, Jul 21 2021

Keywords

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

Crossrefs

Cf. A152691, A239035, A190577, A346515.

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

A346515 a(n) = n*(n+2)*(n+4)*(n+6)*(n+8)*(n+10)*(n+12)*(n+14).

Original entry on oeis.org

0, 2027025, 10321920, 34459425, 92897280, 218243025, 464486400, 916620705, 1703116800, 3011753745, 5109350400, 8365982625, 13284311040, 20534684625, 30996725760, 45808142625, 66421555200, 94670161425, 132843110400, 183771489825, 250925875200, 338526428625, 451666575360
Offset: 0

Views

Author

Lamine Ngom, Jul 21 2021

Keywords

Comments

a(n) can always be expressed as the difference of two squares: x^2 - y^2.
A346514(n) gives the x-values for each product. The y-values being 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) = 8*k^3*n + 28*k^4.
A239035(n) corresponds to a(n,k) in the case k = 1, with related y(n,k) = A346376(n).

Links

Crossrefs

Cf. A239035, A190577, A346514, A346376.

Programs

  • Mathematica
    a[n_] := (n + 14)!!/(n - 2)!!; Array[a, 23, 0] (* Amiram Eldar, Jul 22 2021 *)

Formula

a(n) = A346514(n)^2 - A152691(n+7)^2.
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.