A351683 - OEIS

%I #74 Nov 28 2022 20:13:07

%S 0,1,196,38025,7376656,1431033241,277613072100,53855504954161,

%T 10447690348035136,2026798072013862225,393188378280341236516,

%U 76276518588314186021881,14797251417754671747008400,2870590498525818004733607721,556879759462590938246572889476

%N Squares that are also 4-dimensional pyramidal numbers.

%C This sequence is a quartic divisibility sequence. a(n+1) divides a(m+1) whenever n divides m. This is because this sequence is based on solutions to a special case of the general Jacobi quartic form y^2 = b*x^4 - 2*c*x^2 + 1. - _Thomas Scheuerle_, May 06 2022

%H Dustin Moody, <a href="https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=908330">Division Polynomials for Jacobi Quartic Curves</a>, NIST publication.

%F a(n) = A007655(n)^2.

%F a(2*n - 2) = (a(n) - a(n-1))^2, for n > 1. - _Thomas Scheuerle_, May 06 2022

%F O.g.f.: A(x) = x^2*(1 - x^2)/(1 - 196*x + 390*x^2 - 196*x^3 + x^4). With offset 0, this is the case P1 = 196, P2 = 194, Q = 1 of a 3-parameter family of fourth-order linear divisibility sequences. See A100047 for further details. - _Peter Bala_, Nov 28 2022

%e 196 is a term because 196 = 14^2 is a perfect square and 196 = (2*6 + 5*6^2 + 4*6^3 + 6^4)/12 is the 6th four-dimensional pyramidal number.

%t Select[Table[1/12 (2 n + 5 n^2 + 4 n^3 + n^4), {n, 0, 75000}], IntegerQ[Sqrt[#]]&]

%Y Intersection of A000290 and A002415.

%Y Cf. A007655, A100047.

%K nonn

%O 1,3

%A _Kelvin Voskuijl_, May 05 2022

%E a(11)-a(14) from _Amiram Eldar_, May 05 2022

%E a(15) from _Kelvin Voskuijl_, Jun 05 2022