OFFSET
1,3
COMMENTS
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
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..200
Dustin Moody, Division Polynomials for Jacobi Quartic Curves, NIST publication.
Index entries for linear recurrences with constant coefficients, signature (195,-195,1).
FORMULA
a(n) = A007655(n)^2.
a(2*n - 2) = (a(n) - a(n-1))^2, for n > 1. - Thomas Scheuerle, May 06 2022
O.g.f.: A(x) = x^2*(1 + x)/((1 - x)*(1 - 194*x + x^2)). 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.g.f.: (exp(97*x)*(97*cosh(56*sqrt(3)*x) - 56*sqrt(3)*sinh(56*sqrt(3)*x)) - 96 - exp(x))/96. - Stefano Spezia, Nov 15 2025
EXAMPLE
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.
MATHEMATICA
Select[Table[1/12 (2 n + 5 n^2 + 4 n^3 + n^4), {n, 0, 75000}], IntegerQ[Sqrt[#]]&]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Kelvin Voskuijl, May 05 2022
EXTENSIONS
a(11)-a(14) from Amiram Eldar, May 05 2022
a(15) from Kelvin Voskuijl, Jun 05 2022
STATUS
approved