A348646 a(n) = (72*n + 5)*(1296*n^2 + 153*n + 4).

20, 111881, 818606, 2680067, 6256136, 12106685, 20791586, 32870711, 48903932, 69451121, 95072150, 126326891, 163775216, 207976997, 259492106, 318880415, 386701796, 463516121, 549883262, 646363091, 753515480, 871900301, 1002077426, 1144606727, 1300048076, 1468961345, 1651906406

Offset: 0

Author

Michel Marcus, Oct 27 2021

Keywords

• nonn
• easy
• changed

Comments

a(n) is the entry (2,2) of a family of unimodular matrices none of whose entries is 1 or -1, such that when each entry of the matrix is replaced by its cube, the resulting matrix is also unimodular.

In these matrices, the entries (1,3) and (3,1) = 2; the entries (2,3) and (3,2) = 3; the entry (3,3) = 0.

Links

• Ajai Choudhry, A diophantine problem concerning third order matrices, arXiv:2110.12643 [math.NT], 2021.
• Wikipedia, Unimodular matrix.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000004, A007395, A010701, A348643, A348644, A348645.

Programs

PARI

a(n) = (72*n + 5)*(1296*n^2 + 153*n + 4);

Formula

From Elmo R. Oliveira, Sep 04 2025: (Start)

G.f.: (20 + 111801*x + 371202*x^2 + 76849*x^3)/(x-1)^4.

E.g.f.: (20 + 111861*x + 297432*x^2 + 93312*x^3)*exp(x).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)