A348645 a(n) = (12*n + 1)*(5184*n^2 + 540*n + 13).

13, 74581, 545725, 1786693, 4170733, 8071093, 13861021, 21913765, 32602573, 46300693, 63381373, 84217861, 109183405, 138651253, 172994653, 212586853, 257801101, 309010645, 366588733, 430908613, 502343533, 581266741, 668051485, 763071013, 866698573, 979307413, 1101270781

Offset: 0

Author

Michel Marcus, Oct 27 2021

Keywords

• nonn
• easy
• changed

Comments

a(n) is the entry (2,1) 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, A348646.

Programs

PARI

a(n) = (12*n + 1)*(5184*n^2 + 540*n + 13);

Formula

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

G.f.: (13 + 74529*x + 247479*x^2 + 51227*x^3)/(x-1)^4.

E.g.f.: (13 + 74568*x + 198288*x^2 + 62208*x^3)*exp(x).

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