A186636 a(n) = n*(n^3 + n^2 + 2*n + 1).

0, 5, 34, 129, 356, 805, 1590, 2849, 4744, 7461, 11210, 16225, 22764, 31109, 41566, 54465, 70160, 89029, 111474, 137921, 168820, 204645, 245894, 293089, 346776, 407525, 475930, 552609, 638204, 733381, 838830, 955265, 1083424, 1224069, 1377986, 1545985, 1728900, 1927589, 2142934, 2375841, 2627240

Offset

0,2

Comments

Number of lunar divisors of the number 11111 in base n+1.

From I. J. Kennedy, May 01 2025: (Start)

It appears that Table 10 of the Dismal Arithmetic paper matches the number of equivalence classes, with respect to matrix similarity, of k X k integer matrices under mod b-1 arithmetic. At least that's the case when b-1 is prime and we're dealing with a field GF(p).

For example, there are 805 lunar divisors of 1111_6, and there are 805 equivalence classes of 4 X 4 matrices over GF(5). (End)

Links

Table of n, a(n) for n=0..40.

David Applegate, Marc LeBrun, and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing].

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

From Elmo R. Oliveira, May 24 2026: (Start)

G.f.: x*(5 + 9*x + 9*x^2 + x^3)/(1 - x)^5.

E.g.f.: x*(5 + 12*x + 7*x^2 + x^3)*exp(x).

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

a(n) = n*A100705(n). (End)

Mathematica

Table[n(n^3+n^2+2n+1), {n, 0, 40}] (* Harvey P. Dale, Nov 14 2024 *)

Crossrefs

Cf. A100705.

Sequence in context: A272955 A317297 A135973 * A034224 A167023 A316559

Adjacent sequences: A186633 A186634 A186635 * A186637 A186638 A186639

Keyword

nonn,easy

Author

N. J. A. Sloane, Feb 24 2011

Status

approved