A383854 - OEIS

A383854

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

8, 41, 122, 275, 524, 893, 1406, 2087, 2960, 4049, 5378, 6971, 8852, 11045, 13574, 16463, 19736, 23417, 27530, 32099, 37148, 42701, 48782, 55415, 62624, 70433, 78866, 87947, 97700, 108149, 119318, 131231, 143912, 157385, 171674, 186803, 202796, 219677

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OFFSET

1,1

COMMENTS

Theorem: a(n) counts grid pairs invariant under rotation and reflection with maximal absolute coordinate value n.

LINKS

Paolo Xausa, Table of n, a(n) for n = 1..10000

Ed Pegg Jr, Postings to SeqFan Mailing List, Two grid points with max n, starting May 11 2025.

Ed Pegg Jr, Math StackExchange, Distinct point pair dihedral invariants on a grid, May 13 2025.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

From Stefano Spezia, May 15 2025: (Start)

G.f.: x*(8 + 9*x + 6*x^2 + x^3)/(1 - x)^4.

E.g.f.: 1 + exp(x)*(4*x^3 + 12*x^2 + 9*x - 1). (End)

EXAMPLE

There are 8 invariant pairs of points with maximal absolute coordinate value 1: {{{0,0},{1,0}}, {{0,-1},{0,1}}, {{0,0},{1,1}}, {{0,1},{1,0}}, {{0,1},{1,-1}}, {{1,0},{1,1}}, {{-1,1},{1,-1}}, {{1,-1},{1,1}}}.

MATHEMATICA

A383854[n_] := 4*n^3 + 5*n - 1; Array[A383854, 50] (* or *)

LinearRecurrence[{4, -6, 4, -1}, {8, 41, 122, 275}, 50] (* Paolo Xausa, Jun 30 2025 *)

CROSSREFS

For point pairs in a(n), A014106[n] and A005893[n] count usage of (-n,n) and (0,n).

Sequence in context: A041116 A326286 A250322 * A135797 A171714 A342034

Adjacent sequences: A383851 A383852 A383853 * A383855 A383856 A383857

KEYWORD

nonn,easy

AUTHOR

Ed Pegg Jr, May 12 2025

STATUS

approved