A393192 a(n) = 32*n^3 - 72*n^2 + 60*n - 19.

1, 69, 377, 1117, 2481, 4661, 7849, 12237, 18017, 25381, 34521, 45629, 58897, 74517, 92681, 113581, 137409, 164357, 194617, 228381, 265841, 307189, 352617, 402317, 456481, 515301, 578969, 647677, 721617, 800981, 885961, 976749, 1073537, 1176517, 1285881, 1401821, 1524529, 1654197, 1791017, 1935181

Offset

1,2

Comments

These are figurate numbers approximating the shape of Escher's solid (solid enclosed by the first stellation of the rhombic dodecahedron) using points in a primitive cubic lattice, as such they could be called the primitive cubic Escher-solid numbers. The n-th such figure has 4*n - 3 points along each convex edge connecting vertices.

Links

Derek Delk, Table of n, a(n) for n = 1..10000

Derek Delk, Visualization of Six Terms

Eric Weisstein's World of Mathematics, Escher's Solid.

Wikipedia, First stellation of the rhombic dodecahedron.

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

Formula

a(n) = 3*A057813(2*n-2) - 2*A046142(n).

a(n) = A046142(n) + 12*Sum_{i=2..n} (2*i - 3)*(4*n - 4*i + 3).

a(n) = A046142(n) + 12*Sum_{i=1..n-1} (2*i - 1 + 2*Sum_{j=1..i-1} (2*j - 1)) + (2*i + 2*Sum_{j=1..i-1} 2*j) = A046142(n) + 12*Sum_{i=1..n-1} A001844(i-1) + A001105(i).

a(n) = A046142(n) + 12*A172482(n-2) for n > 1.

a(n) = (4*n - 3)^3 - 8*Sum_{i=1..n-1} (2*i)^3 - (2*i - 1)^3.

a(n) = (4*n - 3)^3 - 8*(4*(n - 1)^3 + 3*(n - 1)^2) = (4*n - 3)^3 - 8*(4*n - 1)*(n - 1)^2.

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

G.f.: x*(19*x^3 + 107*x^2 + 65*x + 1)/(1 - x)^4.

E.g.f.: (32*x^3 + 24*x^2 + 20*x - 19)*exp(x).

Mathematica

A393192[n_] := 4*n*(2*n*(4*n - 9) + 15) - 19; Array[A393192, 50] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {1, 69, 377, 1117}, 50] (* Paolo Xausa, Feb 10 2026 *)

Crossrefs

Cf. A046142, A057813, A172482.

Sequence in context: A158736 A262456 A161486 * A236158 A253342 A253335

Adjacent sequences: A393189 A393190 A393191 * A393193 A393194 A393195

Keyword

nonn,easy

Author

Derek Delk, Feb 04 2026

Status

approved