A393192 - OEIS

%I #16 Feb 12 2026 19:57:48

%S 1,69,377,1117,2481,4661,7849,12237,18017,25381,34521,45629,58897,

%T 74517,92681,113581,137409,164357,194617,228381,265841,307189,352617,

%U 402317,456481,515301,578969,647677,721617,800981,885961,976749,1073537,1176517,1285881,1401821,1524529,1654197,1791017,1935181

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

%C 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.

%H Derek Delk, <a href="/A393192/b393192.txt">Table of n, a(n) for n = 1..10000</a>

%H Derek Delk, <a href="/A393192/a393192.png">Visualization of Six Terms</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EschersSolid.html">Escher's Solid</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/First_stellation_of_the_rhombic_dodecahedron">First stellation of the rhombic dodecahedron</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

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

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

%F 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).

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

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

%F 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.

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

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

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

%t A393192[n_] := 4*n*(2*n*(4*n - 9) + 15) - 19; Array[A393192, 50] (* or *)

%t LinearRecurrence[{4, -6, 4, -1}, {1, 69, 377, 1117}, 50] (* _Paolo Xausa_, Feb 10 2026 *)

%Y Cf. A046142, A057813, A172482.

%K nonn,easy

%O 1,2

%A _Derek Delk_, Feb 04 2026