A380996 a(n) is the number of vertices in the n-fold iterated barycentric subdivision of a triangle (or 2-simplex).
3, 7, 25, 121, 673, 3937, 23425, 140161, 840193, 5039617, 30234625, 181401601, 1088397313, 6530359297, 39182106625, 235092541441, 1410555052033, 8463329918977, 50779978727425, 304679870791681, 1828079221604353, 10968475323334657, 65810851927425025, 394865111539384321
Offset: 0
Examples
a(0)=1 because a triangle has 3 vertices. After one barycentric subdivision, we add 3 vertices at the midpoints of the sides, and 1 vertex at the center, for a(1)=7 vertices total. A second barycentric subdivision adds 12 more midpoints and 6 more centroids, so a(2)=25.
Links
- Wikipedia, Barycentric subdivision.
- Index entries for linear recurrences with constant coefficients, signature (9,-20,12).
Programs
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Mathematica
Table[(6^n + 3*2^n)/2 + 1, {n, 0, 50}] -
Python
def A380996(n): return 3**n+3<
Formula
a(n) = (6^n + 3*2^n)/2 + 1.
From Stefano Spezia, Feb 23 2025: (Start)
G.f.: (3 - 20*x + 22*x^2)/((1 - x)*(1 - 2*x)*(1 - 6*x)).
E.g.f.: exp(x)*(2 + 3*exp(x) + exp(5*x))/2. (End)
a(n) = 3*A074502(n-1) - 2 for n>0. - Hugo Pfoertner, Feb 23 2025
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