%I #7 Jan 03 2020 10:41:15
%S 1,1,2,4,8,13,20,30,45,69,110,184,323,591,1116,2154,4217,8329,16538,
%T 32940,65727,131283,262376,524542,1048853,2097453,4194630,8388960,
%U 16777595,33554839,67109300
%N a(n) = floor((1/16)*(16 + 2^n - 8*n + 8*n^2)).
%C A face number function for a type of exceptional group expansion using Euler's formula V=E-F+2.
%C Derived in Mathematica to give known exceptional group polyhedron sequence: (Platonic solids) e = n*(n - 1); v = f - 2^(n - 3); Solve[v + f - e - 2 == 0, f] Table[Round[{-e, v, f}], {n, 1, 7}] {{0, 1, 1}, {-2, 2, 2}, {-6, 4, 4}, {-12, 6, 8}, {-20, 9, 13}, {-30, 12, 20}, {-42, 14, 30}} Table[Apply[Plus, Round[{-e, v, f}]], {n, 1, 7}]->{2, 2, 2, 2, 2, 2, 2} This result is just a sequence of numbers that seem to work.
%H Colin Barker, <a href="/A130840/b130840.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,7,-2).
%F From _Colin Barker_, Jan 03 2020: (Start)
%F G.f.: x*(1 - 4*x + 6*x^2 - 4*x^3 + x^4 - 3*x^5 + 3*x^6 - x^7) / ((1 - x)^3*(1 - 2*x)).
%F a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4) for n>8.
%F a(n) = 2 + 2^(n-5) - (3*n)/2 + n^2/2 for n>4.
%F (End)
%F E.g.f.: (1/96)*(3*exp(2*x)-6*x-6*x^2-4*x^3-2*x^4+48*exp(x)*(4-2*x+x^2)-195). - _Stefano Spezia_, Jan 03 2020 after _Colin Barker_
%t Table[Round[(1/16)(16 + 2^n - 8 n + 8 n^2)], {n, 0, 30}]
%o (PARI) Vec(x*(1 - 4*x + 6*x^2 - 4*x^3 + x^4 - 3*x^5 + 3*x^6 - x^7) / ((1 - x)^3*(1 - 2*x)) + O(x^40)) \\ _Colin Barker_, Jan 03 2020
%K nonn,easy
%O 1,3
%A _Roger L. Bagula_, Jul 19 2007