A130840 a(n) = floor((1/16)*(16 + 2^n - 8*n + 8*n^2)).

1, 1, 2, 4, 8, 13, 20, 30, 45, 69, 110, 184, 323, 591, 1116, 2154, 4217, 8329, 16538, 32940, 65727, 131283, 262376, 524542, 1048853, 2097453, 4194630, 8388960, 16777595, 33554839, 67109300

Offset

1,3

Comments

A face number function for a type of exceptional group expansion using Euler's formula V=E-F+2.

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.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Formula

From Colin Barker, Jan 03 2020: (Start)

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)).

a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4) for n>8.

a(n) = 2 + 2^(n-5) - (3*n)/2 + n^2/2 for n>4.

(End)

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

Mathematica

Table[Round[(1/16)(16 + 2^n - 8 n + 8 n^2)], {n, 0, 30}]

Prog

(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

Crossrefs

Sequence in context: A164476 A164466 A164487 * A115266 A164508 A308094

Adjacent sequences: A130837 A130838 A130839 * A130841 A130842 A130843

Keyword

nonn,easy

Author

Roger L. Bagula, Jul 19 2007

Status

approved