login
Figurate numbers based on the small stellated dodecahedron: a(n) = n*(21*n^2 - 33*n + 14)/2.
1

%I #30 Sep 08 2022 08:46:22

%S 1,32,156,436,935,1716,2842,4376,6381,8920,12056,15852,20371,25676,

%T 31830,38896,46937,56016,66196,77540,90111,103972,119186,135816,

%U 153925,173576,194832,217756,242411,268860,297166,327392,359601,393856,430220,468756,509527

%N Figurate numbers based on the small stellated dodecahedron: a(n) = n*(21*n^2 - 33*n + 14)/2.

%C The small stellated dodecahedron is a 3D nonconvex regular polyhedron represented by the Schlaefli symbol {5/2, 5}.

%C When truncated, a degenerate dodecahedron is produced. It is then easy to recognize that every small stellated dodecahedron can be constructed by morphing the 12 pentagonal faces of a regular dodecahedron into pentagonal pyramids.

%C The last digits form a cycle of length 20 [1, 2, 6, 6, ..., 1, 2, 6, 6].

%H Colin Barker, <a href="/A318159/b318159.txt">Table of n, a(n) for n = 1..1000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Small_stellated_dodecahedron">Small stellated 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) = A006566(n) + 12*A002411(n-1).

%F a(n) == a(n+20) (mod 10).

%F From _Colin Barker_, Aug 20 2018: (Start)

%F G.f.: x*(1 + 28*x + 34*x^2) / (1 - x)^4.

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

%F (End)

%t Table[(n (14 - 33 n + 21 n^2)) / 2, {n, 45}] (* _Vincenzo Librandi_, Aug 27 2018 *)

%t CoefficientList[Series[(1 + 28*x + 34*x^2) / (1 - x)^4 , {x, 0, 45}], x] (* or *)

%t LinearRecurrence[{4, -6, 4, -1}, {1, 32, 156, 436}, 45] (* _Stefano Spezia_, Sep 02 2018 *)

%o (PARI) Vec(x*(1 + 28*x + 34*x^2) / (1 - x)^4 + O(x^40)) \\ _Colin Barker_, Aug 20 2018

%o (PARI) a(n) = (n*(14 - 33*n + 21*n^2)) / 2 \\ _Colin Barker_, Aug 20 2018

%o (Magma) [n*(21*n^2-33*n+14)/2: n in [1..40]]; // _Vincenzo Librandi_, Aug 27 2018

%Y Cf. A006566, A002411.

%K nonn,easy

%O 1,2

%A _Alejandro J. Becerra Jr._, Aug 19 2018

%E More terms from _Colin Barker_, Aug 20 2018