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%I #63 Sep 09 2022 03:54:15
%S 1,51,245,679,1449,2651,4381,6735,9809,13699,18501,24311,31225,39339,
%T 48749,59551,71841,85715,101269,118599,137801,158971,182205,207599,
%U 235249,265251,297701,332695,370329,410699,453901,500031,549185,601459,656949,715751
%N Small rhombicuboctahedron with faces of centered polygons.
%C The sequence starts with a central dot and expands outward with (n-1) centered polygonal pyramids producing a small rhombicuboctahedron. Each iteration requires the addition of (n-2) edges and (n-1) vertices to complete the centered polygon of each face. [centered triangles (A005448) and centered squares (A001844)]
%H Vincenzo Librandi, <a href="/A193250/b193250.txt">Table of n, a(n) for n = 1..10000</a>
%H OEIS, <a href="/wiki/(Centered_polygons)_pyramidal_numbers"> (Centered_polygons) pyramidal numbers</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Tetrahedral_number">Tetrahedral number</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Triangular_number">Triangular number</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Centered_polygonal_number">Centered polygonal number</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) = 16*n^3 - 24*n^2 + 10*n - 1.
%F G.f.: x*(1+x)*(x^2 + 46*x + 1) / (x-1)^4. - _R. J. Mathar_, Aug 26 2011
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=51, a(2)=245, a(3)=679. - _Harvey P. Dale_, Aug 27 2011
%F E.g.f.: 1 + (-1 + 2*x + 24*x^2 + 16*x^3)*exp(x). - _G. C. Greubel_, Nov 10 2018
%t Table[16n^3-24n^2+10n-1,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,51,245,679},40] (* _Harvey P. Dale_, Aug 27 2011 *)
%o (Excel) copy and paste the following formula =16 *ROW()^3-24 *ROW()^2+10*ROW()-1 fill down to desired size.
%o (Magma) [16*n^3-24*n^2+10*n-1: n in [1..50]]; // _Vincenzo Librandi_, Aug 30 2011
%o (PARI) vector(40, n, 16*n^3-24*n^2+10*n-1) \\ _G. C. Greubel_, Nov 10 2018
%Y Cf. A079414 (partial sums).
%Y Cf. A005448, A001844.
%K nonn,easy
%O 1,2
%A _Craig Ferguson_, Jul 19 2011
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