A193252 - OEIS

%I #65 Sep 08 2022 08:45:58

%S 1,75,365,1015,2169,3971,6565,10095,14705,20539,27741,36455,46825,

%T 58995,73109,89311,107745,128555,151885,177879,206681,238435,273285,

%U 311375,352849,397851,446525,499015,555465,616019,680821,750015,823745,902155,985389,1073591

%N Great 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 great rhombicosidodecahedron. Each iteration requires the addition of (n-2) edge units and (n-1) vertices to complete the centered polygon of each face: centered squares, centered octagons and centered hexagons.

%H Bruno Berselli, <a href="/A193252/b193252.txt">Table of n, a(n) for n = 1..1000</a>

%H OEIS Wiki, <a href="/wiki/(Centered_polygons)_pyramidal_numbers">(Centered polygons) pyramidal numbers</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetrahedral_number">Tetrahedral number</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Triangular_number">Triangular number</a>

%H Wikipedia, <a href="https://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) = 24*n^3 - 36*n^2 + 14*n - 1.

%F G.f.: x*(1+x)*(1+70*x+x^2)/(1-x)^4; a(n) = (2*n-1)*A069190(n). - _Bruno Berselli_, Jul 21 2011

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=75, a(2)=365, a(3)=1015. - _Harvey P. Dale_, Jul 27 2011

%F a(n) = 72 * A000330(n-1) + A005408(n-1). - _Bruce J. Nicholson_, Feb 23 2019

%F E.g.f.: 1 + (-1 + 2*x + 36*x^2 + 24*x^3)*exp(x). - _G. C. Greubel_, Feb 26 2019

%t Table[24n^3-36n^2+14n-1,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,75,365,1015},40] (* _Harvey P. Dale_, Jul 27 2011 *)

%o (Excel) =24*ROW()^3-36*ROW()^2+14*ROW()-1

%o (Magma) A069190:=func<n | 12*n^2-12*n+1>; [(2*n-1)*A069190(n): n in [1..40]];  // _Bruno Berselli_, Jul 21 2011

%o (PARI) for(n=1,40, print1(24*n^3-36*n^2+14*n-1", "));  \\ _Bruno Berselli_, Jul 21 2011

%o (Sage) [24*n^3 -36*n^2 +14*n -1 for n in (1..40)] # _G. C. Greubel_, Feb 26 2019

%o (GAP) List([1..40], n-> 24*n^3 -36*n^2 +14*n -1) # _G. C. Greubel_, Feb 26 2019

%Y First differences in 2*A158591.

%Y Cf. A001844 (centered square numbers), A016754 (centered octagonal numbers), A003215 (centered hexagonal numbers).

%K nonn,easy

%O 1,2

%A _Craig Ferguson_, Jul 19 2011