A007589 - OEIS

%I M5083 #42 Apr 25 2024 09:34:14

%S 0,1,20,50,92,170,284,434,620,842,1100,1394,1724,2090,2492,2930,3404,

%T 3914,4460,5042,5660,6314,7004,7730,8492,9290,10124,10994,11900,12842,

%U 13820,14834,15884,16970,18092,19250

%N Dodecahedral surface numbers: a(0)=0, a(1)=1, a(2)=20, thereafter 2*((3*n-7)^2 + 21).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A007589/b007589.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = Vertices + Edges*(n-2) + Faces*facenumber(n-3) = 20 + 30(n-2) + 12*A000326(n-3). - _Mitch Harris_, Oct 25 2006

%F From _G. C. Greubel_, Aug 30 2019: (Start)

%F G.f.: x*(1 + 17*x - 7*x^2 + x^3 + 24*x^4)/(1-x)^3.

%F E.g.f.: -140 -73*x -12*x^2 + (140 -66*x +18*x^2)*exp(x). (End)

%p seq(coeff(series(x*(1+17*x-7*x^2+x^3+24*x^4)/(1-x)^3, x, n+1), x, n), n = 0 .. 45); # _G. C. Greubel_, Aug 30 2019

%t Join[{0,1,20},2((3Range[3,40]-7)^2+21)] (* _Harvey P. Dale_, Sep 24 2011 *)

%t Join[{0,1,20}, CoefficientList[Series[2*(25-29*x+22*x^2)/(1-x)^3, {x,0,50}], x] ] (* _Vincenzo Librandi_, Jul 07 2012 *)

%o (Magma) I:=[0, 1, 20, 50, 92, 170]; [n le 6 select I[n] else 3*Self(n-1) -3*Self(n-2)+Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Jul 07 2012

%o (PARI) my(x='x+O('x^45)); concat([0], Vec(x*(1+17*x-7*x^2+x^3+24*x^4)/(1-x)^3)) \\ _G. C. Greubel_, Aug 30 2019

%o (Sage)

%o def A007589_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P(x*(1+17*x-7*x^2+x^3+24*x^4)/(1-x)^3).list()

%o A007589_list(45) # _G. C. Greubel_, Aug 30 2019

%o (GAP) a:=[50,92,170];; for n in [4..45] do a[n]:=3*a[n-1]-3*a[n-2]+ a[n-3]; od; Concatenation([0, 1, 20], a); # _G. C. Greubel_, Aug 30 2019

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, _R. K. Guy_

%E Revised definition from _N. J. A. Sloane_, Sep 25 2011