|
%I #37 Sep 08 2022 08:45:13
%S 1,19,64,136,235,361,514,694,901,1135,1396,1684,1999,2341,2710,3106,
%T 3529,3979,4456,4960,5491,6049,6634,7246,7885,8551,9244,9964,10711,
%U 11485,12286,13114,13969,14851,15760,16696,17659,18649,19666,20710,21781
%N a(n) = (27*n^2 + 9*n + 2)/2.
%C Dodecahedral gnomon numbers: first differences of dodecahedral numbers.
%C The sequence is related to other gnomon numbers of polyhedra, known by other more familiar names: triangular numbers (tetrahedral gnomon numbers), hexagonal numbers (cubic gnomon numbers), square pyramidal numbers (octahedral gnomon numbers).
%C A124388 = first differences; second differences = 27. - _Reinhard Zumkeller_, Oct 30 2006
%C Sums of the triangular numbers from A000217(3*n-1) to A000217(3*n+1), with A000217(-1) = 0. - _Bruno Berselli_, Sep 04 2018
%H Vincenzo Librandi, <a href="/A093485/b093485.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = (n+1)*(3*(n+1)-1)*(3*(n+1)-2)/2-n*(3*n-1)*(3*n-2)/2.
%F G.f.: (1 + 16*x + 10*x^2)/(1 - x)^3. - _Colin Barker_, Mar 28 2012
%e a(1) = 19 because (1+1)*(3*(1+1)-1)*(3*(1+1)-2)/2-1*(3*1-1)*(3*1-2)/2 = 2*(6-1)*(6-2)/2 - 1*(3-1)*(3-2)/2 = 20-1 = 19.
%o (Magma) [(27*n^2 + 9*n + 2)/2 : n in [0..50]]; // _Vincenzo Librandi_, Oct 08 2011
%o (Haskell)
%o a093485 n = (9 * n * (3 * n + 1) + 2) `div` 2
%o -- _Reinhard Zumkeller_, Jun 16 2013
%o (PARI) a(n)=(27*n^2+9*n+2)/2 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y Cf. A000217, A000330, A003215, A005901, A006656.
%K nonn,easy
%O 0,2
%A _Michael Joseph Halm_, May 13 2004
%E New definition from _Ralf Stephan_, Dec 01 2004
%E Name corrected and initial term added by _Arkadiusz Wesolowski_, Aug 15 2011
|
