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%I #44 Mar 12 2024 23:41:47
%S 0,1,12,47,120,245,436,707,1072,1545,2140,2871,3752,4797,6020,7435,
%T 9056,10897,12972,15295,17880,20741,23892,27347,31120,35225,39676,
%U 44487,49672,55245,61220,67611,74432,81697,89420,97615,106296,115477,125172
%N Cubic star numbers: a(n) = n^3 + 4*Sum_{i=0..n-1} i^2.
%C Also as a(n) = (1/6)*(14*n^3 - 12*n^2 + 4*n), n>0: structured cubeoctahedral numbers (vertex structure 7); and structured pentagonal anti-diamond numbers (vertex structure 7) (cf. A004466 = alternate vertex) (cf. A100188 = structured anti-diamonds). Cf. A100145 for more on structured polyhedral numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
%C Starting with offset 1 = binomial transform of [1, 11, 24, 14, 0, 0, 0, ...]. - _Gary W. Adamson_, Aug 05 2009
%C This is prime for a(3) = 47. The subsequence of semiprimes begins: 707, 7435, 10897, 20741, 115477, 341797, 825091, 897097, no more through a(100). - _Jonathan Vos Post_, May 27 2010
%D T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
%H Vincenzo Librandi, <a href="/A051673/b051673.txt">Table of n, a(n) for n = 0..1000</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) = n*(n*(7*n-6) + 2)/3.
%F G.f.: x*(1+8*x+5*x^2)/(1-x)^4. - _Bruno Berselli_, May 12 2011
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=12, a(3)=47. - _Harvey P. Dale_, Jul 22 2011
%F From _Reinhard Zumkeller_, Jul 25 2012: (Start)
%F a(n) = A214659(n) - A002378(n).
%F a(n) = Sum_{k=1..n} A214661(n, k), for n > 0 (row sums). (End)
%F E.g.f.: (x/3)*(3 + 15*x + 7*x^2)*exp(x). - _G. C. Greubel_, Mar 10 2024
%e a(51) = 51*(51*(7*51-6)+2)/3 = 304351 = 17 * 17903 is semiprime. - _Jonathan Vos Post_, May 27 2010
%p A051673:=n->n*(n*(7*n-6)+2)/3; seq(A051673(n), n=0..40); # _Wesley Ivan Hurt_, Feb 02 2014
%t Table[n^3+4Sum[i^2,{i,0,n-1}],{n,0,40}] (* or *) LinearRecurrence[ {4,-6,4,-1},{0,1,12,47},40] (* _Harvey P. Dale_, Jul 22 2011 *)
%o (Magma) [n*(n*(7*n-6)+2)/3: n in [0..50]]; // _Vincenzo Librandi_, May 12 2011
%o (PARI) a(n)=n*(n*(7*n-6)+2)/3 \\ _Charles R Greathouse IV_, Oct 07 2015
%o (SageMath) [n*(7*n^2-6*n+2)/3 for n in range(51)] # _G. C. Greubel_, Mar 10 2024
%Y Cf. A002378, A004466, A005915, A051662, A100145, A100188.
%Y Cf. A214659, A214661.
%K easy,nice,nonn
%O 0,3
%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
%E Corrected by _T. D. Noe_, Nov 01 2006, Nov 08 2006
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