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%I #24 Sep 08 2022 08:45:15
%S 1,6,27,84,205,426,791,1352,2169,3310,4851,6876,9477,12754,16815,
%T 21776,27761,34902,43339,53220,64701,77946,93127,110424,130025,152126,
%U 176931,204652,235509,269730,307551,349216
%N Polar structured meta-anti-diamond numbers, the n-th number from a polar structured n-gonal anti-diamond number sequence.
%H Vincenzo Librandi, <a href="/A100188/b100188.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5, 1).
%F a(n) = (1/6)*(2*n^4 - 2*n^2 + 6*n).
%F G.f.: x*(1 + x + 7*x^2 - x^3)/(1-x)^5. - _Colin Barker_, Apr 16 2012
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(1)=1, a(2)=6, a(3)=27, a(4)=84, a(5)=205. - _Harvey P. Dale_, May 11 2016
%F E.g.f.: (3*x + 6*x^2 + 6*x^3 + x^4)*exp(x)/3. - _G. C. Greubel_, Nov 08 2018
%e There are no 1- or 2-gonal anti-diamonds, so 1 and (2n+2) are the first and second terms since all the sequences begin as such.
%t Table[(2n^4-2n^2+6n)/6,{n,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1}, {1,6,27,84,205},40] (* _Harvey P. Dale_, May 11 2016 *)
%o (Magma) [(1/6)*(2*n^4-2*n^2+6*n): n in [1..40]]; // _Vincenzo Librandi_, Aug 18 2011
%o (PARI) vector(40, n, (n^4 -n^2 +3*n)/3) \\ _G. C. Greubel_, Nov 08 2018
%Y Cf. A000578, A000447, A004466, A007588, A063521, A062523 - "polar" structured anti-diamonds; A100189 - "equatorial" structured meta-anti-diamond numbers; A006484 for other structured meta numbers; and A100145 for more on structured numbers.
%K nonn,easy
%O 1,2
%A James A. Record (james.record(AT)gmail.com), Nov 07 2004
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