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A028993 Odd 10-gonal (or decagonal) numbers.

%I A028993 #38 Feb 16 2025 08:32:35
%S A028993 1,27,85,175,297,451,637,855,1105,1387,1701,2047,2425,2835,3277,3751,
%T A028993 4257,4795,5365,5967,6601,7267,7965,8695,9457,10251,11077,11935,12825,
%U A028993 13747,14701,15687,16705,17755,18837,19951,21097,22275,23485,24727,26001,27307,28645
%N A028993 Odd 10-gonal (or decagonal) numbers.
%H A028993 Vincenzo Librandi, <a href="/A028993/b028993.txt">Table of n, a(n) for n = 0..1000</a>
%H A028993 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DecagonalNumber.html">Decagonal Number</a>.
%H A028993 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A028993 a(n) = (2*n+1)*(8*n+1). - _N. J. A. Sloane_
%F A028993 G.f.: -(7*x^2+24*x+1)/(x-1)^3. - _Colin Barker_, Nov 18 2012
%F A028993 Sum_{n>=0} 1/a(n) = (4*log(2) + (sqrt(2)+1)*Pi + 2*sqrt(2)*log(1+sqrt(2)))/12. - _Amiram Eldar_, Feb 27 2022
%F A028993 From _Elmo R. Oliveira_, Oct 27 2024: (Start)
%F A028993 E.g.f.: exp(x)*(1 + 26*x + 16*x^2).
%F A028993 a(n) = A005408(n)*A017077(n) = A001107(2*n+1).
%F A028993 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%p A028993 A028993:=n->(2*n+1)*(8*n+1): seq(A028993(n), n=0..100); # _Wesley Ivan Hurt_, Apr 26 2017
%t A028993 CoefficientList[Series[-(7 x^2 + 24 x + 1)/(x - 1)^3, {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 18 2013 *)
%t A028993 Select[PolygonalNumber[10,Range[100]],OddQ] (* or *) LinearRecurrence[{3,-3,1},{1,27,85},50] (* _Harvey P. Dale_, May 03 2023 *)
%o A028993 (Magma) [(2*n+1)*(8*n+1): n in [0..60]]; // _Vincenzo Librandi_, Oct 18 2013
%o A028993 (PARI) a(n)=(2*n+1)*(8*n+1) \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A028993 Cf. A001107, A005408, A017077, A028994.
%K A028993 nonn,easy
%O A028993 0,2
%A A028993 _Patrick De Geest_