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Second 9-gonal (or nonagonal) numbers: a(n) = n*(7*n+5)/2.
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%I #69 Sep 08 2022 08:45:54

%S 0,6,19,39,66,100,141,189,244,306,375,451,534,624,721,825,936,1054,

%T 1179,1311,1450,1596,1749,1909,2076,2250,2431,2619,2814,3016,3225,

%U 3441,3664,3894,4131,4375,4626,4884,5149,5421,5700,5986,6279,6579,6886

%N Second 9-gonal (or nonagonal) numbers: a(n) = n*(7*n+5)/2.

%C This sequence is a bisection of A118277 (even part).

%C Sequence found by reading the line from 0, in the direction 0, 19... and the line from 6, in the direction 6, 39,..., in the square spiral whose vertices are the generalized 9-gonal numbers A118277. - _Omar E. Pol_, Jul 24 2012

%C The early part of this sequence is a strikingly close approximation to the early part of A100752. - _Peter Munn_, Nov 14 2019

%H Vincenzo Librandi, <a href="/A179986/b179986.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 G.f.: x*(6 + x)/(1 - x)^3.

%F a(n) = Sum_{i=0..(n-1)} A017053(i) for n>0.

%F a(-n) = A001106(n).

%F Sum_{i=0..n} (a(n)+i)^2 = ( Sum_{i=(n+1)..2*n} (a(n)+i)^2 ) + 21*A000217(n)^2 for n>0.

%F a(n) = a(n-1)+7*n-1 for n>0, with a(0)=0. - _Vincenzo Librandi_, Feb 05 2011

%F a(0)=0, a(1)=6, a(2)=19; for n>2, a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - _Harvey P. Dale_, Aug 19 2011

%F a(n) = A174738(7n+5). - _Philippe Deléham_, Mar 26 2013

%F a(n) = A001477(n) + 2*A000290(n) + 3*A000217(n). - _J. M. Bergot_, Apr 25 2014

%F a(n) = A055998(4*n) - A055998(3*n). - _Bruno Berselli_, Sep 23 2016

%F E.g.f.: (x/2)*(12 + 7*x)*exp(x). - _G. C. Greubel_, Aug 19 2017

%t f[n_] := n (7 n + 5)/2; f[Range[0, 60]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 05 2011*)

%t LinearRecurrence[{3, -3, 1}, {0, 6, 19}, 60] (* or *) Array[(#(7# + 5))/2&, 60, 0] (* _Harvey P. Dale_, Aug 19 2011 *)

%t CoefficientList[Series[x (6 + x)/(1 - x)^3, {x, 0, 60}], x] (* _Vincenzo Librandi_, Oct 15 2012 *)

%o (Magma) [n*(7*n+5)/2: n in [0..50]]; // _Bruno Berselli_, Sep 23 2016

%o (Magma) I:=[0, 6, 19]; [n le 3 select I[n] else 3*Self(n-1) -3*Self(n-2) +Self(n-3): n in [1..60]]; // _Vincenzo Librandi_, Oct 15 2012

%o (PARI) a(n)=n*(7*n+5)/2 \\ _Charles R Greathouse IV_, Sep 24 2015

%Y Cf. second k-gonal numbers: A005449 (k=5), A014105 (k=6), A147875 (k=7), A045944 (k=8), this sequence (k=9), A033954 (k=10), A062728 (k=11), A135705 (k=12).

%Y Cf. A001106, A022264, A022265, A024966, A055998, A100752, A174738, A186029, A218471.

%K nonn,easy

%O 0,2

%A _Bruno Berselli_, Jan 13 2011