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%I #53 Sep 08 2022 08:45:25
%S 0,1,6,9,19,24,39,46,66,75,100,111,141,154,189,204,244,261,306,325,
%T 375,396,451,474,534,559,624,651,721,750,825,856,936,969,1054,1089,
%U 1179,1216,1311,1350,1450,1491,1596,1639,1749,1794,1909,1956,2076,2125,2250
%N Generalized 9-gonal (or enneagonal) numbers: m*(7*m - 5)/2 with m = 0, 1, -1, 2, -2, 3, -3, ...
%C Partial sums of A195140. - _Omar E. Pol_, Sep 13 2011
%C The characteristic function starts 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0 , ... and has the generating function f(x,x^6) in terms of Ramanujan's two-variable theta function. See A080995, A010054, A133100 etc. - _Omar E. Pol_, Jul 13 2012
%C Also A179986 and positive terms of A001106 interleaved. - _Omar E. Pol_, Aug 04 2012
%C Sequence provides all integers m such that 56*m + 25 is a square. - _Bruno Berselli_, Oct 07 2015
%H Vincenzo Librandi, <a href="/A118277/b118277.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F a(n) = n*(7*n-5)/2 for positive and negative n.
%F a(n) = (1/16)*(14*n^2 + 14*n - 3 + 3*(-1)^n*(2*n + 1)). - _R. J. Mathar_, Oct 08 2011
%F G.f.: x*(1+5*x+x^2) / ( (1+x)^2*(1-x)^3 ). - _R. J. Mathar_, Oct 08 2011
%F Sum_{n>=1} 1/a(n) = 2*(7 + 5*Pi*tan(3*Pi/14))/25. - _Vaclav Kotesovec_, Oct 05 2016
%F E.g.f.: (1/16)*(3*(1 - 2*x)*exp(-x) + (-3 + 28*x + 14*x^2)*exp(x)). - _G. C. Greubel_, Aug 19 2017
%t n=9; Union[Table[i((n-2)i-(n-4))/2, {i,-30,30}]]
%t LinearRecurrence[{1,2,-2,-1,1},{0,1,6,9,19},60] (* _Harvey P. Dale_, Jun 08 2016 *)
%o (Magma) [7*n^2/8+7*n/8-3/16+3*(-1)^n*(1/16+n/8): n in [0..50]]; // _Vincenzo Librandi_, Oct 10 2011
%o (PARI) a(n)=7*n*(n+1)/8-3/16+3*(-1)^n*(1+2*n)/16 \\ _Charles R Greathouse IV_, Jan 18 2012
%Y Cf. A001106 (9-gonal numbers).
%Y Column 5 of A195152.
%Y Cf. A195140.
%Y Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), this sequence (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
%K nonn,easy
%O 0,3
%A _T. D. Noe_, Apr 21 2006
%E Extended Name by _Omar E. Pol_, Jul 28 2018
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