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a(n) = 36*n + 9.
4

%I #28 Apr 16 2024 10:26:09

%S 9,45,81,117,153,189,225,261,297,333,369,405,441,477,513,549,585,621,

%T 657,693,729,765,801,837,873,909,945,981,1017,1053,1089,1125,1161,

%U 1197,1233,1269,1305,1341,1377,1413,1449,1485,1521,1557,1593,1629,1665,1701

%N a(n) = 36*n + 9.

%C Numbers of the 9th column of positive numbers in the square array of nonnegative and polygonal numbers A139600.

%H Vincenzo Librandi, <a href="/A139609/b139609.txt">Table of n, a(n) for n = 0..5000</a>

%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

%F a(n) = A057145(n+2,9).

%F G.f.: 9*(1+3*x)/(x-1)^2. - _R. J. Mathar_, Jul 28 2016

%F From _Elmo R. Oliveira_, Apr 16 2024: (Start)

%F E.g.f.: 9*exp(x)*(1 + 4*x).

%F a(n) = 2*a(n-1) - a(n-2) for n >= 2.

%F a(n) = 9*A016813(n) = A044102(n) + 9 = A152994(n+1) - A152994(n). (End)

%t Range[9, 7000, 36] (* _Vladimir Joseph Stephan Orlovsky_, Jul 23 2011 *)

%t 36*Range[0,50]+9 (* or *) LinearRecurrence[{2,-1},{9,45},50] (* _Harvey P. Dale_, Jan 04 2018 *)

%o (Magma) [9*(4*n + 1): n in [0..60]]; // _Vincenzo Librandi_, Jul 23 2011

%o (PARI) a(n)=36*n+9 \\ _Charles R Greathouse IV_, Oct 05 2011

%Y Cf. A016813, A044102, A057145, A139600, A152994.

%K easy,nonn

%O 0,1

%A _Omar E. Pol_, Apr 27 2008