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a(n) = (3n+2)^2.
17

%I #54 Sep 08 2022 08:44:41

%S 1,4,25,64,121,196,289,400,529,676,841,1024,1225,1444,1681,1936,2209,

%T 2500,2809,3136,3481,3844,4225,4624,5041,5476,5929,6400,6889,7396,

%U 7921,8464,9025,9604,10201,10816,11449,12100,12769,13456,14161,14884,15625,16384

%N a(n) = (3n+2)^2.

%C If Y is a fixed 2-subset of a (3n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting Y. - _Milan Janjic_, Oct 21 2007

%C The digit root of the sequence, i.e., A010888(a(n)) for n>=0, is a repeating pattern of {4,7,1}, cf. A100402. - _Ram Shankar_, Apr 14 2015

%C With a different offset, partial sums of A298035. - _N. J. A. Sloane_, Jan 22 2018

%H Vincenzo Librandi, <a href="/A016790/b016790.txt">Table of n, a(n) for n = -1..3000</a>

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = A016958(n)/4. - _Zerinvary Lajos_, Jun 30 2009

%F From _Wesley Ivan Hurt_, Apr 14 2015: (Start)

%F G.f.: (4+13*x+x^2)/(1-x)^3.

%F a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). (End)

%F a(n) = a(n-1)+3*(6*n+1). - _Miquel Cerda_, Oct 25 2016

%F a(n) = A016766(n+1)-A016969(n). - _Miquel Cerda_, Oct 26 2016

%F Sum_{n>=0} 1/a(n) = A294967. - _Amiram Eldar_, Nov 12 2020

%p A016790:=n->(3*n+2)^2: seq(A016790(n), n=0..50); # _Wesley Ivan Hurt_, Apr 14 2015

%t (3 Range[0, 50] + 2)^2 (* _Wesley Ivan Hurt_, Apr 14 2015 *)

%o (Magma) [(3*n+2)^2: n in [0..50]]; // _Vincenzo Librandi_, May 06 2011

%o (PARI) vector(50,n,n--;(3*n+2)^2) \\ _Derek Orr_, Apr 14 2015

%Y Cf. A016957, A016958, A294967, A298035.

%K nonn,easy

%O -1,2

%A _N. J. A. Sloane_

%E Notation in formula cleaned up by _R. J. Mathar_, Aug 05 2010

%E Added a(-1)=1 and fixed b-file. Note: this sequence should really be changed to a(n) = (3n-1)^2 and have offset 0. - _N. J. A. Sloane_, Jan 22 2018