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A222170 a(n) = n^2 + 2*floor(n^2/3).

%I A222170 #58 Feb 10 2023 17:13:14
%S A222170 0,1,6,15,26,41,60,81,106,135,166,201,240,281,326,375,426,481,540,601,
%T A222170 666,735,806,881,960,1041,1126,1215,1306,1401,1500,1601,1706,1815,
%U A222170 1926,2041,2160,2281,2406,2535,2666,2801,2940,3081,3226,3375,3526,3681,3840
%N A222170 a(n) = n^2 + 2*floor(n^2/3).
%C A222170 Also, a(n) = n^2 + floor(2*n^2/3), since 2*floor(n^2/3) = floor(2*n^2/3).
%H A222170 Bruno Berselli, <a href="/A222170/b222170.txt">Table of n, a(n) for n = 0..1000</a>
%H A222170 Tadeusz Dorozinskis, <a href="https://geometryka.wordpress.com/2022/12/30/pentagon-polyeder-ii.html">Pentagonpolyhedra Doro</a>
%H A222170 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F A222170 G.f.: x*(1+x)*(1 + 3*x + x^2)/((1 + x + x^2)*(1-x)^3).
%F A222170 a(n) = a(-n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
%F A222170 a(n) = floor(5*n^2/3). - _Wesley Ivan Hurt_, Mar 16 2015
%F A222170 a(n) = a(n-3) + 5*(2n-3) [Tadeusz Dorozinski]. - _Eduard Baumann_, Jan 18 2023
%t A222170 Table[n^2 + 2 Floor[n^2/3], {n, 0, 50}]
%t A222170 CoefficientList[Series[x (1 + x) (1 + 3 x + x^2) / ((1 + x + x^2) (1 - x)^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%t A222170 LinearRecurrence[{2, -1, 1, -2, 1}, {0, 1, 6, 15, 26}, 50] (* _Hugo Pfoertner_, Jan 17 2023 *)
%o A222170 (Magma) [n^2+2*Floor(n^2/3): n in [0..50]];
%o A222170 (Magma) I:=[0,1,6,15,26]; [n le 5 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-3)-2*Self(n-4)+Self(n-5): n in [1..50]]; // _Vincenzo Librandi_, Aug 18 2013
%Y A222170 Subsequence of A008851.
%Y A222170 Cf. A004773 (numbers of the type n+floor(n/3)), A008810 (numbers of the type n^2-2*floor(n^2/3)), A047220 (numbers of the type n+floor(2*n/3)), A184637 (numbers of the type n^2+floor(n^2/3), except the first two).
%K A222170 nonn,easy
%O A222170 0,3
%A A222170 _Bruno Berselli_, Aug 08 2013