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%I #101 Sep 08 2022 08:45:07
%S 0,3,5,14,18,33,39,60,68,95,105,138,150,189,203,248,264,315,333,390,
%T 410,473,495,564,588,663,689,770,798,885,915,1008,1040,1139,1173,1278,
%U 1314,1425,1463,1580,1620,1743,1785,1914,1958,2093,2139,2280,2328,2475
%N Even triangular numbers halved.
%C Set of integers k such that k + (1 + 2 + 3 + 4 + ... + x) = 3*k, where x is sufficiently large. For example, 203 is a term because 203 + (1 + 2 + 3 + 4 + ... +28) = 609 and 609 = 3*203. - _Gil Broussard_, Sep 01 2008
%C Set of all m such that 16*m+1 is a perfect square. - _Gary Detlefs_, Feb 21 2010
%C Integers of the form Sum_{k=0..n} k/2. - _Arkadiusz Wesolowski_, Feb 07 2012
%C Numbers of the form h*(4*h + 1) for h = 0, -1, 1, -2, 2, -3, 3, ... - _Bruno Berselli_, Feb 26 2018
%C Numbers whose distance to nearest square equals their distance to nearest oblong; that is, numbers k such that A053188(k) = A053615(k). - _Lamine Ngom_, Oct 27 2020
%H David A. Corneth, <a href="/A074378/b074378.txt">Table of n, a(n) for n = 0..9999</a>
%H Neville Holmes, <a href="http://web.archive.org/web/20120410031141/http://www.comp.utas.edu.au/users/nholmes/sqncs/gmtrc.htm#A074377">More Geometric Integer Sequences</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F Sum_{n>=0} q^a(n) = (Prod_{n>0} (1-q^n))*(Sum_{n>=0} A035294(n)*q^n).
%F a(n) = n*(n + 1)/4 where n*(n + 1)/2 is even.
%F G.f.: x*(3 + 2*x + 3*x^2)/((1 - x)*(1 - x^2)^2).
%F From _Benoit Jubin_, Feb 05 2009: (Start)
%F a(n) = (2*n + 1)*floor((n + 1)/2).
%F a(2*k) = k*(4*k+1); a(2*k+1) = (k+1)*(4*k+3). (End)
%F a(2*n) = A007742(n), a(2*n-1) = A033991(n). - _Arkadiusz Wesolowski_, Jul 20 2012
%F a(n) = (4*n + 1 - (-1)^n)*(4*n + 3 - (-1)^n)/4^2. - _Peter Bala_, Jan 21 2019
%F a(n) = (2*n+1)*(n+1)*(1+(-1)^(n+1))/4 + (2*n+1)*(n)*(1+(-1)^n)/4. - _Eric Simon Jacob_, Jan 16 2020
%F From _Amiram Eldar_, Jul 03 2020: (Start)
%F Sum_{n>=1} 1/a(n) = 4 - Pi (A153799).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 6*log(2) - 4 (See A016687). (End)
%p a:=n->(2*n+1)*floor((n+1)/2): seq(a(n),n=0..50); # _Muniru A Asiru_, Feb 01 2019
%t 1/2 * Select[PolygonalNumber@ Range[0, 100], EvenQ] (* _Michael De Vlieger_, Jun 01 2017, Version 10.4 *)
%o (PARI) a(n)=(2*n+1)*(n-n\2)
%o (Magma) f:=func<i | i*(4*i+1)>; [0] cat [f(n*m): m in [-1,1], n in [1..25]]; // _Bruno Berselli_, Nov 13 2012
%Y Cf. A011848, A014493, A074377, A035294.
%Y Cf. A010709, A047522. [_Vincenzo Librandi_, Feb 14 2009]
%Y Cf. A266883 (numbers n such that 16*n-15 is a square).
%Y Cf. A016687, A153799.
%Y Cf. A053615, A053188.
%K nonn,easy
%O 0,2
%A _W. Neville Holmes_, Sep 04 2002
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