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%I #62 Mar 15 2022 04:32:38
%S 0,2,4,10,14,24,30,44,52,70,80,102,114,140,154,184,200,234,252,290,
%T 310,352,374,420,444,494,520,574,602,660,690,752,784,850,884,954,990,
%U 1064,1102,1180,1220,1302,1344,1430,1474,1564,1610,1704,1752,1850,1900,2002
%N a(n) = (n+1)*(3*n+1)/4 for n odd, a(n) = n*(3*n+2)/4 for n even.
%C Interleaving of A049450 and A049451 (for n > 0).
%C Also, integer values of k*(k+1)/3. - _Charles R Greathouse IV_, Dec 11 2010
%C The nonzero coefficients of the expansion of f(a) = Product_{k>=1} (1-a^(2k)), see A194159, occur at the terms of the sequence given above, i.e., f(a) = 1 - a^2 - a^4 + a^10 + a^14 - a^24 - a^30 + a^44 + a^52 - a^70 - a^80 + ... = Sum_{n>=0} (-1)^binomial(n+1,2)*a^A152749(n). - _Johannes W. Meijer_, Aug 21 2011
%C Partial sums of A109043. - _Reinhard Zumkeller_, Mar 31 2012
%C Nonnegative k such that 12*k+1 is a square. - _Vicente Izquierdo Gomez_, Jul 22 2013
%C Equivalently, numbers of the form h*(3*h+1), where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ... (see also the fifth comment of A062717). - _Bruno Berselli_, Feb 02 2017
%C For n > 0, a(n-1) is the sum of the largest parts of the partitions of 2n into two even parts. - _Wesley Ivan Hurt_, Dec 19 2017
%H Vincenzo Librandi, <a href="/A152749/b152749.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F From _R. J. Mathar_, Jan 03-06 2009: (Start)
%F G.f.: 2*x*(1+x+x^2)/((1+x)^2*(1-x)^3).
%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) = A003154(n+1)/8 - (-1)^n*A005408(n)/8.
%F a(n) = 2*A001318(n) = ((6*n^2+6*n+1) - (2*n+1)*(-1)^n)/8. (End)
%F From _Amiram Eldar_, Mar 15 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 3 - Pi/sqrt(3).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(log(3)-1). (End)
%p A152749 := proc(n): if type(n,even) then n*(3*n+2)/4 else (n+1)*(3*n+1)/4 fi: end: seq(A152749(n), n=0..51); # _Johannes W. Meijer_, Aug 21 2011
%t Table[If[OddQ[n],(n+1)*(3*n+1)/4,n*(3*n+2)/4],{n,0,60}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 03 2012 *)
%t LinearRecurrence[{1,2,-2,-1,1}, {0, 2, 4, 10, 14}, 50] (* _Vincenzo Librandi_, Feb 22 2012 *)
%t Select[Range[1,1000], IntegerQ[Sqrt[12#+1]]&] (* _Vicente Izquierdo Gomez_, Jul 22 2013 *)
%o (Magma) [IsOdd(n) select (n+1)*(3*n+1)/4 else n*(3*n+2)/4: n in [0..52]];
%o (Magma) f:=func<n | n*(3*n+1)>; [0] cat [f(n*m): m in [-1,1], n in [1..30]]; // _Bruno Berselli_, Nov 13 2012
%o (Haskell)
%o a152749 n = a152749_list !! (n-1)
%o a152749_list = scanl1 (+) a109043_list
%o -- _Reinhard Zumkeller_, Mar 31 2012
%Y Cf. A049450 (n*(3*n-1)), A049451 (n*(3*n+1)), A153383 (12n+1 is not prime).
%K nonn,easy
%O 0,2
%A _Vincenzo Librandi_, Dec 31 2009
%E Edited, typo corrected and extended by _Klaus Brockhaus_, Jan 02 2009
%E Leading term a(0)=0 added by _Johannes W. Meijer_, Aug 21 2011
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