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A014736 Squares of odd triangular numbers.

%I A014736 #31 Jul 02 2025 16:01:55
%S A014736 1,9,225,441,2025,3025,8281,11025,23409,29241,53361,64009,105625,
%T A014736 123201,189225,216225,314721,354025,494209,549081,741321,815409,
%U A014736 1071225,1168561,1500625,1625625,2047761,2205225,2732409,2927521
%N A014736 Squares of odd triangular numbers.
%H A014736 Vincenzo Librandi, <a href="/A014736/b014736.txt">Table of n, a(n) for n = 0..1000</a>
%H A014736 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-4,-6,6,4,-4,-1,1).
%F A014736 a(n) = A014493(n+1)^2. - _Vincenzo Librandi_, Mar 23 2012
%F A014736 From _G. C. Greubel_, Jul 24 2019: (Start)
%F A014736 G.f.: x*(1 + 8*x + 212*x^2 + 184*x^3 + 726*x^4 + 184*x^5 + 212*x^6 + 8*x^7 + x^8)/((1 - x)^5*(1 + x)^4).
%F A014736 E.g.f.: (1 + x + 5*x^2 + 20*x^3 + 4*x^4)*cosh(x) - x*(1 - 17*x - 12*x^2 - 4*x^3)* sinh(x) - 1. (End)
%F A014736 From _Amiram Eldar_, Mar 06 2022: (Start)
%F A014736 Sum_{n>=0} 1/a(n) = (3*Pi-8)*Pi/4.
%F A014736 Sum_{n>=0} (-1)^n/a(n) = 4*(G - log(2)), where G is Catalan's constant (A006752). (End)
%t A014736 Select[Accumulate[Range[70]],OddQ]^2 (* _Harvey P. Dale_, Mar 22 2012 *)
%o A014736 (Magma) [((2*n-1)*(2*n-1-(-1)^n))^2/4: n in [1..30]]; // _Vincenzo Librandi_, Mar 23 2012
%o A014736 (PARI) vector(30, n, ((2*n-1)*(2*n-1-(-1)^n))^2/4) \\ _G. C. Greubel_, Jul 24 2019
%o A014736 (Sage) [((2*n-1)*(2*n-1-(-1)^n))^2/4 for n in (1..30)] # _G. C. Greubel_, Jul 24 2019
%o A014736 (GAP) List([1..30], n-> ((2*n-1)*(2*n-1-(-1)^n))^2/4); # _G. C. Greubel_, Jul 24 2019
%o A014736 (Scala) ((1 to 78).scanLeft(0)(_ + _)).filter(_ % 2 == 1).map(n => n * n) // _Alonso del Arte_, Jul 24 2019
%Y A014736 Cf. A000217, A006752, A014493, A014738.
%K A014736 nonn,easy
%O A014736 0,2
%A A014736 _Mohammad K. Azarian_
%E A014736 More terms from _James Sellers_