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A014736
Squares of odd triangular numbers.
2
1, 9, 225, 441, 2025, 3025, 8281, 11025, 23409, 29241, 53361, 64009, 105625, 123201, 189225, 216225, 314721, 354025, 494209, 549081, 741321, 815409, 1071225, 1168561, 1500625, 1625625, 2047761, 2205225, 2732409, 2927521
OFFSET
0,2
LINKS
FORMULA
a(n) = A014493(n+1)^2. - Vincenzo Librandi, Mar 23 2012
From G. C. Greubel, Jul 24 2019: (Start)
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).
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)
From Amiram Eldar, Mar 06 2022: (Start)
Sum_{n>=0} 1/a(n) = (3*Pi-8)*Pi/4.
Sum_{n>=0} (-1)^n/a(n) = 4*(G - log(2)), where G is Catalan's constant (A006752). (End)
MATHEMATICA
Select[Accumulate[Range[70]], OddQ]^2 (* Harvey P. Dale, Mar 22 2012 *)
PROG
(Magma) [((2*n-1)*(2*n-1-(-1)^n))^2/4: n in [1..30]]; // Vincenzo Librandi, Mar 23 2012
(PARI) vector(30, n, ((2*n-1)*(2*n-1-(-1)^n))^2/4) \\ G. C. Greubel, Jul 24 2019
(Sage) [((2*n-1)*(2*n-1-(-1)^n))^2/4 for n in (1..30)] # G. C. Greubel, Jul 24 2019
(GAP) List([1..30], n-> ((2*n-1)*(2*n-1-(-1)^n))^2/4); # G. C. Greubel, Jul 24 2019
(Scala) ((1 to 78).scanLeft(0)(_ + _)).filter(_ % 2 == 1).map(n => n * n) // Alonso del Arte, Jul 24 2019
CROSSREFS
KEYWORD
nonn,easy
EXTENSIONS
More terms from James A. Sellers
STATUS
approved