A014736 Squares of odd triangular numbers.

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

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

Cf. A000217, A006752, A014493, A014738.

Sequence in context: A036896 A120319 A057530 * A017558 A159939 A167038

Adjacent sequences: A014733 A014734 A014735 * A014737 A014738 A014739

Keyword

nonn,easy

Author

Mohammad K. Azarian

Extensions

More terms from James A. Sellers

Status

approved