A084367 a(n) = n*(2*n+1)^2.

0, 9, 50, 147, 324, 605, 1014, 1575, 2312, 3249, 4410, 5819, 7500, 9477, 11774, 14415, 17424, 20825, 24642, 28899, 33620, 38829, 44550, 50807, 57624, 65025, 73034, 81675, 90972, 100949, 111630, 123039, 135200, 148137, 161874

Offset: 0

Author

Charlie Marion, Jun 22 2003

Keywords

• easy
• nonn

Examples

			a(3) = 147 since 147 = 3*7^2.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A006752, A014105, A016754.

Programs

Magma

I:=[0, 9, 50, 147]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 04 2012

Mathematica

CoefficientList[Series[x*(9+14*x+x^2)/(1-x)^4,{x,0,50}],x] (* Vincenzo Librandi, Jul 04 2012 *)

Formula

a(n) = n*( n*(2*n+1)+1 + n*(2*n+1)+2 + ... + n*(2*n+1)+2*n ).

a(n) = n*A016754(n); n*a(n) = A014105(n)^2.

G.f.: x*(9+14*x+x^2)/(1-x)^4. - Colin Barker, Jun 30 2012

a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - Vincenzo Librandi, Jul 04 2012

Sum_{n>=1} 1/a(n) = 4 - 2*log(2) - Pi^2/4. - Amiram Eldar, Jul 21 2020

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/2 + log(2) + 2*G - 4, where G is Catalan's constant (A006752). - Amiram Eldar, Feb 08 2022

E.g.f.: exp(x)*x*(9 + 16*x + 4*x^2). - Stefano Spezia, Sep 27 2023