A050492 Thickened cube numbers: a(n) = n*(n^2 + (n-1)^2) + (n-1)*2*n*(n-1).

1, 14, 63, 172, 365, 666, 1099, 1688, 2457, 3430, 4631, 6084, 7813, 9842, 12195, 14896, 17969, 21438, 25327, 29660, 34461, 39754, 45563, 51912, 58825, 66326, 74439, 83188, 92597, 102690, 113491, 125024, 137313, 150382, 164255, 178956

Offset: 1

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 27 1999

Keywords

• nonn
• easy
• nice

Comments

In other words, positive integers k such that 2*k - 1 is a perfect cube. - Altug Alkan, Apr 15 2016

a(n) represents the first term in a sum of (2*n - 1)^3 consecutive integers which equals (2*n - 1)^6. - Patrick J. McNab, Dec 24 2016

Examples

			       * *      *      * *
a(2) =  *   +  * *  +   *  = 14.
       * *      *      * *
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A001844, A046092, A050533.

Programs

Magma

[n*(4*n^2-6*n+3): n in [1..40]]; // Vincenzo Librandi, Oct 03 2011

Mathematica

Table[n(n^2+(n-1)^2)+(n-1)2n(n-1),{n,40}] (* or *) LinearRecurrence[ {4,-6,4,-1},{1,14,63,172},40] (* Harvey P. Dale, Oct 02 2011 *)

PARI

a(n)=n*(4*n^2-6*n+3) \\ Charles R Greathouse IV, Nov 10 2015

Formula

a(n) = n*(4*n^2-6*n+3).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(1)=1, a(2)=14, a(3)=63, a(4)=172. - Harvey P. Dale, Oct 02 2011

G.f.: x*(1+10*x+13*x^2)/(1-4*x+6*x^2-4*x^3+x^4). - Colin Barker, Jan 04 2012

a(n) = ((2n-1)^3 + 1)/2. - Dave Durgin, May 07 2014

E.g.f.: x*(4*x^2 + 6*x + 1)*exp(x). - G. C. Greubel, Apr 15 2016