A051673 Cubic star numbers: a(n) = n^3 + 4*Sum_{i=0..n-1} i^2.

0, 1, 12, 47, 120, 245, 436, 707, 1072, 1545, 2140, 2871, 3752, 4797, 6020, 7435, 9056, 10897, 12972, 15295, 17880, 20741, 23892, 27347, 31120, 35225, 39676, 44487, 49672, 55245, 61220, 67611, 74432, 81697, 89420, 97615, 106296, 115477, 125172

Offset: 0

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

Keywords

• easy
• nice
• nonn

Comments

Also as a(n) = (1/6)*(14*n^3 - 12*n^2 + 4*n), n>0: structured cubeoctahedral numbers (vertex structure 7); and structured pentagonal anti-diamond numbers (vertex structure 7) (cf. A004466 = alternate vertex) (cf. A100188 = structured anti-diamonds). Cf. A100145 for more on structured polyhedral numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Starting with offset 1 = binomial transform of [1, 11, 24, 14, 0, 0, 0, ...]. - Gary W. Adamson, Aug 05 2009

This is prime for a(3) = 47. The subsequence of semiprimes begins: 707, 7435, 10897, 20741, 115477, 341797, 825091, 897097, no more through a(100). - Jonathan Vos Post, May 27 2010

Examples

			a(51) = 51*(51*(7*51-6)+2)/3 = 304351 = 17 * 17903 is semiprime. - _Jonathan Vos Post_, May 27 2010
		

References

• T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

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. A002378, A004466, A005915, A051662, A100145, A100188.

Cf. A214659, A214661.

Programs

Magma

[n*(n*(7*n-6)+2)/3: n in [0..50]]; // Vincenzo Librandi, May 12 2011

Maple

A051673:=n->n*(n*(7*n-6)+2)/3; seq(A051673(n), n=0..40); # Wesley Ivan Hurt, Feb 02 2014

Mathematica

Table[n^3+4Sum[i^2,{i,0,n-1}],{n,0,40}] (* or *) LinearRecurrence[ {4,-6,4,-1},{0,1,12,47},40] (* Harvey P. Dale, Jul 22 2011 *)

PARI

a(n)=n*(n*(7*n-6)+2)/3 \\ Charles R Greathouse IV, Oct 07 2015

SageMath

[n*(7*n^2-6*n+2)/3 for n in range(51)] # G. C. Greubel, Mar 10 2024

Formula

a(n) = n*(n*(7*n-6) + 2)/3.

G.f.: x*(1+8*x+5*x^2)/(1-x)^4. - Bruno Berselli, May 12 2011

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=12, a(3)=47. - Harvey P. Dale, Jul 22 2011

From Reinhard Zumkeller, Jul 25 2012: (Start)

a(n) = A214659(n) - A002378(n).

a(n) = Sum_{k=1..n} A214661(n, k), for n > 0 (row sums). (End)

E.g.f.: (x/3)*(3 + 15*x + 7*x^2)*exp(x). - G. C. Greubel, Mar 10 2024

Extensions

Corrected by T. D. Noe, Nov 01 2006, Nov 08 2006