|
| |
|
|
A051673
|
|
Cubic star numbers: a(n) = n^3 + 4*Sum_{i=0..n-1} i^2.
|
|
8
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
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
For n > 0: a(n) = sum of n-th row of triangle A214661; a(n)=A214659(n)-A002378(n). - Reinhard Zumkeller, Jul 25 2012
|
|
|
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).
|
|
|
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(0)=0, a(1)=1, a(2)=12, a(3)=47, a(n)=4*a(n-1)-6*a(n-2)+ 4*a(n-3)- a(n-4). - Harvey P. Dale, Jul 22 2011
|
|
|
EXAMPLE
|
a(51) = 51*(51*(7*51-6)+2)/3 = 304351 = 17 * 17903 is semiprime. - Jonathan Vos Post, May 27 2010
|
|
|
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 *)
|
|
|
PROG
|
(MAGMA) [n*(n*(7*n-6)+2)/3: n in [0..50]]; // Vincenzo Librandi, May 12 2011
(PARI) a(n)=n*(n*(7*n-6)+2)/3 \\ Charles R Greathouse IV, Oct 07 2015
|
|
|
CROSSREFS
|
Cf. A005915, A051662.
Sequence in context: A022281 A244803 A024183 * A030623 A030624 A002612
Adjacent sequences: A051670 A051671 A051672 * A051674 A051675 A051676
|
|
|
KEYWORD
|
easy,nice,nonn
|
|
|
AUTHOR
|
Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
|
|
|
EXTENSIONS
|
Corrected by T. D. Noe, Nov 01 2006, Nov 08 2006
|
|
|
STATUS
|
approved
|
| |
|
|
