|
| |
|
|
A081266
|
|
Staggered diagonal of triangular spiral in A051682.
|
|
20
|
|
|
|
0, 6, 21, 45, 78, 120, 171, 231, 300, 378, 465, 561, 666, 780, 903, 1035, 1176, 1326, 1485, 1653, 1830, 2016, 2211, 2415, 2628, 2850, 3081, 3321, 3570, 3828, 4095, 4371, 4656, 4950, 5253, 5565, 5886, 6216, 6555, 6903, 7260, 7626, 8001, 8385, 8778, 9180
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
Staggered diagonal of triangular spiral in A051682, between (0,4,17) spoke and (0,7,23) spoke.
Binomial transform of (0, 6, 9, 0, 0, 0, ...).
If Y is a fixed 3-subset of a (3n+1)-set X then a(n) is the number of (3n-1)-subsets of X intersecting Y. - Milan Janjic, Oct 28 2007
|
|
|
LINKS
|
Muniru A Asiru, Table of n, a(n) for n = 0..10000
Milan Janjic, Two Enumerative Functions
M. Janjic and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Amelia Carolina Sparavigna, The groupoid of the Triangular Numbers and the generation of related integer sequences, Politecnico di Torino, Italy (2019).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
|
|
|
FORMULA
|
a(n) = 6*C(n,1) + 9*C(n,2).
a(n) = 3*n*(3*n+1)/2.
G.f.: (6*x+3*x^2)/(1-x)^3.
a(n) = A000217(3*n); a(2*n) = A144314(n). - Reinhard Zumkeller, Sep 17 2008
a(n) = 3*A005449(n). - R. J. Mathar, Mar 27 2009
a(n) = 9*n+a(n-1)-3 for n>0, a(0)=0. - Vincenzo Librandi, Aug 08 2010
a(n) = A218470(9n+5). - Philippe Deléham, Mar 27 2013
a(n) = Sum_{k=0..3n} (-1)^(n+k)*k^2. - Bruno Berselli, Aug 29 2013
E.g.f.: 3*exp(x)*x*(4 + 3*x)/2. - Stefano Spezia, Jun 06 2021
|
|
|
EXAMPLE
|
a(1)=9*1+0-3=6, a(2)=9*2+6-3=21, a(3)=9*3+21-3=45.
For n=3, a(3) = -0^2+1^2-2^2+3^2-4^2+5^2-6^2+7^2-8^2+9^2 = 45.
|
|
|
MAPLE
|
seq(binomial(3*n+1, 2), n=0..45); # Zerinvary Lajos, Jan 21 2007
a:=n->sum(j, j=0..n): seq(a(3*n), n=0..45); # Zerinvary Lajos, Apr 30 2007
|
|
|
MATHEMATICA
|
s=0; lst={s}; Do[s+=n++ +6; AppendTo[lst, s], {n, 0, 7!, 9}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)
LinearRecurrence[{3, -3, 1}, {0, 6, 21}, 50] (* Harvey P. Dale, Aug 29 2015 *)
|
|
|
PROG
|
(PARI) a(n)=3*n*(3*n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017
(GAP) List([0..50], n->Binomial(3*n+1, 2)); # Muniru A Asiru, Feb 28 2019
|
|
|
CROSSREFS
|
Cf. A000217, A000290, A005449, A014105, A022266, A033585, A062725, A144312, A144314, A218470.
Sequence in context: A180857 A119868 A175729 * A087863 A212656 A051941
Adjacent sequences: A081263 A081264 A081265 * A081267 A081268 A081269
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
Paul Barry, Mar 15 2003
|
|
|
STATUS
|
approved
|
| |
|
|
