A194715 15 times triangular numbers.

0, 15, 45, 90, 150, 225, 315, 420, 540, 675, 825, 990, 1170, 1365, 1575, 1800, 2040, 2295, 2565, 2850, 3150, 3465, 3795, 4140, 4500, 4875, 5265, 5670, 6090, 6525, 6975, 7440, 7920, 8415, 8925, 9450, 9990, 10545, 11115, 11700, 12300, 12915, 13545, 14190, 14850, 15525

Offset

0,2

Comments

Sequence found by reading the line from 0, in the direction 0, 15, ... and the same line from 0, in the direction 0, 45, ..., in the square spiral whose vertices are the generalized 17-gonal numbers.

Sum of the numbers from 7n to 8n. - Wesley Ivan Hurt, Dec 23 2015

Also the number of 4-cycles in the (n+6)-triangular honeycomb obtuse knight graph. - Eric W. Weisstein, Jul 28 2017

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.

Eric Weisstein's World of Mathematics, Graph Cycle.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = 15*n*(n+1)/2 = 15*A000217(n) = 5*A045943(n) = 3*A028895(n) = A069128(n+1) - 1.

From Wesley Ivan Hurt, Dec 23 2015: (Start)

G.f.: 15*x/(1-x)^3.

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>2.

a(n) = Sum_{i=7n..8n} i. (End)

From Amiram Eldar, Feb 21 2023: (Start)

Sum_{n>=1} 1/a(n) = 2/15.

Sum_{n>=1} (-1)^(n+1)/a(n) = (4*log(2) - 2)/15.

Product_{n>=1} (1 - 1/a(n)) = -(15/(2*Pi))*cos(sqrt(23/15)*Pi/2).

Product_{n>=1} (1 + 1/a(n)) = (15/(2*Pi))*cos(sqrt(7/15)*Pi/2). (End)

Maple

A194715:=n->15*n*(n+1)/2: seq(A194715(n), n=0..60); # Wesley Ivan Hurt, Dec 23 2015

Mathematica

15*Accumulate[Range[0, 60]] (* Harvey P. Dale, Feb 12 2012 *)
Table[15 n (n + 1)/2, {n, 0, 60}] (* Wesley Ivan Hurt, Dec 23 2015 *)
15 Binomial[Range[20], 2] (* Eric W. Weisstein, Jul 28 2017 *)
15 PolygonalNumber[Range[0, 20]] (* Eric W. Weisstein, Jul 28 2017 *)

Prog

(Magma) [15*n*(n+1)/2: n in [0..50]]; // Vincenzo Librandi, Oct 04 2011
(PARI) a(n)=15*n*(n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017

Crossrefs

Cf. A000217, A028895, A035008, A045943, A069128, A163756.

Cf. A001105 (3-cycles in the triangular honeycomb obtuse knight graph), A290391 (5-cycles), A290392 (6-cycles). - Eric W. Weisstein, Jul 29 2017

Sequence in context: A066763 A164788 A033849 * A060536 A014634 A303857

Adjacent sequences: A194712 A194713 A194714 * A194716 A194717 A194718

Keyword

nonn,easy

Author

Omar E. Pol, Oct 03 2011

Status

approved