This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A194715 #49 Feb 16 2025 08:33:15
%S A194715 0,15,45,90,150,225,315,420,540,675,825,990,1170,1365,1575,1800,2040,
%T A194715 2295,2565,2850,3150,3465,3795,4140,4500,4875,5265,5670,6090,6525,
%U A194715 6975,7440,7920,8415,8925,9450,9990,10545,11115,11700,12300,12915,13545,14190,14850,15525
%N A194715 15 times triangular numbers.
%C A194715 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.
%C A194715 Sum of the numbers from 7*n to 8*n. - _Wesley Ivan Hurt_, Dec 23 2015
%C A194715 Also the number of 4-cycles in the (n+6)-triangular honeycomb obtuse knight graph. - _Eric W. Weisstein_, Jul 28 2017
%H A194715 Vincenzo Librandi, <a href="/A194715/b194715.txt">Table of n, a(n) for n = 0..10000</a>
%H A194715 M. Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.
%H A194715 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>.
%H A194715 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A194715 a(n) = 15*n*(n+1)/2 = 15*A000217(n) = 5*A045943(n) = 3*A028895(n) = A069128(n+1) - 1.
%F A194715 From _Wesley Ivan Hurt_, Dec 23 2015: (Start)
%F A194715 G.f.: 15*x/(1-x)^3.
%F A194715 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
%F A194715 a(n) = Sum_{i=7*n..8*n} i. (End)
%F A194715 From _Amiram Eldar_, Feb 21 2023: (Start)
%F A194715 Sum_{n>=1} 1/a(n) = 2/15.
%F A194715 Sum_{n>=1} (-1)^(n+1)/a(n) = (4*log(2) - 2)/15.
%F A194715 Product_{n>=1} (1 - 1/a(n)) = -(15/(2*Pi))*cos(sqrt(23/15)*Pi/2).
%F A194715 Product_{n>=1} (1 + 1/a(n)) = (15/(2*Pi))*cos(sqrt(7/15)*Pi/2). (End)
%F A194715 E.g.f.: 15*exp(x)*x*(2 + x)/2. - _Elmo R. Oliveira_, Dec 25 2024
%p A194715 A194715:=n->15*n*(n+1)/2: seq(A194715(n), n=0..60); # _Wesley Ivan Hurt_, Dec 23 2015
%t A194715 15*Accumulate[Range[0, 60]] (* _Harvey P. Dale_, Feb 12 2012 *)
%t A194715 Table[15 n (n + 1)/2, {n, 0, 60}] (* _Wesley Ivan Hurt_, Dec 23 2015 *)
%t A194715 15 Binomial[Range[20], 2] (* _Eric W. Weisstein_, Jul 28 2017 *)
%t A194715 15 PolygonalNumber[Range[0, 20]] (* _Eric W. Weisstein_, Jul 28 2017 *)
%o A194715 (Magma) [15*n*(n+1)/2: n in [0..50]]; // _Vincenzo Librandi_, Oct 04 2011
%o A194715 (PARI) a(n)=15*n*(n+1)/2 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A194715 Cf. A000217, A028895, A035008, A045943, A069128, A163756.
%Y A194715 Cf. A001105 (3-cycles in the triangular honeycomb obtuse knight graph), A290391 (5-cycles), A290392 (6-cycles). - _Eric W. Weisstein_, Jul 29 2017
%K A194715 nonn,easy
%O A194715 0,2
%A A194715 _Omar E. Pol_, Oct 03 2011