# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: id:a049598
Showing 1-1 of 1

%I A049598 #74 Feb 16 2025 08:32:40
%S A049598 0,12,36,72,120,180,252,336,432,540,660,792,936,1092,1260,1440,1632,
%T A049598 1836,2052,2280,2520,2772,3036,3312,3600,3900,4212,4536,4872,5220,
%U A049598 5580,5952,6336,6732,7140,7560,7992,8436,8892,9360,9840,10332,10836,11352
%N A049598 12 times triangular numbers.
%C A049598 a(n-1) is the Wiener index of the helm graph H(n) (n>=3). The graph H(n) is obtained from an n-wheel graph (on n+1 nodes) by adjoining a pendant edge at each node of the cycle. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph. The Wiener polynomial of H(n) is (1/2)*n*t*((n-3)t^3 + 2(n-2)t^2 + (n+3)t + 6). - _Emeric Deutsch_, Sep 28 2010
%C A049598 Also sequence found by reading the line from 0, in the direction 0, 12, ..., and the same line from 0, in the direction 0, 36, ..., in the square spiral whose vertices are the generalized tetradecagonal numbers A195818. Axis perpendicular to A195158 in the same spiral. - _Omar E. Pol_, Sep 29 2011
%C A049598 Also the Wiener index of the (n+1)-gear graph. - _Eric W. Weisstein_, Sep 08 2017
%H A049598 G. C. Greubel, <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL0EwNDk1OTgvYjA0OTU5OC50eHQ">Table of n, a(n) for n = 0..5000</a>
%H A049598 B. E. Sagan, Y-N. Yeh and P. Zhang, <a href="/load/view.php?a=aHR0cDovL3VzZXJzLm1hdGgubXN1LmVkdS91c2Vycy9zYWdhbi9QYXBlcnMvT2xkL3dwZy1wdWIucGRm">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., Vol. 60 (1996), pp. 959-969.
%H A049598 Leo Tavares, <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL0EwNDk1OTgvYTA0OTU5OC5qcGc">Illustration: Centroid Stars</a>.
%H A049598 Eric Weisstein's World of Mathematics, <a href="/load/view.php?a=aHR0cHM6Ly9tYXRod29ybGQud29sZnJhbS5jb20vR2VhckdyYXBoLmh0bWw">Gear Graph</a>.
%H A049598 Eric Weisstein's World of Mathematics, <a href="/load/view.php?a=aHR0cHM6Ly9tYXRod29ybGQud29sZnJhbS5jb20vSGVsbUdyYXBoLmh0bWw">Helm Graph</a>.
%H A049598 Eric Weisstein's World of Mathematics, <a href="/load/view.php?a=aHR0cHM6Ly9tYXRod29ybGQud29sZnJhbS5jb20vV2llbmVySW5kZXguaHRtbA">Wiener Index</a>.
%H A049598 <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL2luZGV4L1JlYyNvcmRlcl8wMw">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A049598 a(n) = 6*n*(n+1).
%F A049598 G.f.: 12*x/(1-x)^3.
%F A049598 a(n) = 12*A000217(n). - _Omar E. Pol_, Dec 11 2008
%F A049598 a(n) = 12*n + a(n-1) (with a(0)=0). - _Vincenzo Librandi_, Aug 06 2010
%F A049598 a(n) = A003154(n+1) - 1. - _Omar E. Pol_, Oct 03 2011
%F A049598 a(n) = A032528(2*n+1) - 1. - _Adriano Caroli_, Jul 19 2013
%F A049598 a(n) = A001844(n) + A073577(n). - _Bruce J. Nicholson_, Aug 06 2017
%F A049598 E.g.f.: 6*x*(x+2)*exp(x). - _G. C. Greubel_, Aug 23 2017
%F A049598 From _Amiram Eldar_, Feb 15 2022: (Start)
%F A049598 Sum_{n>=1} 1/a(n) = 1/6.
%F A049598 Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/3 - 1/6. (End)
%F A049598 From _Amiram Eldar_, Feb 21 2023: (Start)
%F A049598 Product_{n>=1} (1 - 1/a(n)) = -(6/Pi)*cos(sqrt(5/3)*Pi/2).
%F A049598 Product_{n>=1} (1 + 1/a(n)) = (6/Pi)*cos(Pi/(2*sqrt(3))). (End)
%e A049598 a(1) = 12*1 + 0 = 12;
%e A049598 a(2) = 12*2 + 12 = 36;
%e A049598 a(3) = 12*3 + 36 = 72.
%t A049598 12 * Accumulate[Range[0, 50]] (* _Harvey P. Dale_, Feb 05 2013 *)
%t A049598 (* Start from _Eric W. Weisstein_, Sep 08 2017 *)
%t A049598 Table[6 n (n + 1), {n, 0, 20}]
%t A049598 12 PolygonalNumber[3, Range[0, 20]]
%t A049598 12 Binomial[Range[20], 2]
%t A049598 LinearRecurrence[{3, -3, 1}, {12, 36, 72}, {0, 20}]
%t A049598 (* End *)
%o A049598 (PARI) a(n)=6*n*(n+1) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A049598 Cf. A000217, A001844, A003154, A027468, A035008, A073577.
%K A049598 nonn,easy
%O A049598 0,2
%A A049598 Joe Keane (jgk(AT)jgk.org)

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE