cp's OEIS Frontend

A238410 a(n) = floor((3(n-1)^2 + 1)/2).

%I A238410 #63 Mar 12 2023 04:20:01
%S A238410 0,2,6,14,24,38,54,74,96,122,150,182,216,254,294,338,384,434,486,542,
%T A238410 600,662,726,794,864,938,1014,1094,1176,1262,1350,1442,1536,1634,1734,
%U A238410 1838,1944,2054,2166,2282,2400,2522,2646,2774,2904,3038,3174,3314,3456,3602,3750,3902,4056,4214,4374,4538,4704
%N A238410 a(n) = floor((3(n-1)^2 + 1)/2).
%C A238410 a(n) = the eccentric connectivity index of the path P[n] on n vertices. The eccentric connectivity index of a simple connected graph G is defined to be the sum over all vertices i of G of the product E(i)D(i), where E(i) is the eccentricity and D(i) is the degree of vertex i. For example, a(4)=14 because the vertices of P[4] have degrees 1,2,2,1 and eccentricities 3,2,2,3; we have 1*3 + 2*2 + 2*2 + 1*3 = 14.
%C A238410 From _Paul Curtz_, Feb 23 2023: (Start)
%C A238410 East spoke of the hexagonal spiral using A004526 with a single 0:
%C A238410 .
%C A238410             43  42  42  41  41  40
%C A238410           43  28  28  27  27  26  40
%C A238410         44  29  17  16  16  15  26  39
%C A238410       44  29  17   8   8   7  15  25  39
%C A238410     45  30  18   9   3   2   7  14  25  38
%C A238410   45  30  18   9   3   0---2---6--14--24--38-->
%C A238410     31  19  10   4   1   1   6  13  24  37
%C A238410       31  19  10   4   5   5  13  23  37
%C A238410         32  20  11  11  12  12  23  36
%C A238410           32  20  21  21  22  22  36
%C A238410             33  33  34  34  35  35
%C A238410 .
%C A238410 Other spokes are A032528, A005449, A227017, A045943, A005448, A212959, A000326, A282513, A143689, and A104249. (End)
%H A238410 Matthew House, <a href="/A238410/b238410.txt">Table of n, a(n) for n = 1..10000</a>
%H A238410 M. J. Morgan, S. Mukwembi and H. C. Swart, <a href="http://dx.doi.org/10.1016/j.disc.2009.12.013">On the eccentric connectivity index of a graph</a>, Discrete Math., 311, 2011, 1229-1234.
%H A238410 B. Zhou and Zh. Du, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match63/n1/match63n1_181-198.pdf">On eccentric connectivity index</a>, Comm. Math. Comp. Chem. (MATCH), 63, 2010, 181-198.
%H A238410 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A238410 a(n) = (3*n)^2/6 for n even and a(n) = ((3*n)^2 + 3)/6 for n odd. - _Miquel Cerda_, Jun 17 2016
%F A238410 From _Ilya Gutkovskiy_, Jun 17 2016: (Start)
%F A238410 G.f.: 2*x^2*(1 + x + x^2)/((1 - x)^3*(1 + x)).
%F A238410 a(n) = (6*n^2 - 12*n + 7 + (-1)^n)/4.
%F A238410 a(n) = 2* A077043(n-1). (End)
%F A238410 a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - _Matthew House_, Feb 15 2017
%F A238410 Sum_{n>=2} 1/a(n) = Pi^2/36 + tanh(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)). - _Amiram Eldar_, Mar 12 2023
%p A238410 a := proc (n) options operator, arrow: floor((3/2)*(n-1)^2+1/2) end proc: seq(a(n), n = 1 .. 70);
%t A238410 Table[Floor[(3(n-1)^2+1)/2],{n,80}]  (* or *) LinearRecurrence[{2,0,-2,1},{0,2,6,14},80] (* _Harvey P. Dale_, Apr 30 2022 *)
%o A238410 (PARI) a(n)=(3*(n-1)^2 + 1)\2 \\ _Charles R Greathouse IV_, Feb 15 2017
%Y A238410 Cf. A004526, A077043.
%Y A238410 (Cf. A056105, A056107.)
%K A238410 nonn,easy
%O A238410 1,2
%A A238410 _Emeric Deutsch_, Feb 27 2014