A143938 -id:A143938 - cp's OEIS frontend

A193390 The hyper-Wiener index of a benzenoid consisting of a straight-line chain of n hexagons (s=2; see the Gutman et al. reference).

42, 215, 680, 1661, 3446, 6387, 10900, 17465, 26626, 38991, 55232, 76085, 102350, 134891, 174636, 222577, 279770, 347335, 426456, 518381, 624422, 745955, 884420, 1041321, 1218226, 1416767, 1638640, 1885605, 2159486, 2462171, 2795612, 3161825, 3562890, 4000951, 4478216

Offset: 1

Emeric Deutsch, Jul 25 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
A. A. Dobrynin, I. Gutman, S. Klavzar, P. Zigert, Wiener Index of Hexagonal Systems, Acta Applicandae Mathematicae 72 (2002), pp. 247-294.
I. Gutman, S. Klavzar, M. Petkovsek, and P. Zigert, On Hosoya polynomials of benzenoid graphs, Comm. Math. Comp. Chem. (MATCH), 43, 2001, 49-66.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A143937, A143938.

[(8*n^4 + 32*n^3 + 46*n^2 + 37*n + 3)/3: n in [1..30]]; // Vincenzo Librandi, Jul 26 2011

a := proc (n) options operator, arrow: (8/3)*n^4+(32/3)*n^3+(46/3)*n^2+(37/3)*n+1 end proc; seq(a(n), n = 1 .. 35);

a(n)=(8*n^4+32*n^3+46*n^2+37*n)/3+1 \\ Charles R Greathouse IV, Jul 26 2011

a(n) = (8*n^4 + 32*n^3 + 46*n^2 + 37*n + 3)/3.
The Wiener-Hosoya polynomial is W(n,t) = (2*(t+1)*t^(2*n+2) - t^3 - 2*t^2 - 3*t + n*(t-1)*(t^2+1)*(t^2-t-4)+2)/(1-t)^2.
G.f.: x*(42 + 5*x + 25*x^2 - 9*x^3 + x^4)/(1-x)^5. - Bruno Berselli, Jul 27 2011

A193400 Hyper-Wiener index of a benzenoid consisting of a chain of n hexagons characterized by the encoding s = 1133 (see the Gutman et al. reference, Sec. 5).

Original entry on oeis.org

42, 215, 636, 1513, 2862, 5211, 8352, 13229, 19314, 28063, 38532, 52785, 69366, 91043, 115752, 147061, 182202, 225639, 273804, 332153, 396222, 472555, 555696, 653373, 759042, 881711, 1013652, 1165249, 1327494, 1512243, 1709112, 1931525, 2167626, 2432503, 2712732

Offset: 1

Emeric Deutsch, Jul 25 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
A. A. Dobrynin, I. Gutman, S. Klavzar, P. Zigert, Wiener Index of Hexagonal Systems, Acta Applicandae Mathematicae 72 (2002), pp. 247-294.
I. Gutman, S. Klavzar, M. Petkovsek, and P. Zigert, On Hosoya polynomials of benzenoid graphs, Comm. Math. Comp. Chem. (MATCH), 43, 2001, 49-66.
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A143937, A143938, A193391, A193392, A193393, A193394, A193395, A193396, A193397, A193398, A193399.

[(6*n^4 + 40*n^3 + 114*n^2 + 16*n - 45 + (-1)^n*(6*n^2 +20*n -63))/4: n in [1..40]]; // Vincenzo Librandi, Jul 26 2011

a := proc (n) options operator, arrow: (3/2)*n^4+10*n^3+(57/2)*n^2+4*n-45/4+(1/4)*(-1)^n*(6*n^2+20*n-63) end proc: seq(a(n), n = 1 .. 35);

a(n)=(6*n^4+40*n^3+114*n^2+16*n-45+(-1)^n*(6*n^2+20*n-63))/4 \\ Charles R Greathouse IV, Jul 28 2011

a(n) = ( 6*n^4 +40*n^3 +114*n^2 +16*n -45 +(-1)^n*(6*n^2 +20*n -63) )/4.

G.f.: x*(42+131*x+122*x^2+63*x^3-146*x^4+25*x^5+78*x^6-27*x^7)/((1+x)^3*(1-x)^5). - Bruno Berselli, Jul 27 2011

A245830 The Szeged index of a benzenoid consisting of a linear chain of n hexagons.

Original entry on oeis.org

54, 243, 656, 1381, 2506, 4119, 6308, 9161, 12766, 17211, 22584, 28973, 36466, 45151, 55116, 66449, 79238, 93571, 109536, 127221, 146714, 168103, 191476, 216921, 244526, 274379, 306568, 341181, 378306, 418031

Offset: 1

Emeric Deutsch, Aug 07 2014

			a(1)=54; indeed, the benzenoid consists of 1 hexagon and each of its six edges contributes 3*3 towards the Szeged index; 6*9 = 54.

M. V. Diudea, I. Gutman, J. Lorentz, Molecular Topology, Nova Science Publishers, Huntington, NY (2001).

G. C. Greubel, Table of n, a(n) for n = 1..1000
I. Gutman, S. Klavzar, An algorithm for the calculation of the Szeged index of benzenoid hydrocarbons, preprint.
I. Gutman, S. Klavzar, An algorithm for the calculation of the Szeged index of benzenoid hydrocarbons, J. Chem. Inf. Comput. Sci., 35, 1995, 1011-1014.
I. Gutman, P. V. Khadikar, T. Khaddar, Wiener and Szeged indices of benzenoid hydrocarbons containing a linear polyacene fragment, Commun. Math. Chem. (MATCH), 35, 1997, 105-116.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A143938.

a := proc (n) options operator, arrow: (44/3)*n^3+24*n^2+(43/3)*n+1 end proc: seq(a(n), n = 1 .. 30);

LinearRecurrence[{4, -6, 4, -1}, {54, 243, 656, 1381}, 100] (* or *) Table[(44*n^3 + 72*n^2 + 43*n + 3)/3, {n,1,100}] (* G, C, Greubel, Dec 08 2016 *)

Vec(z*(54 + 27*z + 8*z^2 - z^3)/(1-z)^4 + O(x^50)) \\ G. C. Greubel, Dec 08 2016

a(n) = (44*n^3 + 72*n^2 + 43*n + 3)/3.
G.f.: z*(54 + 27*z + 8*z^2 - z^3)/(1-z)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - G. C. Greubel, Dec 08 2016

A248095 Triangle read by rows: TR(m,n) is the Wiener index of the hexagonal trapezium T(m,n), defined in the He et al. reference (1 <= n <= m).

Original entry on oeis.org

27, 109, 210, 279, 566, 822, 569, 1182, 1816, 2328, 1011, 2130, 3370, 4540, 5433, 1637, 3482, 5612, 7772, 9707, 11130, 2479, 5310, 8670, 12224, 15653, 18622, 20748, 3569, 7686, 12672, 18096, 23559, 28662, 32974, 36000, 4939, 10682, 17746, 25588

Offset: 1

Emeric Deutsch, Nov 15 2014

m denotes the number of hexagons in the bottom row, while n is the number of rows of hexagons.
TR(m,1) = A143938(m) = Wiener index of a linear chain of m hexagons.
TR(n,n) = A033544(n) = Wiener index of an n-hexagonal triangle.

			Row 1 is 27; indeed T(1,1) is just one hexagon ABCDEF; it has 6 distances equal to 1 (= number of edges), 6 distances equal to 2 (AC, BD, CE, DA, EA, FB), and 3 distances equal to 3 (AD, BE, CF); 6*1 + 6*2 + 3*3 = 27.
Triangle starts:
  27;
  109, 210;
  279, 566, 822;
  569, 1182, 1816, 2328;

Q. H. He, J. Z. Gu, S. J. Xu, and W. H. Chan, Hosoya polynomials of hexagonal triangles and trapeziums, MATCH, Commun. Math. Comput. Chem. 72, 2014, 835-843.

Cf. A143938, A033544.

/* As triangle */ [[(4*m^3*(n + 1)^2 + 2*m^2*(3 + 11*n + 6*n^2 - 2*n^3))/3 + ((n*(28 + 45*n - 35*n^2 - 8*n^4)+20*m*(1 + 9*n + 6*n^2 - 4*n^3 + n^4)) / 30): n in [1..m]]: m in [1.. 15]]; // Vincenzo Librandi, Nov 16 2014

TR := proc (m, n) options operator, arrow: (4/3)*m^3*(n+1)^2+(2/3)*m^2*(3+11*n+6*n^2-2*n^3)+(1/30)*n*(28+45*n-35*n^2-8*n^4)+(2/3)*m*(1+9*n+6*n^2-4*n^3+n^4) end proc: for m to 10 do seq(TR(m, n), n = 1 .. m) end do; # yields sequence in triangular form
G := factor(sum(sum(TR(i, j)*z^i*t^j, j = 1 .. i), i = 1 .. infinity));

TR(m,n) = (4*m^3*(n + 1)^2 + 2*m^2*(3 + 11*n + 6*n^2 - 2*n^3))/3 + (n*(28 + 45*n - 35*n^2 - 8*n^4)+20*m*(1 + 9*n + 6*n^2 - 4*n^3 + n^4))/30; see Corollary 3,7 in the He et al. reference.

The reader can get the lengthy expression of the bivariate g.f. G by activating the Maple program for TR(m,n) and then activating the Maple program for G.

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A193390 The hyper-Wiener index of a benzenoid consisting of a straight-line chain of n hexagons (s=2; see the Gutman et al. reference).

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A193400 Hyper-Wiener index of a benzenoid consisting of a chain of n hexagons characterized by the encoding s = 1133 (see the Gutman et al. reference, Sec. 5).

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A245830 The Szeged index of a benzenoid consisting of a linear chain of n hexagons.

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A248095 Triangle read by rows: TR(m,n) is the Wiener index of the hexagonal trapezium T(m,n), defined in the He et al. reference (1 <= n <= m).

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