A143937 -id:A143937 - 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

A228597 The Wiener index of the graph obtained by applying Mycielski's construction to a benzenoid consisting of a linear chain of n hexagons.

Original entry on oeis.org

141, 445, 941, 1629, 2509, 3581, 4845, 6301, 7949, 9789, 11821, 14045, 16461, 19069, 21869, 24861, 28045, 31421, 34989, 38749, 42701, 46845, 51181, 55709, 60429, 65341, 70445, 75741, 81229, 86909, 92781, 98845, 105101, 111549, 118189

Offset: 1

Emeric Deutsch, Aug 27 2013

D. B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001, p. 205.

G. C. Greubel, Table of n, a(n) for n = 1..1000
R. Balakrishnan, S. F. Raj, The Wiener number of powers of the Mycielskian, Discussiones Math. Graph Theory, 30, 2010, 489-498 (see Theorem 2.1).
B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A143937.

[96*n^2+16*n+29: n in [1..40]]; // Vincenzo Librandi, Dec 09 2016

a := proc (n) options operator, arrow: 96*n^2+16*n+29 end proc: seq(a(n), n = 1 .. 35);

LinearRecurrence[{3, -3, 1}, {141, 445, 941}, 100] (* or *) Table[96*n^2 + 16*n + 29 , {n,1,100}] (* G. C. Greubel, Dec 08 2016 *)

Vec(x*(141+22*x+29*x^2)/(1-x)^3 + O(x^50)) \\ G. C. Greubel, Dec 08 2016

a(n) = 96*n^2 + 16*n + 29.
G.f.: x*(141+22*x+29*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - G. C. Greubel, Dec 08 2016

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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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A228597 The Wiener index of the graph obtained by applying Mycielski's construction to a benzenoid consisting of a linear chain of n hexagons.

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