A143938 -id:A143938 - cp's OEIS frontend

A143937 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in a benzenoid consisting of a linear chain of n hexagons (1 <= k <= 2n+1).

6, 6, 3, 11, 14, 12, 6, 2, 16, 22, 21, 14, 10, 6, 2, 21, 30, 30, 22, 18, 14, 10, 6, 2, 26, 38, 39, 30, 26, 22, 18, 14, 10, 6, 2, 31, 46, 48, 38, 34, 30, 26, 22, 18, 14, 10, 6, 2, 36, 54, 57, 46, 42, 38, 34, 30, 26, 22, 18, 14, 10, 6, 2, 41, 62, 66, 54, 50, 46, 42, 38, 34, 30, 26, 22

Offset: 1

Emeric Deutsch, Sep 06 2008

The entries in row n are the coefficients of the Wiener polynomial of the benzenoid consisting of a linear chain of n hexagons.
Sum of entries in row n is (2*n+1)*(4*n+1) = A014634(n).
Sum_{k=1..2n+1} k*T(n,k) = A143938(n) is the Wiener index of a benzenoid consisting of a linear chain of n hexagons.

			T(1,2)=6 because in a hexagon there are 6 distances equal to 2.
Triangle starts:
   6,  6,  3;
  11, 14, 12,  6,  2;
  16, 22, 21, 14, 10,  6,  2;
  21, 30, 30, 22, 18, 14, 10,  6,  2;

A. A. Dobrynin, I. Gutman, S. Klavzar, and P. Zigert, Wiener index of hexagonal systems, Acta Applicandae Mathematicae 72 (2002), pp. 247-294.
B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Cf. A014634, A143938.

T:=proc(n,k) if 2*n+1 < k then 0 elif k = 1 then 5*n+1 elif k = 3 then 9*n-6 elif `mod`(k, 2) = 0 then 8*n-4*k+6 else 8*n-4*k+6 end if end proc: for n to 8 do seq(T(n,k),k=1..2*n+1) end do; # yields sequence in triangular form

For 1 <= k <= 2n+1, T(n,k) is given by T(n,1) = 5*n+1, T(n,3) = 9*n - 6, T(n,2*p+1) = 8*n-8*p+2, T(n,2*p) = 8*n-8*p+6.

G.f.: q*z*(6+6*q-z+2*q*z+3*q^2+q^2*z^2-q^4*z)/((1-q^2*z)*(1-z)^2).

A193391 Wiener index of a benzenoid consisting of a spiral chain of n hexagons (s=1; see the Gutman et al. reference).

Original entry on oeis.org

27, 109, 271, 529, 899, 1397, 2039, 2841, 3819, 4989, 6367, 7969, 9811, 11909, 14279, 16937, 19899, 23181, 26799, 30769, 35107, 39829, 44951, 50489, 56459, 62877, 69759, 77121, 84979, 93349, 102247, 111689, 121691, 132269, 143439, 155217, 167619, 180661

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 (4,-6,4,-1).

Cf. A143937, A143938, A193392.

[(27 - 26*n + 72*n^2 +8*n^3)/3: n in [1..30]]; // Vincenzo Librandi, Jul 26 2011

a(n)=(8*n^3+72*n^2-26*n)/3+9 \\ Charles R Greathouse IV, Jul 26 2011

a(n) = (27 - 26*n + 72*n^2 + 8*n^3)/3.

G.f.: x*(27 + x - 3*x^2 - 9*x^3)/(1-x)^4. - Bruno Berselli, Jul 27 2011

A193392 Hyper-Wiener index of a benzenoid consisting of a spiral chain of n hexagons (s=1; see the Gutman et al. reference).

Original entry on oeis.org

42, 215, 636, 1401, 2622, 4427, 6960, 10381, 14866, 20607, 27812, 36705, 47526, 60531, 75992, 94197, 115450, 140071, 168396, 200777, 237582, 279195, 326016, 378461, 436962, 501967, 573940, 653361, 740726, 836547, 941352, 1055685, 1180106, 1315191, 1461532

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, A193391.

[(2*n^4 + 28*n^3 + 154*n^2 - 169*n + 111)/3: n in [1..40]]; // Vincenzo Librandi, Jul 26 2011

a := proc (n) options operator, arrow; (2/3)*n^4+(28/3)*n^3+(154/3)*n^2-(169/3)*n+37 end proc: seq(a(n), n = 1 .. 35);

a(n)=(2*n^4+28*n^3+154*n^2-169*n)/3+37 \\ Charles R Greathouse IV, Jul 26 2011

a(n) = (2*n^4 + 28*n^3 + 154*n^2 - 169*n + 111)/3.

G.f.: x*(42 + 5*x - 19*x^2 - 49*x^3 + 37*x^4)/(1-x)^5. - Bruno Berselli, Jul 27 2011

A193393 Wiener index of a benzenoid consisting of a zig-zag chain of n hexagons (s=13; see the Gutman et al. reference).

Original entry on oeis.org

27, 109, 271, 545, 963, 1557, 2359, 3401, 4715, 6333, 8287, 10609, 13331, 16485, 20103, 24217, 28859, 34061, 39855, 46273, 53347, 61109, 69591, 78825, 88843, 99677, 111359, 123921, 137395, 151813, 167207, 183609, 201051, 219565, 239183, 259937, 281859, 304981, 329335, 354953

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 (4,-6,4,-1).

Cf. A143937, A143938, A193391.

[(16*n^3 + 24*n^2 + 62*n - 21)/3: n in [1..40]]; // Vincenzo Librandi, Jul 26 2011

a(n)=(16*n^3+24*n^2+62*n)/3-7 \\ Charles R Greathouse IV, Jul 26 2011

a(n) = (16*n^3 + 24*n^2 + 62*n - 21)/3.

G.f.: x*(27 + x - 3*x^2 + 7*x^3)/(1-x)^4. - Bruno Berselli, Jul 27 2011

A193394 Hyper-Wiener index of a benzenoid consisting of a zig-zag chain of n hexagons (s=13; see the Gutman et al. reference).

Original entry on oeis.org

42, 215, 636, 1513, 3118, 5787, 9920, 15981, 24498, 36063, 51332, 71025, 95926, 126883, 164808, 210677, 265530, 330471, 406668, 495353, 597822, 715435, 849616, 1001853, 1173698, 1366767, 1582740, 1823361, 2090438, 2385843, 2711512, 3069445, 3461706, 3890423, 4357788

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, A193391, A193392, A193393.

[(8*n^4 + 24*n^3 + 28*n^2 + 147*n - 81)/3: n in [1..40]]; // Vincenzo Librandi, Jul 26 2011

a := n-> (8/3)*n^4+8*n^3+(28/3)*n^2+49*n-27: seq(a(n), n = 1 .. 35);

LinearRecurrence[{5,-10,10,-5,1},{42,215,636,1513,3118},40] (* Harvey P. Dale, Dec 10 2021 *)

a(n)=(8*n^2+24*n+28)*n^2/3+49*n-27 \\ Charles R Greathouse IV, Jul 28 2011

a(n) = (8*n^4 + 24*n^3 + 28*n^2 + 147*n - 81)/3.

G.f.: x*(42 + 5*x - 19*x^2 + 63*x^3 - 27*x^4)/(1-x)^5. - Bruno Berselli, Jul 27 2011

A193395 Wiener index of a benzenoid consisting of a double-step zig-zag chain of n hexagons (n >= 2, s = 2123; see the Gutman et al. reference).

Original entry on oeis.org

109, 271, 553, 971, 1573, 2375, 3425, 4739, 6365, 8319, 10649, 13371, 16533, 20151, 24273, 28915, 34125, 39919, 46345, 53419, 61189, 69671, 78913, 88931, 99773, 111455, 124025, 137499, 151925, 167319, 183729, 201171, 219693, 239311, 260073, 281995, 305125, 329479, 355105

Offset: 2

Emeric Deutsch, Jul 25 2011

Vincenzo Librandi, Table of n, a(n) for n = 2..10000
A. A. Dobrynin, I. Gutman, S. Klavzar, and 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 (3,-2,-2,3,-1).

Cf. A143937, A143938, A193391, A193392, A193393, A193394.

[(16*n^3 + 24*n^2 + 74*n +6*(-1)^n - 51)/3: n in [2..40]]; // Vincenzo Librandi, Jul 26 2011

a := proc (n) options operator, arrow; (16/3)*n^3+8*n^2+(74/3)*n+2*(-1)^n-17 end proc: seq(a(n), n = 2 .. 40);

Table[(16n^3+24n^2+74n+6(-1)^n-51)/3,{n,2,40}] (* or *) LinearRecurrence[ {3,-2,-2,3,-1},{109,271,553,971,1573},40] (* Harvey P. Dale, Apr 08 2020 *)

a(n)=(16*n^3+24*n^2+74*n+6*(-1)^n)/3-17 \\ Charles R Greathouse IV, Jul 28 2011

a(n) = (16*n^3 + 24*n^2 + 74*n +6*(-1)^n - 51)/3.

G.f.: x^2*(109 - 56*x - 42*x^2 + 72*x^3 - 19*x^4)/((1+x)*(1-x)^4). - Bruno Berselli, Jul 27 2011

A193396 Hyper-Wiener index of a benzenoid consisting of a double-step zig-zag chain of n hexagons (n >= 2, s = 2123; see the Gutman et al. reference).

Original entry on oeis.org

215, 636, 1557, 3162, 5875, 10008, 16113, 24630, 36239, 51508, 71245, 96146, 127147, 165072, 210985, 265838, 330823, 407020, 495749, 598218, 715875, 850056, 1002337, 1174182, 1367295, 1583268, 1823933, 2091010, 2386459, 2712128, 3070105, 3462366, 3891127, 4358492

Offset: 2

Emeric Deutsch, Jul 25 2011

Vincenzo Librandi, Table of n, a(n) for n = 2..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 (4,-5,0,5,-4,1).

Cf. A143937, A143938, A193391, A193392, A193393, A193394, A193395.

[(8*n^4 + 24*n^3 + 28*n^2 + 213*n + 33*(-1)^n - 246)/3: n in [2..40]]; // Vincenzo Librandi, Jul 26 2011

a := proc (n) options operator, arrow: (8/3)*n^4+8*n^3+(28/3)*n^2+71*n+11*(-1)^n-82 end proc: seq(a(n), n = 2 .. 35);

a(n)=(8*n^4+24*n^3+28*n^2+213*n+33*(-1)^n-246)/3 \\ Charles R Greathouse IV, Jul 28 2011

a(n) = (8*n^4 + 24*n^3 + 28*n^2 + 213*n + 33*(-1)^n - 246)/3.

G.f.: x^2*(215 - 224*x + 88*x^2 + 114*x^3 - 63*x^4 - 2*x^5)/((1+x)*(1-x)^5). - Bruno Berselli, Jul 27 2011

A193397 Wiener index of a benzenoid consisting of a double-step spiral chain of n hexagons (n>=2, s=21; see the Gutman et al. reference).

Original entry on oeis.org

109, 271, 553, 955, 1541, 2279, 3265, 4435, 5917, 7615, 9689, 12011, 14773, 17815, 21361, 25219, 29645, 34415, 39817, 45595, 52069, 58951, 66593, 74675, 83581, 92959, 103225, 113995, 125717, 137975, 151249, 165091, 180013, 195535, 212201, 229499, 248005, 267175, 287617

Offset: 2

Emeric Deutsch, Jul 25 2011

Vincenzo Librandi, Table of n, a(n) for n = 2..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 (2,1,-4,1,2,-1).

Cf. A143937, A143938, A193391, A193392, A193393, A193394, A193395, A193396.

[4*n^3 + 20*n^2 - 12*n + 2*(-1)^n*(n-2) + 21: n in [2..40]]; // Vincenzo Librandi, Jul 26 2011

a := proc (n) options operator, arrow: 4*n^3+20*n^2-12*n+2*(-1)^n*(n-2)+21 end proc: seq(a(n), n = 2 .. 40);

Table[4n^3+20n^2-12n+2(-1)^n(n-2)+21,{n,2,40}] (* or *) LinearRecurrence[ {2,1,-4,1,2,-1},{109,271,553,955,1541,2279},39] (* Harvey P. Dale, Aug 26 2011 *)

a(n) = 4*n^3 + 20*n^2 - 12*n + 2*(-1)^n*(n-2) + 21.
G.f.: x^2*(109+53*x-98*x^2+14*x^3+53*x^4-35*x^5)/((1+x)^2*(1-x)^4). - Bruno Berselli, Jul 27 2011
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6), with a(2)=109, a(3)=271, a(4)=553, a(5)=955, a(6)=1541, a(7)=2279. - Harvey P. Dale, Aug 26 2011

A193398 Hyper-Wiener index of a benzenoid consisting of a double-step spiral chain of n hexagons (n >= 2, s = 21; see the Gutman et al. reference).

Original entry on oeis.org

215, 636, 1557, 3018, 5555, 8968, 14225, 20790, 30159, 41364, 56525, 74146, 97067, 123168, 156105, 193038, 238535, 288940, 349829, 416634, 496035, 582456, 683777, 793318, 920255, 1056708, 1213245, 1380690, 1571099, 1773904, 2002745, 2245566, 2517687, 2805468

Offset: 2

Emeric Deutsch, Jul 25 2011

Vincenzo Librandi, Table of n, a(n) for n = 2..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.

[(3/2)*n^4+12*n^3+(3/2)*n^2*(-1)^n+(73/2)*n^2+6*n*(-1)^n-79*n+(83/4)*(-1)^(n+1)+439/4: n in [2..40]]; // Vincenzo Librandi, Jul 26 2011

a := n -> (3/2)*n^4+12*n^3+(3/2)*n^2*(-1)^n+(73/2)*n^2+6*n*(-1)^n-79*n+(83/4)*(-1)^(n+1)+439/4: seq(a(n), n = 2 .. 35);

LinearRecurrence[{2,2,-6,0,6,-2,-2,1},{215,636,1557,3018,5555,8968,14225,20790},40] (* Harvey P. Dale, Aug 30 2017 *)

a(n)=(6*n^4+48*n^3+146*n^2-316*n+439+(-1)^n*(6*n^2+24*n-83))/4 \\ Charles R Greathouse IV, Jul 28 2011

a(n) = (6*n^4 + 48*n^3 + 146*n^2 - 316*n + 439 + (-1)^n*(6*n^2 + 24*n - 83))/4.

G.f.: x^2*(215 + 206*x - 145*x^2 - 78*x^3 + 221*x^4 - 126*x^5 - 99*x^6 + 94*x^7)/((1+x)^3*(1-x)^5). - Bruno Berselli, Jul 26 2011

A193399 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

27, 109, 271, 545, 931, 1493, 2199, 3145, 4267, 5693, 7327, 9329, 11571, 14245, 17191, 20633, 24379, 28685, 33327, 38593, 44227, 50549, 57271, 64745, 72651, 81373, 90559, 100625, 111187, 122693, 134727

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 (2,1,-4,1,2,-1).

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

[4*n^3 + 16*n^2 + 8*n + 2*(-1)^n*(n - 2) - 3: n in [1..40]]; // Vincenzo Librandi, Jul 26 2011

a := proc (n) options operator, arrow: 4*n^3+16*n^2+8*n+2*(-1)^n*(n-2)-3 end proc: seq(a(n), n = 1 .. 40);

a(n)=4*n^3+16*n^2+8*n+2*(-1)^n*(n-2)-3 \\ Charles R Greathouse IV, Jul 28 2011

a(n) = 4*n^3 + 16*n^2 + 8*n + 2*(-1)^n*(n - 2) - 3.

G.f.: x*(27 + 55*x + 26*x^2 + 2*x^3 - 21*x^4 + 7*x^5)/((1+x)^2*(1-x)^4). - Bruno Berselli, Jul 27 2011

cp's OEIS Frontend

A143937 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in a benzenoid consisting of a linear chain of n hexagons (1 <= k <= 2n+1).

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A193391 Wiener index of a benzenoid consisting of a spiral chain of n hexagons (s=1; see the Gutman et al. reference).

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A193392 Hyper-Wiener index of a benzenoid consisting of a spiral chain of n hexagons (s=1; see the Gutman et al. reference).

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A193393 Wiener index of a benzenoid consisting of a zig-zag chain of n hexagons (s=13; see the Gutman et al. reference).

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A193394 Hyper-Wiener index of a benzenoid consisting of a zig-zag chain of n hexagons (s=13; see the Gutman et al. reference).

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A193395 Wiener index of a benzenoid consisting of a double-step zig-zag chain of n hexagons (n >= 2, s = 2123; see the Gutman et al. reference).

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Crossrefs

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Magma

Maple

Mathematica

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A193396 Hyper-Wiener index of a benzenoid consisting of a double-step zig-zag chain of n hexagons (n >= 2, s = 2123; see the Gutman et al. reference).

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Links

Crossrefs

Programs

Magma

Maple

PARI

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A193397 Wiener index of a benzenoid consisting of a double-step spiral chain of n hexagons (n>=2, s=21; see the Gutman et al. reference).

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