A263226 -id:A263226 - cp's OEIS frontend

A263231 a(n) = n(25n - 39)/2.

0, -7, 11, 54, 122, 215, 333, 476, 644, 837, 1055, 1298, 1566, 1859, 2177, 2520, 2888, 3281, 3699, 4142, 4610, 5103, 5621, 6164, 6732, 7325, 7943, 8586, 9254, 9947, 10665, 11408, 12176, 12969, 13787, 14630, 15498, 16391, 17309, 18252, 19220

Offset: 0

Emeric Deutsch, Oct 14 2015

For n>=3, a(n) = the hyper-Wiener index of the Jahangir graph J_{2,n}. The Jahangir graph J_{2,n} is a connected graph consisting of a cycle graph C(2*n) and one additional center vertex that is adjacent to n vertices of C(2*n) at distances 2 to each other on C(2*n).

The Hosoya polynomial of J_{2,n} is 3*n*x + n*(n+3)*x^2/2 + n*(n-2)*x^3 + n*(n-3)*x^4/2.

M. R. Farahani, Hosoya polynomial and Wiener index of Jahangir graphs J_{2,m}, Pacific J. Appl. Math, 7 (3), 2015.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A263226, A263227, A263228, A263229, A263230.

a263231 n = n * (25 * n - 39) `div` 2
a263231_list = 0 : -7 : 11 : zipWith (+) a263231_list
   (map (* 3) $ tail $ zipWith (-) (tail a263231_list) a263231_list)
-- Reinhard Zumkeller, Nov 04 2015

Maple
```
seq((25*n^2 - 39*n)/2, n=0..40);
```
Mathematica
```
Table[n (25 n - 39)/2, {n, 0, 40}]
```
PARI
```
vector(50, n, n--; n*(25*n-39)/2)
```

concat(0, Vec(x*(32*x-7)/(1-x)^3 + O(x^100))) \\ Altug Alkan, Oct 18 2015

G.f.: x*(32*x-7)/(1-x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

A263227 a(n) = n(67n - 89)/2.

Original entry on oeis.org

0, -11, 45, 168, 358, 615, 939, 1330, 1788, 2313, 2905, 3564, 4290, 5083, 5943, 6870, 7864, 8925, 10053, 11248, 12510, 13839, 15235, 16698, 18228, 19825, 21489, 23220, 25018, 26883, 28815, 30814, 32880, 35013, 37213, 39480, 41814, 44215, 46683, 49218, 51820

Offset: 0

Emeric Deutsch, Oct 12 2015

For n>=3, a(n) = the hyper-Wiener index of the Jahangir graph J_{3,n}. The Jahangir graph J_{3,n} is a connected graph consisting of a cycle graph C(3n) and one additional center vertex that is adjacent to n vertices of C(3n) at distances 3 to each other on C(3n).

The Hosoya polynomial of J_(3,n) is 4nx + (1/2)n(n+9)x^2 + 2n(n-1)x^3 + n(2n-5)x^4.

M. R. Farahani, The Wiener index and Hosoya polynomial of a class of Jahangir graphs J_{3,m}, Fundamental J. Math. and Math. Sci., 3 (1), 91-96, 2015.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A049598, A263226, A263228, A263229, A263231.

[n*(67*n-89)/2: n in [0..40]]; // Bruno Berselli, Oct 15 2015

Maple
```
seq((1/2)*n*(67*n-89), n = 0 .. 40);
```

Table[n (67 n - 89)/2, {n, 0, 40}] (* Vincenzo Librandi, Oct 13 2015 *)

vector(50, n, n--; n*(67*n-89)/2) \\ Altug Alkan, Oct 12 2015

G.f.: x*(-11+78*x)/(1-x)^3. - Vincenzo Librandi, Oct 13 2015

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Oct 13 2015

A263228 a(n) = 2n(16*n - 13).

Original entry on oeis.org

0, 6, 76, 210, 408, 670, 996, 1386, 1840, 2358, 2940, 3586, 4296, 5070, 5908, 6810, 7776, 8806, 9900, 11058, 12280, 13566, 14916, 16330, 17808, 19350, 20956, 22626, 24360, 26158, 28020, 29946, 31936, 33990, 36108, 38290, 40536, 42846, 45220, 47658, 50160

Offset: 0

Emeric Deutsch, Oct 13 2015

For n>=3, a(n) = the Wiener index of the Jahangir graph J_{4,n}. The Jahangir graph J_{4,n} is a connected graph consisting of a cycle graph C(4*n) and one additional center vertex that is adjacent to n vertices of C(4*n) at distances 4 to each other on C(4*n). In the Farahani reference the expression in Theorem 2 is accidentally incorrect; it should be 2*m*(16*m - 13).

The Hosoya polynomial of J_{4,n} is 5*n*x + n*(n+11)*x^2/2 + n*(2*n+1)*x^3 + n*(3*n-4)*x^4 + 2*n*(n-2)*x^5 + n*(n-3)*x^6/2 (see the Farahani reference, p. 234, last line; however, the expression in Theorem 1, p. 233, is accidentally incorrect).

M. R. Farahani, Hosoya polynomial and of Jahangir graphs J_{4,m}, Global J. Math, 3 (1), 232-236, 2015.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A049598, A263226, A263227, A263229, A263231.

[2*n*(16*n-13): n in [0..60]]; // Vincenzo Librandi, Oct 15 2015

Maple
```
seq(32*n^2 - 26*n, n=0..40);
```

Table[2 n (16 n - 13), {n, 0, 40}] (* Bruno Berselli, Oct 15 2015 *)

vector(50, n, n--; 2*n*(16*n-13)) \\ Altug Alkan, Oct 15 2015

G.f.: 2*x*(3+29*x)/(1-x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

A263229 a(n) = 4n(21*n - 26).

Original entry on oeis.org

0, -20, 128, 444, 928, 1580, 2400, 3388, 4544, 5868, 7360, 9020, 10848, 12844, 15008, 17340, 19840, 22508, 25344, 28348, 31520, 34860, 38368, 42044, 45888, 49900, 54080, 58428, 62944, 67628, 72480, 77500, 82688, 88044, 93568, 99260, 105120, 111148, 117344, 123708, 130240

Offset: 0

Emeric Deutsch, Oct 13 2015

For n>=3, a(n) = the hyper-Wiener index of the Jahangir graph J_{4,n}.

See A263228 for more comments.

M. R. Farahani, Hosoya polynomial and of Jahangir graphs J_{4,m}, Global J. Math, 3 (1), 232-236, 2015.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A049598, A263226, A263227, A263228, A263231.

[4*n*(21*n-26): n in [0..20]]; // Vincenzo Librandi, Oct 15 2015

Maple
```
seq(84*n^2 - 104*n, n=0..40);
```

Table[4 n (21 n - 26), {n, 0, 40}] (* Bruno Berselli, Oct 15 2015 *)

vector(50, n, n--; 4*n*(21*n-26)) \\ Altug Alkan, Oct 15 2015

G.f.: 4*x*(47*x-5)/(1-x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

cp's OEIS Frontend

A263231 a(n) = n*(25*n - 39)/2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Formula

A263227 a(n) = n*(67*n - 89)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A263228 a(n) = 2*n*(16*n - 13).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A263229 a(n) = 4*n*(21*n - 26).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A263231 a(n) = n(25n - 39)/2.

A263227 a(n) = n(67n - 89)/2.

A263228 a(n) = 2n(16*n - 13).

A263229 a(n) = 4n(21*n - 26).