A139576 -id:A139576 - cp's OEIS frontend

A139579 a(n) = 2n^2 + 15n.

0, 17, 38, 63, 92, 125, 162, 203, 248, 297, 350, 407, 468, 533, 602, 675, 752, 833, 918, 1007, 1100, 1197, 1298, 1403, 1512, 1625, 1742, 1863, 1988, 2117, 2250, 2387, 2528, 2673, 2822, 2975, 3132, 3293, 3458, 3627, 3800, 3977, 4158, 4343, 4532, 4725, 4922, 5123, 5328, 5537

Offset: 0

Omar E. Pol, May 19 2008

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

Cf. A014105, A014106, A033537, A130861, A139576, A139577, A139578, A139580, A139581.

LinearRecurrence[{3,-3,1},{0,17,38},50] (* Stefano Spezia, Oct 21 2023 *)

a(n)=2*n^2+15*n \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + 4*n + 13; a(0) = 0. - Vincenzo Librandi, Nov 24 2010
From Stefano Spezia, Oct 21 2023: (Start)
O.g.f.: x*(17 - 13*x)/(1 - x)^3.
E.g.f.: exp(x)*x*(17 + 2*x). (End)
From Amiram Eldar, Nov 10 2023: (Start)
Sum_{n>=1} 1/a(n) = 182144/675675 - 2*log(2)/15.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/15 - Pi/30 + 67952/675675. (End)

A139577 a(n) = n(2n + 11).

Original entry on oeis.org

0, 13, 30, 51, 76, 105, 138, 175, 216, 261, 310, 363, 420, 481, 546, 615, 688, 765, 846, 931, 1020, 1113, 1210, 1311, 1416, 1525, 1638, 1755, 1876, 2001, 2130, 2263, 2400, 2541, 2686, 2835, 2988, 3145, 3306, 3471, 3640, 3813, 3990

Offset: 0

Omar E. Pol, May 19 2008

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014105, A014106, A033537, A130861, A139576, A139578, A139579, A139580, A139581.

s=0;lst={s};Do[s+=n++ +13;AppendTo[lst, s], {n, 0, 7!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 19 2008 *)
Table[n(2n+11),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,13,30},50] (* Harvey P. Dale, Mar 17 2019 *)

a(n)=n*(2*n+11) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 2*n^2 + 11*n.
a(n) = a(n-1) + 4*n + 9 (with a(0)=0). - Vincenzo Librandi, Nov 24 2010
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: x*(13 - 9*x)/(1-x)^3.
E.g.f.: exp(x)*x*(13 + 2*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A139578 a(n) = n(2n + 13).

Original entry on oeis.org

0, 15, 34, 57, 84, 115, 150, 189, 232, 279, 330, 385, 444, 507, 574, 645, 720, 799, 882, 969, 1060, 1155, 1254, 1357, 1464, 1575, 1690, 1809, 1932, 2059, 2190, 2325, 2464, 2607, 2754, 2905, 3060, 3219, 3382, 3549, 3720, 3895, 4074, 4257, 4444, 4635, 4830, 5029

Offset: 0

Omar E. Pol, May 19 2008

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014105, A014106, A033537, A130861, A139576, A139577, A139579, A139580, A139581.

s=0;lst={s};Do[s+=n++ +15;AppendTo[lst, s], {n, 0, 7!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 19 2008 *)
Table[n(2n+13),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,15,34},50] (* Harvey P. Dale, Nov 22 2014 *)

a(n)=n*(2*n+13) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 2*n^2 + 13*n.
a(n) = a(n-1) + 4*n + 11 (with a(0)=0). - Vincenzo Librandi, Nov 24 2010
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: x*(15 - 11*x)/(1 - x)^3.
E.g.f.: exp(x)*x*(15 + 2*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A139580 a(n) = n(2n + 17).

Original entry on oeis.org

0, 19, 42, 69, 100, 135, 174, 217, 264, 315, 370, 429, 492, 559, 630, 705, 784, 867, 954, 1045, 1140, 1239, 1342, 1449, 1560, 1675, 1794, 1917, 2044, 2175, 2310, 2449, 2592, 2739, 2890, 3045, 3204, 3367, 3534, 3705, 3880, 4059

Offset: 0

Omar E. Pol, May 19 2008

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014105, A014106, A033537, A130861, A139576, A139577, A139578, A139579, A139581.

a[n_]:=Sum[4*i+15, {i, 1, n}]; (* Vladimir Joseph Stephan Orlovsky, Dec 04 2008 *)
Table[n(2n+17),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,19,42},50] (* Harvey P. Dale, Jul 07 2024 *)

a(n)=n*(2*n+17) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 2*n^2 + 17*n.
a(n) = a(n-1) + 4*n + 15; a(0) = 0. - Vincenzo Librandi, Nov 24 2010
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: x*(19 - 15*x)/(1-x)^3.
E.g.f.: exp(x)*x*(19 + 2*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A139581 a(n) = n(2n + 19).

Original entry on oeis.org

0, 21, 46, 75, 108, 145, 186, 231, 280, 333, 390, 451, 516, 585, 658, 735, 816, 901, 990, 1083, 1180, 1281, 1386, 1495, 1608, 1725, 1846, 1971, 2100, 2233, 2370, 2511, 2656, 2805, 2958, 3115, 3276, 3441, 3610, 3783, 3960, 4141

Offset: 0

Omar E. Pol, May 19 2008

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014105, A014106, A033537, A130861, A139576, A139577, A139578, A139579, A139580.

Table[n(2n+19),{n,0,50}]  (* Harvey P. Dale, Jan 26 2011 *)

a(n)=n*(2*n+19) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 2*n^2 + 19*n.
a(n) = a(n-1) + 4*n + 17 (with a(0)=0). - Vincenzo Librandi, Nov 24 2010
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: x*(21 - 17*x)/(1-x)^3.
E.g.f.: exp(x)*x*(21 + 2*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A277979 a(n) = 4n^2 + 18n.

Original entry on oeis.org

0, 22, 52, 90, 136, 190, 252, 322, 400, 486, 580, 682, 792, 910, 1036, 1170, 1312, 1462, 1620, 1786, 1960, 2142, 2332, 2530, 2736, 2950, 3172, 3402, 3640, 3886, 4140, 4402, 4672, 4950, 5236, 5530, 5832, 6142, 6460, 6786, 7120, 7462, 7812, 8170, 8536, 8910, 9292, 9682

Offset: 0

Emeric Deutsch, Nov 08 2016

For n>=3, a(n) is the first Zagreb index of the double-wheel graph DW[n]. The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
The double-wheel graph DW[n] consists of two cycles C[n], whose vertices are connected to an additional vertex.
The M-polynomial of the double-wheel graph DW[n] is M(DW[n],x,y)= 2*n*x^3*y^3 + 2*n*x^3*y^{2*n}.
4*a(n) + 81 is a square. - Bruno Berselli, May 08 2018

			a(3) = 90. Indeed, the double-wheel graph DW[3] has 6 edges with end-point degrees 3,3 and 6 edges with end-point degrees 3,6. Then the first Zagreb index is 6*6 + 6*9 = 90.

E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A139576, A277980.

Subsequence of A028569.

[4*n^2+18*n: n in [0..50]]; // Vincenzo Librandi, Nov 09 2016

Maple
```
seq(4*n^2+18*n,n = 0..50);
```

Table[4 n^2 + 18 n, {n, 0, 50}] (* Vincenzo Librandi, Nov 09 2016 *)
LinearRecurrence[{3,-3,1},{0,22,52},50] (* Harvey P. Dale, Mar 01 2022 *)

a(n)=4*n^2+18*n \\ Charles R Greathouse IV, Jun 17 2017

O.g.f.: 2*x*(11 - 7*x)/(1 - x)^3.
E.g.f.: 2*x*(11 + 2*x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A139576(n).

cp's OEIS Frontend

A139579 a(n) = 2*n^2 + 15*n.

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A139577 a(n) = n*(2*n + 11).

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A139578 a(n) = n*(2*n + 13).

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A139580 a(n) = n*(2*n + 17).

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A139581 a(n) = n*(2*n + 19).

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A277979 a(n) = 4*n^2 + 18*n.

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A139579 a(n) = 2n^2 + 15n.

A139577 a(n) = n(2n + 11).

A139578 a(n) = n(2n + 13).

A139580 a(n) = n(2n + 17).

A139581 a(n) = n(2n + 19).

A277979 a(n) = 4n^2 + 18n.