A343560 -id:A343560 - cp's OEIS frontend

A178242 Numerator of n(5+n)/((n+1)(n+4)).

0, 3, 7, 6, 9, 25, 33, 21, 26, 63, 75, 44, 51, 117, 133, 75, 84, 187, 207, 114, 125, 273, 297, 161, 174, 375, 403, 216, 231, 493, 525, 279, 296, 627, 663, 350, 369, 777, 817, 429, 450, 943, 987, 516, 539, 1125, 1173, 611, 636, 1323, 1375, 714, 741, 1537, 1593

Offset: 0

Paul Curtz, Dec 20 2010

Sequence of differences denominator(n) - numerator(n) = 1,2,2,1... = A014695(n).

Denominator: A160050(n+2).

			The reduced fractions are 0, 3/5, 7/9, 6/7, 9/10, 25/27, 33/35, 21/22, 26/27, 63/65, 75/77, 44/45, ..

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Wikipedia, Quasi-polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Cf. A001793, A014695, A028557, A060819, A160050, A176027.

[Floor(n*(n+5)*((-1)^((2*n-(-1)^n-3)/4)+3)/8) : n in [0..50]]; // Vincenzo Librandi, Oct 08 2011

A178242 := proc(n) n*(5+n)/(n+1)/(n+4) ;  numer(%) ;end proc:
seq(A178242(n),n=0..80) ; # R. J. Mathar, Dec 20 2010

f[n_] := n/GCD[n, 4]; Array[f[#] f[# + 5] &, 50, 0]
Table[Numerator[n*(5+n)/((n+1)*(n+4))], {n,0,50}] (* G. C. Greubel, Sep 21 2018 *)

vector(50, n, n--; numerator(n*(5+n)/((n+1)*(n+4)))) \\ G. C. Greubel, Sep 21 2018

a(n) = numerator(A176027(n)/A001793(n+1)).
a(n) = A060819(n)*A060819(n+5).
a(n) = +3*a(n-1) -6*a(n-2) +10*a(n-3) -12*a(n-4) +12*a(n-5) -10*a(n-6) +6*a(n-7) -3*a(n-8) +a(n-9).
a(n) = 3*a(n-4) -3*a(n-8) +a(n-12).
G.f.:  x*(-3+2*x-3*x^2-3*x^3+x^7) / ( (x-1)^3*(x^2+1)^3 ).
a(n) = n*(n+5)*((-1)^((2*n-(-1)^n-3)/4)+3)/8 = n*(n+5)*(3-i^(n*(n+1)))/8, where i=sqrt(-1); also a(n) = a(n-4)*A028557(n)/A028557(n-4) for n>4. - Bruno Berselli, Dec 30 2010
From Peter Bala, Aug 07 2022: (Start)
a(n) = numerator of n*(n+5)/4.
a(n) is quasi-polynomial in n: a(4*n) = n*(4*n+5) = A343560(n+1); a(4*n+1) = (2*n+3)*(4*n+1); a(4*n+2) = (2*n+1)*(4*n+7); a(4*n+3) = (n+2)*(4*n+3) = A180863(n+2). (End)
Sum_{n>=1} 1/a(n) = 112/75 - Pi/10. - Amiram Eldar, Aug 16 2022

A180863 Wiener index of the n-sun graph.

Original entry on oeis.org

6, 21, 44, 75, 114, 161, 216, 279, 350, 429, 516, 611, 714, 825, 944, 1071, 1206, 1349, 1500, 1659, 1826, 2001, 2184, 2375, 2574, 2781, 2996, 3219, 3450, 3689, 3936, 4191, 4454, 4725, 5004, 5291, 5586, 5889, 6200, 6519, 6846, 7181, 7524, 7875, 8234, 8601

Offset: 2

Emeric Deutsch, Sep 28 2010

The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.
The Wiener polynomial of the n-sun graph is (1/2)*nt[(n-3)t^2+2(n-1)t+n+3].
Comment from Zachary Dove, Apr 19 2021 (Start)
   (Cf. A343560)
On a square lattice, place the nonnegative integers at lattice points forming a spiral as follows: place "0" at the origin; then move one step downward (i.e., in the negative y direction) and place "1" at the lattice point reached; then turn 90 degrees in either direction and place a "2" at the next lattice point; then make another 90-degree turn in the same direction and place a "3" at the lattice point; etc. The terms of the sequence will lie parallel to the positive x-axis, located within the first quadrant, as seen in the example below:
.
  99  64--65--66--67--68--69--70--71--72
   |   |                               |
  98  63  36--37--38--39--40--41--42  73
   |   |   |                       |   |
  97  62  35  16--17--18--19--20  43  74
   |   |   |   |               |   |   |
  96  61  34  15   4---5--*6**21**44**75*
   |   |   |   |   |       |   |   |   |
  95  60  33  14   3   0   7  22  45  76
   |   |   |   |   |   |   |   |   |   |
  94  59  32  13   2---1   8  23  46  77
   |   |   |   |           |   |   |   |
  93  58  31  12--11--10---9  24  47  78
   |   |   |                   |   |   |
  92  57  30--29--28--27--26--25  48  79
   |   |                           |   |
  91  56--55--54--53--52--51--50--49  80
   |                                   |
  90--89--88--87--86--85--84--83--82--81
(End)

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60 (1996), 959-969.
Eric Weisstein's World of Mathematics, Sun Graph.
Eric Weisstein's World of Mathematics, Wiener Index.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Maple
```
seq(n*(4*n-5), n = 2 .. 50);
```

(* Start from Eric W. Weisstein, Sep 07 2017, adapted to new offset *)
Table[n (4 n - 5), {n, 2, 20}]
LinearRecurrence[{3, -3, 1}, {6, 21, 44}, 20]
CoefficientList[Series[(-6 - 3 x + x^2)/(-1 + x)^3, {x, 0, 20}], x]
(* End *)

a(n)=n*(4*n-5) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = n*(4*n-5).
G.f.: x^2*(-6-3*x+x^2)/(x-1)^3. - Colin Barker, Oct 31 2012, adapted to new offset Sep 29 2021
a(n) = 3*a(n-1) - 3*a(n-2) + a(n). - Eric W. Weisstein, Sep 07 2017
a(n) = A033954(n-1)-1 = A033951(n-1) -1. -R. J. Mathar, Sep 29 2021
Sum_{n>=2} 1/a(n) = 1/5 - Pi/10 + 3*log(2)/5. - Amiram Eldar, Apr 16 2022
E.g.f.: exp(x)*(-x + 4*x^2) + x. - Nikolaos Pantelidis, Feb 10 2023

a(2)=6 prefixed by R. J. Mathar, Sep 29 2021

A386486 a(0) = 1; thereafter a(n) = 4n^2 - 3n + 2.

Original entry on oeis.org

1, 3, 12, 29, 54, 87, 128, 177, 234, 299, 372, 453, 542, 639, 744, 857, 978, 1107, 1244, 1389, 1542, 1703, 1872, 2049, 2234, 2427, 2628, 2837, 3054, 3279, 3512, 3753, 4002, 4259, 4524, 4797, 5078, 5367, 5664, 5969, 6282, 6603, 6932, 7269, 7614, 7967, 8328, 8697, 9074, 9459, 9852, 10253, 10662, 11079, 11504, 11937, 12378

Offset: 0

N. J. A. Sloane, Aug 27 2025

Differs from A001107, A054552, and A343560 only by a small constant, but has its own entry because of an important geometric application (which will be added soon).

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

Cf. A001107, A054552, A343560.

A386486[n_] := If[n == 0, 1, (4*n - 3)*n + 2]; Array[A386486, 100, 0] (* or *)
LinearRecurrence[{3, -3, 1}, {1, 3, 12, 29}, 100] (* Paolo Xausa, Aug 27 2025 *)

From Elmo R. Oliveira, Sep 02 2025: (Start)
G.f.: (1 + 6*x^2 + x^3)/(1 - x)^3.
E.g.f.: exp(x)*(2 + x + 4*x^2) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

cp's OEIS Frontend

A178242 Numerator of n(5+n)/((n+1)(n+4)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A180863 Wiener index of the n-sun graph.

Views

Author

Keywords

Comments

Links

Programs

Maple

Mathematica

PARI

Formula

Extensions

A386486 a(0) = 1; thereafter a(n) = 4n^2 - 3n + 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A178242 Numerator of n*(5+n)/((n+1)*(n+4)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A180863 Wiener index of the n-sun graph.

Views

Author

Keywords

Comments

Links

Programs

Maple

Mathematica

PARI

Formula

Extensions

A386486 a(0) = 1; thereafter a(n) = 4*n^2 - 3*n + 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A178242 Numerator of n(5+n)/((n+1)(n+4)).

A386486 a(0) = 1; thereafter a(n) = 4n^2 - 3n + 2.