A338109 -id:A338109 - cp's OEIS frontend

A002939 a(n) = 2n(2*n-1).

0, 2, 12, 30, 56, 90, 132, 182, 240, 306, 380, 462, 552, 650, 756, 870, 992, 1122, 1260, 1406, 1560, 1722, 1892, 2070, 2256, 2450, 2652, 2862, 3080, 3306, 3540, 3782, 4032, 4290, 4556, 4830, 5112, 5402, 5700, 6006, 6320, 6642, 6972, 7310, 7656, 8010, 8372

Offset: 0

N. J. A. Sloane

Write 0,1,2,... in a spiral; sequence gives numbers on one of 4 diagonals (see Example section).
For n>1 this is the Engel expansion of cosh(1), A118239. - Benoit Cloitre, Mar 03 2002
a(n) = A125199(n,n) for n>0. - Reinhard Zumkeller, Nov 24 2006
Central terms of the triangle in A195437: a(n+1) = A195437(2*n,n). - Reinhard Zumkeller, Nov 23 2011
For n>2, the terms represent the sums of those primitive Pythagorean triples with hypotenuse (H) one unit longer than the longest side (L), or H = L + 1. - Richard R. Forberg, Jun 09 2015
For n>1, a(n) is the perimeter of a Pythagorean triangle with an odd leg 2*n-1. - Agola Kisira Odero, Apr 26 2016
From Rigoberto Florez, Nov 07 2020 : (Start)
A338109(n)/a(n+1) is the Kirchhoff index of the join of the disjoint union of two complete graphs on n vertices with the empty graph on n+1 vertices.
Equivalently, the graph can be described as the graph on 3*n + 1 vertices with labels 0..3*n and with i and j adjacent iff iff  i+j> 0 mod 3.
A338588(n)/a(n+1) is the Kirchhoff index of the disjoint union of two complete graphs each on n and n+1 vertices with the empty graph on n+1 vertices.
Equivalently, the graph can be described as the graph on 3*n + 2 vertices with labels 0..3*n+1 and with i and j adjacent iff  i+j> 0 mod 3.
These graphs are cographs. (End)
a(n), n>=1, is the number of paths of minimum length (length=2) from the origin to the cross polytope of size 2 in Z^n (column 2 in A371064). - Shel Kaphan, Mar 09 2024

			G.f. = 2*x + 12*x^2 + 30*x^3 + 56*x^4 + 90*x^5 + 132*x^6 + 182*x^7 + 240*x^8 + ...
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 in any of the four cardinal directions 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 along one of the diagonals, 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
.
[Edited by _Jon E. Schoenfield_, Jan 01 2017]

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
H-Y. Ching, R. Florez, and A. Mukherjee, Families of Integral Cographs within a Triangular Arrays, arXiv:2009.02770 [math.CO], 2020.
A. M. Nemirovsky et al., Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108.
R. Tijdeman, Some applications of Diophantine approximation, pp. 261-284 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.
Eric Weisstein's World of Mathematics, Kirchhoff Index
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Sequences from spirals: A001107, A002939, A007742, A033951-A033953, A033954, A033989-A033991, A002943, A033996, A033988.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. A016789, A017041, A017485, A125202.
Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488 (this sequence is the case k=8). - Bruno Berselli, Jun 10 2013
Cf. A017089 (first differences), A268684 (partial sums), A010050 (partial products).
Cf. A371064.

a002939 n = (* 2) . a000384
a002939_list = scanl1 (+) a017089_list
-- Reinhard Zumkeller, Jun 08 2015

[2*n*(2*n-1): n in [0..50]]; // Vincenzo Librandi, Jul 26 2011

A002939:=n->2*n*(2*n-1): seq(A002939(n), n=0..100); # Wesley Ivan Hurt, Jan 28 2017

Table[2*n*(2*n-1), {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Oct 25 2008 *)
2#(2#-1)&/@Range[0,50]  (* Harvey P. Dale, Mar 06 2011 *)

a(n)=2*binomial(2*n,2) \\ Charles R Greathouse IV, Jul 25 2011

a=lambda n: 2*n*(2*n-1) # Indranil Ghosh, Jan 01 2017

Sum_{n >= 1} 1/a(n) = log(2) (cf. Tijdeman).
Log(2) = Sum_{n >= 1} ((1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + (1/7 - 1/8) + ...) = Sum_{n >= 0} (-1)^n/(n+1). Log(2) = Integral_{x=0..1} 1/(1+x) dx. - Gary W. Adamson, Jun 22 2003
a(n) = A000384(n)*2. - Omar E. Pol, May 14 2008
From R. J. Mathar, Apr 23 2009: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 2*x*(1+3*x)/(1-x)^3. (End)
a(n) = a(n-1) + 8*n - 6 (with a(0)=0). - Vincenzo Librandi, Nov 12 2010
a(n) = A118729(8n+1). - Philippe Deléham, Mar 26 2013
Product_{k=1..n} a(k) = (2n)! = A010050(n). - Tony Foster III, Sep 06 2015
E.g.f.: 2*x*(1 + 2*x)*exp(x). - Ilya Gutkovskiy, Apr 29 2016
a(n) = A002943(-n) for all n in Z. - Michael Somos, Jan 28 2017
0 = 12 + a(n)*(-8 + a(n) - 2*a(n+1)) + a(n+1)*(-8 + a(n+1)) for all n in Z. - Michael Somos, Jan 28 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 - log(2)/2. - Amiram Eldar, Jul 31 2020

A338104 Number of spanning trees in the join of the disjoint union of two complete graphs each on n vertices with the empty graph on n+1 vertices.

Original entry on oeis.org

1, 4, 1200, 2074464, 10883911680, 128615328600000, 2881502756476710912, 109416128865750000000000, 6508595325997684722663161856, 572150341080161420030586961966080, 71062412455566037275496151040000000000

Offset: 0

Rigoberto Florez, Oct 10 2020

nonn

Equivalently, the graph can be described as the graph on 3*n + 1 vertices with labels 0..3*n and with i and j adjacent iff A011655(i + j) = 1.

These graphs are cographs.

			The adjacency matrix of the graph associated with n = 2 is: (compare A204437)
  [0, 1, 1, 0, 1, 1, 0]
  [1, 0, 0, 1, 1, 0, 1]
  [1, 0, 0, 1, 0, 1, 1]
  [0, 1, 1, 0, 1, 1, 0]
  [1, 1, 0, 1, 0, 0, 1]
  [1, 0, 1, 1, 0, 0, 1]
  [0, 1, 1, 0, 1, 1, 0]
a(2) = 1200 because the graph has 1200 spanning trees.

H-Y. Ching, R. Florez, and A. Mukherjee, Families of Integral Cographs within a Triangular Arrays, arXiv:2009.02770 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Spanning Tree

Cf. A011655, A204437, A338109.

Table[(n + 1)*(2 n)^n*(2 n + 1)^(2 (n - 1)), {n, 1, 10}]

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

A338527 Number of spanning trees in the join of the disjoint union of two complete graphs each on n and n+1 vertices with the empty graph on n+1 vertices.

Original entry on oeis.org

24, 13500, 34420736, 239148450000, 3520397039081472, 94458953432730437824, 4179422085120000000000000, 283894102615246085842939590912, 28059580711858187192007680000000000, 3870669526565955444680027453177986243584

Offset: 1

Rigoberto Florez, Nov 07 2020

nonn

Equivalently, the graph can be described as the graph on 3*n + 2 vertices with labels 0..3*n+1 and with i and j adjacent iff i+j> 0 mod 3.

These graphs are cographs.

			The adjacency matrix of the graph associated with n = 2 is: (compare A204437)
 [0, 1, 0, 0, 0, 1, 1, 1]
 [1, 0, 0, 0, 0, 1, 1, 1]
 [0, 0, 0, 1, 1, 1, 1, 1]
 [0, 0, 1, 0, 1, 1, 1, 1]
 [0, 0, 1, 1, 0, 1, 1, 1]
 [1, 1, 1, 1, 1, 0, 0, 0]
 [1, 1, 1, 1, 1, 0, 0, 0]
 [1, 1, 1, 1, 1, 0, 0, 0]
a(2) = 13500 because the graph has 13500 spanning trees.

H-Y. Ching, R. Florez, and A. Mukherjee, Families of Integral Cographs within a Triangular Arrays, arXiv:2009.02770 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Spanning Tree

Cf. A338104, A338109.

Table[(n + 1)*(2 n + 2)^n*(2 n + 1)^(2 n - 1), {n, 1, 10}]

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

Offset changed by Georg Fischer, Nov 03 2023

A338588 a(n)/A002939(n+1) is the Kirchhoff index of the disjoint union of two complete graphs each on n and n+1 vertices with the empty graph on n+1 vertices.

Original entry on oeis.org

2, 77, 334, 881, 1826, 3277, 5342, 8129, 11746, 16301, 21902, 28657, 36674, 46061, 56926, 69377, 83522, 99469, 117326, 137201, 159202, 183437, 210014, 239041, 270626, 304877, 341902, 381809, 424706, 470701, 519902

Offset: 0

Rigoberto Florez, Nov 07 2020

Equivalently, the graph can be described as the graph on 3*n + 2 vertices with labels 0..3*n+1 and with i and j adjacent iff i+j>0 mod 3.
These graphs are cographs.
The initial term a(0) = 2 has been included to agree with the formula. For the graph, is not defined.

			The adjacency matrix of the graph associated with n = 2 is:
  [0, 1, 0, 0, 0, 1, 1, 1]
  [1, 0, 0, 0, 0, 1, 1, 1]
  [0, 0, 0, 1, 1, 1, 1, 1]
  [0, 0, 1, 0, 1, 1, 1, 1]
  [0, 0, 1, 1, 0, 1, 1, 1]
  [1, 1, 1, 1, 1, 0, 0, 0]
  [1, 1, 1, 1, 1, 0, 0, 0]
  [1, 1, 1, 1, 1, 0, 0, 0].
a(2) = 334 because the Kirchhoff index of the graph is 334/30=334/A002939(3).
The first few Kirchhoff indices (n >= 1) as reduced fractions are 77/12, 167/15, 881/56, 913/45, 3277/132, 2671/91, 8129/240, 5873/153, 16301/380, 10951/231.

H-Y. Ching, R. Florez, and A. Mukherjee, Families of Integral Cographs within a Triangular Arrays, arXiv:2009.02770 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Kirchhoff Index
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A338527, A338104, A338109.

Mathematica
```
Table[(18n^3+37n^2+20n+2), {n,0,30}]
```

a(n) = 18*n^3 + 37*n^2 + 20*n + 2.
G.f.: (2 + 69*x + 38*x^2 - x^3)/(x - 1)^4.
E.g.f.: exp(x)*(2 + 75*x + 91*x^2 + 18*x^3). - Stefano Spezia, Nov 08 2020
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Nov 08 2020

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A002939 a(n) = 2*n*(2*n-1).

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A338104 Number of spanning trees in the join of the disjoint union of two complete graphs each on n vertices with the empty graph on n+1 vertices.

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A338527 Number of spanning trees in the join of the disjoint union of two complete graphs each on n and n+1 vertices with the empty graph on n+1 vertices.

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A338588 a(n)/A002939(n+1) is the Kirchhoff index of the disjoint union of two complete graphs each on n and n+1 vertices with the empty graph on n+1 vertices.

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A002939 a(n) = 2n(2*n-1).