A055267 -id:A055267 - cp's OEIS frontend

A055273 a(n) = 3*a(n-1) - a(n-2) with a(0) = 1, a(1) = 8.

1, 8, 23, 61, 160, 419, 1097, 2872, 7519, 19685, 51536, 134923, 353233, 924776, 2421095, 6338509, 16594432, 43444787, 113739929, 297775000, 779585071, 2040980213, 5343355568, 13989086491, 36623903905, 95882625224, 251023971767, 657189290077, 1720543898464

Offset: 0

Barry E. Williams, May 28 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Cf. A000032, A000045, A055267.

a:=[1,8];; for n in [3..30] do a[n]:=3*a[n-1]-a[n-2]; od; Print(a); # Muniru A Asiru, Dec 29 2018

[Fibonacci(2*n+2) + 5*Fibonacci(2*n): n in [0..30]]; // Vincenzo Librandi, Dec 25 2018

seq(coeff(series((1+5*x)/(1-3*x+x^2),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Dec 29 2018

Table[3Fibonacci[2n+2]-Fibonacci[2n-3], {n,0,20}] (* Rigoberto Florez, Dec 24 2018 *)
LinearRecurrence[{3, -1}, {1, 8}, 30] (* Vincenzo Librandi, Dec 25 2018 *)

vector(30, n, fibonacci(2*n) + 5*fibonacci(2*n-2) ) \\ G. C. Greubel, Jan 17 2020

[fibonacci(2*n+2) +5*fibonacci(2*n) for n in (0..30)] # G. C. Greubel, Jan 17 2020

a(n) = (8*(((3 + sqrt(5))/2)^n - ((3 - sqrt(5))/2)^n) - (((3 + sqrt(5))/2)^(n - 1) - ((3 - sqrt(5))/2)^(n - 1)))/sqrt(5).
G.f.: (1 + 5*x)/(1 - 3*x + x^2).
From Rigoberto Florez, Dec 24 2018: (Start)
a(n) = Fibonacci(2n+2) + 5*Fibonacci(2n),
a(n) = 3*Fibonacci(2n+2) - Fibonacci(2n-3). (End)
E.g.f.: (1/5)*exp(3*x/2)*(5*cosh(sqrt(5)*x/2) + 13*sqrt(5)*sinh(sqrt(5)*x/2)). - Franck Maminirina Ramaharo, Dec 26 2018
a(n) = ChebyshevT(n, 3/2) + (13/2)*ChebyshevU(n-1, 3/2) = ChebyshevU(n, 3/2) + 5*ChebyshevU(n-1, 3/2). - G. C. Greubel, Jan 17 2020

A141751 Triangle, read by rows, where T(n,k) = [T(n-1,k-1)T(n-1,k) + 1]/T(n-2,k-1) for 0=0 and T(n,0) = Fibonacci(2n-1) for n>=1.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 13, 13, 8, 4, 1, 34, 34, 21, 11, 5, 1, 89, 89, 55, 29, 14, 6, 1, 233, 233, 144, 76, 37, 17, 7, 1, 610, 610, 377, 199, 97, 45, 20, 8, 1, 1597, 1597, 987, 521, 254, 118, 53, 23, 9, 1, 4181, 4181, 2584, 1364, 665, 309, 139, 61, 26, 10, 1

Offset: 0

Paul D. Hanna, Jul 04 2008

			Generating rule.
Given nonzero elements W, X, Y, Z, relatively arranged like so:
.. W .....
.. X Y ...
.... Z ...
then Z = (X*Y + 1)/W.
Triangle begins:
1;
1, 1;
2, 2, 1;
5, 5, 3, 1;
13, 13, 8, 4, 1;
34, 34, 21, 11, 5, 1;
89, 89, 55, 29, 14, 6, 1;
233, 233, 144, 76, 37, 17, 7, 1;
610, 610, 377, 199, 97, 45, 20, 8, 1;
1597, 1597, 987, 521, 254, 118, 53, 23, 9, 1;
4181, 4181, 2584, 1364, 665, 309, 139, 61, 26, 10, 1; ...

Paul D. Hanna, Table of n, a(n) for n = 0..1080, as a flattened table of rows 0..45

Cf. row sums: A141752; columns: A001519, A001906, A002878, A054486, A054492, A055267, A055273, A055849.

PARI
```
T(n,k)=if(n
				
```

T(n,k)=fibonacci(2*(n-k))*k+fibonacci(2*(n-k)-1)
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))

T(n,k) = Fibonacci(2*(n-k)-1) + k*Fibonacci(2*(n-k)) for 0<=k<=n.

A335807 Number of vertices in the n-th simplex graph of the complete graph on three vertices (K_3).

Original entry on oeis.org

3, 8, 21, 54, 140, 365, 954, 2496, 6533, 17102, 44772, 117213, 306866, 803384, 2103285, 5506470, 14416124, 37741901, 98809578, 258686832, 677250917, 1773065918, 4641946836, 12152774589, 31816376930, 83296356200, 218072691669, 570921718806, 1494692464748, 3913155675437

Offset: 0

James Turbett, Aug 14 2020

The simplex k(G) of an undirected graph G is itself a graph that has one vertex for each clique in G. Two vertices are connected by an edge in k(G) if their corresponding cliques differ by the presence or absence of a single vertex in G. For the purposes of finding a simplex graph, the empty set of zero vertices is considered a 0-clique in G, and each individual vertex in G is considered a 1-clique in G.
If we define the simplex graph of a graph G to be k(G), then we can define the 2nd simplex graph to be k(k(G)), which is the simplex of the simplex of the graph G, and the 3rd simplex graph to be k(k(k(G))), and so on. We can also define the 0th simplex graph to be the graph itself.
Using recurrence relations and the initial values for order and size of the 1st simplex graph of a graph G, it is possible to calculate the orders (and sizes) of the n-th simplex graphs using an algorithm, which I provided as a JavaScript program.
The order, |V|(n), of the n-th simplex graph of a graph G follows this recurrence relation: |V|(n) = |V|(n - 1) + |E|(n - 1) + 1, where |E|(n - 1) is the size of the (n-1)-th simplex graph of G and |V|(n-1) is the order of the (n-1)-th simplex graph of G.
The size, |E|(n), of the n-th simplex graph of a graph G follows this recurrence relation: |E|(n) = |V|(n - 1) + 2*|E|(n - 1), where |V|(n-1) is the order of the (n-1)-th simplex graph of G and |E|(n-1) is the size of the (n-1)-th simplex graph of G.

			The simplex graph of K_3 will have 8 vertices: 1 for the 0-clique in K_3, 3 for the 1-cliques in K_3, 3 for the 2-cliques in K_3, and 1 for the 3-clique in K_3.
It also has size 12. In the simplex graph, each vertex corresponding to a 1-clique in K_3 links to the simplex graph vertex corresponding to a 0-clique in K_3, creating 3 edges. Also, each simplex graph vertex corresponding to a 2-clique in K_3 links to 2 simplex graph vertices corresponding to a 1-clique in K_3, producing 6 more edges. Finally, the single vertex in the simplex graph corresponding to the 3-clique in K_3 links to each simplex graph vertex corresponding to a 2-clique in K_3, producing 3 more edges, for a total of 12 edges in the 1st simplex graph of K_3.
Using the recurrence relation, we can calculate |V|(2) = |V|(1) + |E|(1) + 1 = 8 + 12 + 1 = 21. Therefore, the 2nd simplex graph of K_3 will have 21 vertices.

Wikipedia, Simplex Graph
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

//Define order and size of 1st simplex graph of K_3
var order = [8]
var size = [12]//Calculate the orders of the 1st through (numberOfTerms + 1)th simplex graphs of K_3
function simplexSequenceOrder (numberOfTerms) {
for (var i = 1; i <= numberOfTerms; i++) {
order[i] = order[i-1] + size[i-1] + 1;
size [i] = order[i-1] + 2*size[i-1];
}
return order.join();
}

Vec((3 - 4*x + x^2 - x^3) / ((1 - x)*(1 - 3*x + x^2)) + O(x^28)) \\ Colin Barker, Aug 16 2020

From Colin Barker, Aug 15 2020: (Start)
G.f.: (3 - 4*x + x^2 - x^3) / ((1 - x)*(1 - 3*x + x^2)).
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3) for n>3.
(End)
a(n) = (10 + (5 - 11*sqrt(5))*(1/2*(3 - sqrt(5)))^n + (1/2*(3 + sqrt(5)))^n*(5 + 11*sqrt(5)))/10 for n > 0. - Stefano Spezia, Aug 15 2020
a(n) = 3*a(n-1) - a(n-2) - 1 for n >= 3. - James Turbett, Aug 18 2020
a(n) = 1+A055267(n), n>0. - R. J. Mathar, Mar 06 2022

cp's OEIS Frontend

A055273 a(n) = 3*a(n-1) - a(n-2) with a(0) = 1, a(1) = 8.

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A141751 Triangle, read by rows, where T(n,k) = [T(n-1,k-1)T(n-1,k) + 1]/T(n-2,k-1) for 0=0 and T(n,0) = Fibonacci(2n-1) for n>=1.

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A335807 Number of vertices in the n-th simplex graph of the complete graph on three vertices (K_3).

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JavaScript

PARI

Formula

cp's OEIS Frontend

A055273 a(n) = 3*a(n-1) - a(n-2) with a(0) = 1, a(1) = 8.

Views

Author

Keywords

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A141751 Triangle, read by rows, where T(n,k) = [T(n-1,k-1)*T(n-1,k) + 1]/T(n-2,k-1) for 0=0 and T(n,0) = Fibonacci(2*n-1) for n>=1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A335807 Number of vertices in the n-th simplex graph of the complete graph on three vertices (K_3).

Views

Author

Keywords

Comments

Examples

Links

Programs

JavaScript

PARI

Formula

A141751 Triangle, read by rows, where T(n,k) = [T(n-1,k-1)T(n-1,k) + 1]/T(n-2,k-1) for 0=0 and T(n,0) = Fibonacci(2n-1) for n>=1.