A171496 -id:A171496 - cp's OEIS frontend

A171495 a(n) = 3*a(n-1)+4 for n > 0; a(0) = 6.

6, 22, 70, 214, 646, 1942, 5830, 17494, 52486, 157462, 472390, 1417174, 4251526, 12754582, 38263750, 114791254, 344373766, 1033121302, 3099363910, 9298091734, 27894275206, 83682825622, 251048476870, 753145430614, 2259436291846

Offset: 0

Klaus Brockhaus, Dec 10 2009

nonn

Binomial transform of A171494; second binomial transform of A010726.

Inverse binomial transform of A171496.

Index entries for linear recurrences with constant coefficients, signature (4, -3).

Equals 2*A171498.

Cf. A010726 (repeat 6, 10), A171494, A171496.

NestList[3#+4&,6,30] (* Harvey P. Dale, Aug 25 2019 *)

{m=25; v=concat([6], vector(m-1)); for(n=2, m, v[n]=3*v[n-1]+4); v}

a(n) = 2*(4*3^n-1).

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

A308700 a(n) = n * 2^(n - 2) * (2^(n - 1) - 1).

Original entry on oeis.org

0, 0, 2, 18, 112, 600, 2976, 14112, 65024, 293760, 1308160, 5761536, 25153536, 109025280, 469704704, 2013143040, 8589672448, 36506664960, 154617643008, 652832538624, 2748773826560, 11544861081600, 48378488553472, 202310091276288, 844424829468672, 3518436999168000

Offset: 0

Stefano Spezia, Jun 17 2019

Given a pseudo-graph P of the set X = {1, 2, ..., n}, defined as a graph represented by the discrete topology on the set X (the power set of X), for n > 0, a(n) is the number of edges of the topological graph arising by deleting loops in P (see Theorem 3.3 in Kozae et al.).

			For n = 3, the set X = {1,2,3},
  the power set 2^X = {{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, X} and the pseudo-graph P represented by 2^X has the following edges, here grouped into...
  simple loops:
  {1} --- {1}, {2} --- {2}, {3} --- {3} for a total of 3.
  double loops:
  {1,2} --- {1,2}, {1,3} --- {1,3}, {2,3} --- {2,3} for a total of 6 simple loops.
  triple loop:
  X --- X for a total of 3 simple loops.
  simple edges:
  {1} --- {1,2}, {1} --- {1,3}, {1} --- X, {2} --- {1,2}, {2} --- {2,3}, {2} --- X, {3} --- {1,3}, {3} --- {2,3}, {3} --- X, {1,2} --- {1,3}, {1,2} --- {2,3}, {1,3} --- {2,3} for a total of 12.
  double edges:
  {1,2} --- X, {1,3} --- X, {2,3} --- X for a total of 6 simple edges.
  By deleting the loops in P, there remain a total of a(3) = 12 + 6 = 18 edges for the topological graph arising from P.

A. M. Kozae, A. A. El Atik, A. Elrokh and M. Atef, New types of graphs induced by topological spaces, Journal of Intelligent & Fuzzy Systems, vol. 36, no. 6 (2019), pp. 5125-5134; on Research Gate.
Index entries for linear recurrences with constant coefficients, signature (12,-52,96,-64).

Cf. A000225, A001787, A005843, A006516, A036239, A082134, A171476, A171496.

Cf. A082134 (total number of edges of the pseudo-graph P).

Flat(List([0..25], n->n*2^(n-2)*(2^(n-1)-1)))

Magma
```
[n*2^(n-2)*(2^(n-1)-1): n in [0..25]];
```

a:=n->n*2^(n-2)*(2^(n-1)-1): seq(a(n),n=0..25);

Table[n 2^(n - 2)(2^(n - 1) - 1), {n, 0, 31}]

makelist(n*2^(n-2)*(2^(n-1)-1), n, 0, 25);

PARI
```
a(n)=n*2^(n-2)*(2^(n-1)-1);
```

O.g.f.: -2 * x^2 * (-1 + 3*x)/((-1 + 2*x)^2 * (-1 + 4*x)^2).
E.g.f.: (1/2) * exp(2*x) * (-1 + exp(2*x)) * x.
a(n) = 12 * a(n - 1) - 52*a(n - 2) + 96*a(n - 3) - 64*a(n - 4) for n > 3.
a(n) = n * 2^(n - 2) * (2^(n - 1) - 1).
Lim_{n -> infinity} a(n)/a(n - 1) = 4.
a(n) = A082134(n) - A001787(n).
a(n) = A005843(A001787(n)) * A000225(n - 1).
a(n) = n * A006516(n - 1).
a(n) = n * A171476(n - 2).
a(n) = n * A171496(n - 3).

cp's OEIS Frontend

A171495 a(n) = 3*a(n-1)+4 for n > 0; a(0) = 6.

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