A191341 a(n) = 4^n - 2*2^n + 3.

3, 11, 51, 227, 963, 3971, 16131, 65027, 261123, 1046531, 4190211, 16769027, 67092483, 268402691, 1073676291, 4294836227, 17179607043, 68718952451, 274876858371, 1099509530627, 4398042316803, 17592177655811, 70368727400451, 281474943156227, 1125899839733763

Offset

1,1

Comments

Also the number of dominating sets for the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Apr 24 2017

For n > 1, a(n) is the largest integer such that the binary representations of a(n)-1 and a(n)+1 both contain exactly n 0's and n 1's.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..500 (corrected by Ray Chandler, Jan 19 2019)

Eric Weisstein's World of Mathematics, Dominating Set

Eric Weisstein's World of Mathematics, Complete Bipartite Graph

Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Formula

a(n) = 4^n - 2^(n+1) + 3. - Nathaniel Johnston, May 30 2011

a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3). - Eric W. Weisstein, Jun 29 2017

G.f.: (3 -10*x +16*x^2)/((1-x)*(1-2*x)*(1-4*x)). - R. J. Mathar, Jun 02 2011 ( corrected by G. C. Greubel, Feb 10 2019 )

E.g.f.: -2 + 3*exp(x) - 2*exp(2*x) + exp(4*x). - G. C. Greubel, Feb 10 2019

Mathematica

Table[3 - 2^(1+n) + 4^n, {n, 20}] (* Eric W. Weisstein, Apr 24 2017 *)
With[{r = Range[10^5]}, 2 + SplitBy[Cases[Transpose[{Partition[Tally[#][[All, 2]] & /@ IntegerDigits[r, 2], 2, 1, 1], r}], {{{n_, n_}, {n_, n_}}, p_} :> {n, p}], First][[All, -1, -1]]] (* Eric W. Weisstein, Apr 24 2017 *)
LinearRecurrence[{7, -14, 8}, {3, 11, 51}, 20] (* Eric W. Weisstein, Jun 29 2017 *)
CoefficientList[Series[(3-10x+16x^2)/((1-x)(1-2x)(1-4x)), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 29 2017 *)

Prog

(PARI) a(n)=4^n-2*2^n+3 \\ Charles R Greathouse IV, Jun 08 2011
(Magma) [4^n - 2^(n+1) + 3: n in [1..30]]; // Vincenzo Librandi, Jun 09 2011
(Sage) [(2^n-1)^2+2 for n in (1..30)] # G. C. Greubel, Feb 10 2019
(GAP) List([1..30], n -> (2^n-1)^2+2); # G. C. Greubel, Feb 10 2019

Crossrefs

Cf. A031443 (digitally balanced numbers), A191292, A191296.

Sequence in context: A145144 A367560 A284702 * A359146 A199212 A113283

Adjacent sequences: A191338 A191339 A191340 * A191342 A191343 A191344

Keyword

nonn,easy

Author

Juri-Stepan Gerasimov, May 30 2011

Extensions

Definition changed to closed-form formula and original definition clarified and moved to comment by Eric W. Weisstein, Apr 24 2017

Status

approved