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
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).
Programs
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GAP
List([1..30], n -> (2^n-1)^2+2); # G. C. Greubel, Feb 10 2019
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Magma
[4^n - 2^(n+1) + 3: n in [1..30]]; // Vincenzo Librandi, Jun 09 2011
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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 *) -
PARI
a(n)=4^n-2*2^n+3 \\ Charles R Greathouse IV, Jun 08 2011
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Sage
[(2^n-1)^2+2 for n in (1..30)] # G. C. Greubel, Feb 10 2019
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
Extensions
Definition changed to closed-form formula and original definition clarified and moved to comment by Eric W. Weisstein, Apr 24 2017
Comments