|
%I #57 Sep 12 2023 08:30:02
%S 1,4,9,19,39,79,159,319,639,1279,2559,5119,10239,20479,40959,81919,
%T 163839,327679,655359,1310719,2621439,5242879,10485759,20971519,
%U 41943039,83886079,167772159,335544319,671088639,1342177279,2684354559
%N a(n) = 5*2^(n-1) - 1, n>0, with a(0)=1.
%C A153894 is a better version of this sequence. - _N. J. A. Sloane_, Feb 07 2009
%C Equals binomial transform of [1, 3, 2, 3, 2, 3, 2, ...]. - _Gary W. Adamson_, May 11 2008
%H G. C. Greubel, <a href="/A052549/b052549.txt">Table of n, a(n) for n = 0..1000</a>
%H David Eppstein, <a href="https://arxiv.org/abs/1804.07396">Making Change in 2048</a>, arXiv:1804.07396 [cs.DM], 2018.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=486">Encyclopedia of Combinatorial Structures 486</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F G.f.: (1 + x - x^2)/((1-2*x)*(1-x)).
%F a(n) = 2*a(n-1) + 1, for n>1, with a(0)=1 and a(1)=4.
%F E.g.f.: (5*exp(2*x) - 2*exp(x) - 1)/2. - _G. C. Greubel_, May 07 2019
%F a(n) = 1 + A000225(n-1) + A000225(n+1) for n > 0. - _Gennady Eremin_, Sep 08 2023
%p spec := [S,{S=Prod(Sequence(Union(Z,Z)),Union(Z,Sequence(Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t {1}~Join~Array[5*2^(# -1)-1 &,30] (* _Michael De Vlieger_, Jul 18 2018 *)
%t LinearRecurrence[{3,-2}, {1,4,9}, 30] (* _G. C. Greubel_, May 07 2019 *)
%o (PARI) vector(30, n, n--; if(n==0, 1, 5*2^(n-1) -1)) \\ _G. C. Greubel_, May 07 2019
%o (Magma) [n eq 0 select 1 else 5*2^(n-1) -1: n in [0..30]]; // _G. C. Greubel_, May 07 2019
%o (Sage) [1]+[5*2^(n-1) -1 for n in (1..30)] # _G. C. Greubel_, May 07 2019
%o (GAP) Concatenation([1], List([1..30], n-> 5*2^(n-1) -1)) # _G. C. Greubel_, May 07 2019
%o (Python)
%o a052549 = [1] + [(5<<(n-1))-1 for n in range(1, 30)]
%o print(a052549) # _Gennady Eremin_, Sep 10 2023
%Y Cf. A000225, A153894.
%K easy,nonn
%O 0,2
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E More terms from _James A. Sellers_, Jun 06 2000
|
