A052968 a(n) = 1 + 2^(n-1) + n for n > 0, a(0) = 2.

2, 3, 5, 8, 13, 22, 39, 72, 137, 266, 523, 1036, 2061, 4110, 8207, 16400, 32785, 65554, 131091, 262164, 524309, 1048598, 2097175, 4194328, 8388633, 16777242, 33554459, 67108892, 134217757, 268435486, 536870943, 1073741856, 2147483681

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

O. M. Cain proves that the number of consecutive zeros in the decimal expansion of 5^(n+2^n+2), that is 5^a(n+1), is nondecreasing and grows to infinity as m gets bigger. See link. - Michel Marcus, Nov 07 2019

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• O. M. Cain, The Exceptional Selfcondensability of Powers of Five, arXiv:1910.13829 [math.HO], 2019. See Theorem 10.4. p. 13.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1040
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Programs

Magma

I:=[2, 3, 5, 8]; [n le 4 select I[n] else 4*Self(n-1)-5*Self(n-2)+2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 22 2012

Maple

spec := [S,{S=Union(Sequence(Prod(Sequence(Z),Z)),Prod(Sequence(Z),Sequence(Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);

Mathematica

CoefficientList[Series[(-2+5*x-3*x^2+x^3)/(-1+2*x)/(-1+x)^2,{x,0,40}],x] (* Vincenzo Librandi, Jun 22 2012 *)
Join[{2},Table[1+2^(n-1)+n,{n,40}]] (* or *) LinearRecurrence[{4,-5,2},{2,3,5,8},40] (* Harvey P. Dale, Feb 10 2018 *)

Formula

G.f.: (-2 + 5*x - 3*x^2 + x^3)/(-1+2*x)/(-1+x)^2

Recurrence: {a(3)=8, a(2)=5, a(1)=3, a(0)=2, 2*a(n)-a(n+1)-n=0}.

Extensions

More terms from James Sellers, Jun 05 2000