A062510 a(n) = 2^n + (-1)^(n+1).

0, 3, 3, 9, 15, 33, 63, 129, 255, 513, 1023, 2049, 4095, 8193, 16383, 32769, 65535, 131073, 262143, 524289, 1048575, 2097153, 4194303, 8388609, 16777215, 33554433, 67108863, 134217729, 268435455, 536870913, 1073741823, 2147483649

Offset: 0

Author

Jason Earls, Jun 24 2001

Keywords

• easy
• nonn

Comments

The identity 2 = 2^2/3 + 2^3/(3*3) - 2^4/(3*3*9) - 2^5/(3*3*9*15) + + - - can be viewed as a generalized Engel-type expansion of the number 2 to the base 2. Compare with A014551. - Peter Bala, Nov 13 2013

References

• D. M. Burton, Elementary Number Theory, Allyn and Bacon, Inc. Boston, MA, 1976, p. 29.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• G. Everest, Y. Puri and T. Ward, Integer sequences counting periodic points, arXiv:math/0204173 [math.NT], 2002.
• Index entries for linear recurrences with constant coefficients, signature (1,2).

Crossrefs

Cf. A102345, A105723.

Programs

Magma

[2^n + (-1)^(n+1): n in [0..40]]; // Vincenzo Librandi, Aug 14 2011

Mathematica

LinearRecurrence[{1,2},{0,3}, 30] (* or *) Table[2^n - (-1)^n, {n,0,30}] (* G. C. Greubel, Jan 15 2018 *)

PARI

for(n=0,22,print(2^n+(-1)^(n+1)))

Formula

a(n) = 3*A001045(n). - Paul Curtz, Jan 17 2008

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

G.f.: Q(0) where Q(k)= 1 - 1/(4^k - 2*x*16^k/(2*x*4^k - 1/(1 + 1/(2*4^k - 8*x*16^k/(4*x*4^k + 1/Q(k+1) ))))); (continued fraction). - Sergei N. Gladkovskii, Apr 13 2013

E.g.f.: (exp(3*x) - 1)*exp(-x). - Ilya Gutkovskiy, Nov 20 2016

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jul 06 2001