A070313 a(n) = 2^n - (2*n+1).

0, -1, -1, 1, 7, 21, 51, 113, 239, 493, 1003, 2025, 4071, 8165, 16355, 32737, 65503, 131037, 262107, 524249, 1048535, 2097109, 4194259, 8388561, 16777167, 33554381, 67108811, 134217673, 268435399, 536870853, 1073741763, 2147483585

Offset: 0

Author

N. J. A. Sloane, May 16 2002

Keywords

• sign
• easy

Comments

Binomial transform of (-1)^n! + !n. - Paul Barry, May 13 2004

This appears as the exponent in Krotov, who writes on p. 2: "in general, two extended Hamming codes can intersect in 2^(2^m - 2m - 1) elements." - Jonathan Vos Post, Jan 13 2013

Primes appear at positions n = 4, 7, 8, 28, 32, 81, 669, 1108, ... (A344781). - R. J. Mathar, Jan 22 2013

a(n) is the total number of dollars lost when using the Martingale method (bet $1, if win then continue to bet $1, if lose then double next bet) for n trials of a wager with exactly one win, n-1 losses. For the case with exactly one loss, n-1 wins, see A165900. - Max Winnick, Jun 28 2022

Links

• Denis Krotov, A partition of the hypercube into cosets of maximally nonparallel Hamming codes, arXiv:1210.0010v1 [cs.IT], Sep 28, 2012.
• D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 19 (1968), 8-16.
• D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 19 (1968), 8-16. [Annotated scanned copy]
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Crossrefs

Second diagonal of A046739.

Cf. A344781.

Programs

Mathematica

lst={};s=-1;Do[s+=s+n;AppendTo[lst, s], {n, 1, 5!, 2}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 18 2008 *)
Table[2^n-(2n+1),{n,0,40}] (* Harvey P. Dale, Feb 13 2024 *)

Maxima

makelist(2^n - (2*n+1),n,0,20); /* Martin Ettl, Jan 25 2013 */

PARI

a(n)=2^n-(2*n+1) \\ Charles R Greathouse IV, Oct 07 2015

Formula

E.g.f.: (exp(x))^2 - exp(x) - 2*x*exp(x). - Paul Barry, May 13 2004

From Colin Barker, Mar 21 2012: (Start)

a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).

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