A293948 - OEIS

%I #28 Oct 29 2025 03:54:29

%S 1,1,3,5,11,20,41,82,163,334,659,1350,2667,5430,10763,21782,43307,

%T 87254,173867,349270,697003,1397590,2791595,5591382,11174571,22367574,

%U 44716715,89474390,178907819,357905750,715721387,1431639382,2863082155,5726590294,11452754603,22906426710,45811935915,91625837910

%N Arises in a coin-weighing problem.

%C Number of coins that can be processed using n weighings with a "Light-Real" coin with the starting state unknown. - _Robert Price_, Nov 29 2017

%H Robert Price, <a href="/A293948/b293948.txt">Table of n, a(n) for n = 0..201</a>

%H Tanya Khovanova and Konstantin Knop, <a href="https://arxiv.org/abs/1611.09201">Coins that Change Their Weights</a>, arXiv:1611.09201 [math.CO], 2016. See Section 4.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,5,-10,-8,16,4,-8).

%F From _Robert Israel_, Nov 29 2017: (Start)

%F a(n) = (2^(n+1)+1)/3 + (1/2-n/8)*2^(n/2) if n>2 is even.

%F a(n) = (2^(n+1)+2)/3 - 2^((n+1)/2)/4 if n>1 is odd.

%F G.f.: (1-x-4*x^2+4*x^3+4*x^4-5*x^5+6*x^7-2*x^8-4*x^9)/((1-x^2)*(1-2*x)*(1-2*x^2)^2). (End)

%F E.g.f.: (8*cosh(x) + 16*cosh(2*x) + 12*cosh(sqrt(2)*x) + 16*sinh(x) + 16*sinh(2*x) - 3*sqrt(2)*(2 + x)*sinh(sqrt(2)*x) - 6*(2 + 2*x + x^2))/24. - _Stefano Spezia_, Oct 28 2025

%p f:= proc(n) if n::even then (2^(n+1)+1)/3 + (1/2-n/8)*2^(n/2)

%p else (2^(n+1)+2)/3 - 2^((n+1)/2)/4 fi end proc:

%p 1,1,3,seq(f(i),i=3..100); # _Robert Israel_, Nov 29 2017

%t Jacobsthal[n_] := (2^n - (-1)^n)/3;

%t lst = {1, 1, 3, 5};

%t For[k = 2, k <= 100, k++,

%t     AppendTo[lst, (Jacobsthal[2*k + 2] - (k - 2)*2^(k - 1) + 1)/2];

%t     AppendTo[lst, (Jacobsthal[2*k + 3] - 2^k + 1)/2]];

%t lst

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Oct 28 2017

%E Terms a(9) and beyond from _Robert Price_, Nov 29 2017