A124793 - OEIS

%I #40 Feb 23 2024 06:55:26

%S 1,5,35,157,1123,5021,35939,160669,1150051,5141405,36801635,164524957,

%T 1177652323,5264798621,37684874339,168473555869,1205915978851,

%U 5391153787805,38589311323235,172516921209757,1234857962343523

%N Numbers in a perpendicular plane intersecting a 3D clockwise spiral produced by powers of 2.

%C The general formula for powers of k integer is a(n) = k^((1/4)*(10*n - 7 - (-1)^n)) + k^((1/4)*(10*n - 1 + (-1)^n)) - a(n-1), with a(0)=1 and where k is an integer value. If we replace k with "i" or "-i" where i=sqrt(-1), we get a periodic complex sequence (period 8).

%H Charles R Greathouse IV, <a href="/A124793/b124793.txt">Table of n, a(n) for n = 1..1329</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,32,32).

%F a(n) = 2^((1/4)*(10*n - 7 - (-1)^n)) + 2^((1/4)*(10*n - 1 + (-1)^n)) - a(n-1), with a(0)=1.

%F From _Colin Barker_, Jul 07 2012: (Start)

%F a(n) = -a(n-1) + 32*a(n-2) + 32*a(n-3).

%F G.f.: x*(1+2*x)*(1+4*x)/((1+x)*(1-32*x^2)). (End)

%F a(2n) = 3/31 + 19*32^n/124, a(2n+1) = -3/31 + 136*32^n/124. [_R. J. Mathar_, Jul 10 2012]

%e Write powers of 2 in a sort of 3D clockwise spiral. After the initial 1 (2^0) move right till 2^1=2 (practically only one step); then move down till 2^2=4 (3,4); then left till 2^3=8 (5,6,7,8). When writing number 5 we are in the same column of 1 so 5 is the second number of the sequence. Then move up till 2^4=16. Then move up perpendicularly to the plane till 2^5=32 and again right till 2^6=64. The number 35 is in the sequence because it lies in the same line as 1 and 5. The process continues down, left, up, perpendicular, right and so on.

%p P:=proc(n) local a,i,x,y; a:=1; print(a); for i from 1 by 1 to n do x:=1/4*(10*i-7-(-1)^i); y:=1/4*(10*i-1+(-1)^i); a:=2^x+2^y-a; print(a); od; end: P(100);

%t LinearRecurrence[{-1, 32, 32}, {1, 5, 35}, 25] (* _Paolo Xausa_, Feb 23 2024 *)

%Y Cf. A108981, A001107.

%K easy,nonn

%O 1,2

%A _Paolo P. Lava_ and _Giorgio Balzarotti_, Jun 27 2007