A210098 - OEIS

%I #20 Sep 08 2022 08:46:01

%S 0,1,1,1,1,2,1,-1,-3,-5,-4,-11,-13,-7,23,86,87,199,415,799,-152,-4159,

%T -8063,-17047,-38727,-155366,-142471,445015,2309453,7627979,13609844,

%U 81138437,187790979,142104721,-1743980081,-12357952274,-25547499185,-134098256401

%N Somos-5 sequence variant: a(n) = (a(n-1) * a(n-4) - a(n-2) * a(n-3)) / a(n-5), a(0) = 0, a(1) = a(2) = a(3) = a(4) = 1, a(5) = 2.

%C This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m).

%H Alois P. Heinz, <a href="/A210098/b210098.txt">Table of n, a(n) for n = 0..200</a>

%H K. S. Brown, <a href="http://www.mathpages.com/home/kmath162/kmath162.htm">A Quasi-Periodic Sequence</a>

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%F a(n) = -a(-n), a(n) * a(n-5) = a(n-1) * a(n-4) - a(n-2) * a(n-3) for all n in Z.

%F a(n+4) * a(n-4) = a(n+2) * a(n-2) - a(n) * a(n), a(n+2) * a(n-2) = (2 - (-1)^n) * a(n+1) * a(n-1) - a(n) * a(n) for all n in Z.

%e G.f. = x + x^2 + x^3 + x^4 + 2*x^5 + x^6 - x^7 - 3*x^8 - 5*x^9 - 4*x^10 + ...

%p a:= proc(n) a(n):= `if`(n<6, [0, 1$4, 2][n+1],

%p       (a(n-1)*a(n-4) -a(n-2)*a(n-3)) / a(n-5))

%p     end:

%p seq (a(n), n=0..40);  # _Alois P. Heinz_, Oct 20 2012

%t Join[{0},RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==1,a[5]==2,a[n] == (a[n-1]a[n-4]-a[n-2]a[n-3])/a[n-5]},a,{n,40}]] (* _Harvey P. Dale_, Oct 20 2012 *)

%o (PARI) {a(n) = my(v, m); if( n==0, 0, m = abs(n); sign(n) * if( m<6, 1 + (m>4), v = vector( m, i, 1); v[5] = 2; for( i=6, m, v[i] = (v[i-1] * v[i-4] - v[i-2] * v[i-3]) / v[i-5]); v[m]))};

%o (Magma) I:=[1,1,1,1,2]; [n le 5 select I[n] else (Self(n-1)*Self(n-4) - Self(n-2)*Self(n-3))/Self(n-5): n in [1..50]]; // _G. C. Greubel_, Aug 11 2018

%Y Cf. A006721.

%K sign

%O 0,6

%A _Michael Somos_, Mar 17 2012