The On-Line Encyclopedia of Integer Sequences (OEIS)

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A210098

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.
(history; published version)

#20 by Charles R Greathouse IV at Thu Sep 08 08:46:01 EDT 2022

PROG

(MAGMAMagma) 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

Discussion

Thu Sep 08

08:46

OEIS Server: https://oeis.org/edit/global/2944

#19 by Michael Somos at Sat Aug 11 18:13:42 EDT 2018

STATUS

editing

approved

#18 by Michael Somos at Sat Aug 11 18:13:21 EDT 2018

FORMULA

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.

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.

PROG

(PARI) {a(n) = localmy(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]))};

STATUS

proposed

editing

Discussion

Sat Aug 11

18:13

Michael Somos: Light edits.

#17 by G. C. Greubel at Sat Aug 11 14:41:36 EDT 2018

STATUS

editing

proposed

#16 by G. C. Greubel at Sat Aug 11 14:41:32 EDT 2018

EXAMPLE

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 + ...

PROG

(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

STATUS

approved

editing

#15 by Charles R Greathouse IV at Fri Nov 13 16:13:53 EST 2015

STATUS

editing

approved

#14 by Charles R Greathouse IV at Fri Nov 13 16:13:45 EST 2015

LINKS

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

PROG

(PARI) {a(n) = local(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]))}

STATUS

approved

editing

#13 by Alois P. Heinz at Sat Oct 20 17:04:30 EDT 2012

STATUS

editing

approved

#12 by Alois P. Heinz at Sat Oct 20 17:04:19 EDT 2012

LINKS

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

#11 by Alois P. Heinz at Sat Oct 20 17:03:57 EDT 2012

LINKS

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

MAPLE

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

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

end:

seq (a(n), n=0..40); # Alois P. Heinz, Oct 20 2012