login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A018896
a(n) = ( a(n-1)*a(n-7) + a(n-4)^2 ) / a(n-8); a(0) = ... = a(7) = 1.
8
1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 9, 18, 34, 93, 180, 348, 724, 3033, 9666, 24986, 83761, 261033, 1023728, 3923791, 26128126, 105734485, 381740209, 1895904805, 14058722881, 97964968321, 517832518189, 4364261070929, 25225712161101, 181840424632390
OFFSET
0,9
COMMENTS
From Vladimir Shevelev, Apr 04 2016: (Start)
For k >= 0, an infinite sequence {a(k,n)} of Somos's sequences (n>=0) is:
a(k,0) = a(k,1)= ... = a(k,2*k+1) = 1;
and then for n >= 2*k+2,
a(k,n) = (a(k,n-1)*a(k,n-2*k-1) + a(k,n-k-1)^2)/a(k,n-2*k-2).
In particular, {a(0,n)}=A006125, {a(1,n)}=A006720, {a(2,n)}=A102276, {a(3,n)}=A018896.
One can prove that the sequence {a(k,n)} has the first 4k+2 simple differences: 2k+1 zeros, after that k+1 1's and after that k consecutive squares, beginning with 2^2.
Further we have nontrivial differences. The first of them for k=0,1,2,... are 6, 16, 33, 59, 96, 146, 211, 293, 394, 516, ... that is, {k^3/3 + 5*k^2/2 + 43*k/6 + 6}.
(End)
LINKS
Mohamed Bensaid, Sato tau functions and construction of Somos sequence, arXiv:2409.05911 [math.NT], 2024. See p. 7.
David Gale, Mathematical Entertainments, Mathematical Intelligencer, volume 18, number 3, Summer 1996, page 25.
Eric Weisstein's World of Mathematics, Somos Sequence
MAPLE
f:= proc(n) option remember;
if n <= 7 then 1 else
(procname(n-1)*procname(n-7)+procname(n-4)^2)/procname(n-8)
fi
end proc:
seq(f(n), n=0..50); # Robert Israel, Apr 04 2016
MATHEMATICA
RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==a[5]==a[6]==a[7]==a[8]==1, a[n]==(a[n-1]a[n-7]+ a[n-4]^2)/a[n-8]}, a[n], {n, 50}] (* Harvey P. Dale, May 02 2011 *)
k = 3; Set[#, 1] & /@ Map[a[k, #] &, Range[0, 2 k + 1]]; a[k_, n_] /; n >= 2 k + 2 := (a[k, n - 1] a[k, n - 2 k - 1] + a[k, n - k - 1]^2)/ a[k, n - 2 k - 2]; Table[a[k, n], {n, 0, 35}] (* Michael De Vlieger, Apr 04 2016 *)
PROG
(Haskell)
a018896 n = a018896_list !! n
a018896_list = replicate 8 1 ++ f 8 where
f x = ((a018896 (x - 1) * a018896 (x - 7) + a018896 (x - 4) ^ 2)
`div` a018896 (x - 8)) : f (x + 1)
-- Reinhard Zumkeller, Oct 01 2012
(Magma) [n le 8 select 1 else (Self(n-1)*Self(n-7)+Self(n-4)^2 ) / Self(n-8): n in [1..40]]; // Vincenzo Librandi, Dec 08 2016
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Harvey P. Dale, May 02 2011
STATUS
approved