A018896 a(n) = ( a(n-1)*a(n-7) + a(n-4)^2 ) / a(n-8); a(0) = ... = a(7) = 1.

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

Author

Michael Somos

Keywords

• nonn
• nice

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

• T. D. Noe, Table of n, a(n) for n = 0..100
• 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
• Index entries for two-way infinite sequences

Crossrefs

Cf. A006125, A006720, A102276.

Programs

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

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 *)

Extensions

More terms from Harvey P. Dale, May 02 2011