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A078495 a(n) = (a(n-1) * a(n-6) + a(n-3) * a(n-4)) / a(n-7) (a variant of Somos-7).

%I A078495 #61 Feb 18 2024 08:25:19
%S A078495 1,1,1,1,1,1,1,2,3,4,6,12,24,72,144,288,864,3456,10368,41472,124416,
%T A078495 497664,2985984,17915904,71663616,429981696,2579890176,20639121408,
%U A078495 185752092672,1486016741376,8916100448256,106993205379072
%N A078495 a(n) = (a(n-1) * a(n-6) + a(n-3) * a(n-4)) / a(n-7) (a variant of Somos-7).
%C A078495 From _Vladimir Shevelev_, Apr 16 2016: (Start)
%C A078495 For k >= 0, an infinite sequence {b(k,n)} of Somos's sequences (n >= 0) is:
%C A078495 b(k,0) = b(k,1) = ... = b(k,2*k+2) = 1;
%C A078495 and then for n >= 2*k+3,
%C A078495 b(k,n) = (b(k,n-1)*b(k,n-2*k-2) + b(k,n-k-1)*b(k,n-k-2))/b(k,n-2*k-3).
%C A078495 In particular, {b(0,n)} is essentially A060656, {b(1,n)}=A006721, {a(2,n)}=A078495.
%C A078495 One can prove that the sequence {b(k,n)} has the first 4*(k+1) simple differences: 2k+2 zeros, after that k+1 1's and after that k+1 consecutive doubled triangular numbers (A000217), beginning with 2.
%C A078495 Further we have nontrivial differences. The first of them for k=0,1,2,... are 12, 26, 48, 80, 124, 182, 256, 348, 460, 594, ..., that is, {k^3/3 + 3*k^2 + 32*k/3 + 12}.
%C A078495 (End)
%D A078495 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%H A078495 T. D. Noe, <a href="/A078495/b078495.txt">Table of n, a(n) for n = 0..100</a>
%H A078495 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%F A078495 a(n) = 144 * a(n-6) * a(n-10) / a(n-16), a(n) = a(6-n) for all n in Z.
%t A078495 RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==a[4]==a[5]==a[6]==1,a[n] == (a[n-1]*a[n-6]+a[n-3]*a[n-4])/a[n-7]},a,{n,40}] (* _Harvey P. Dale_, Apr 20 2012 *)
%o A078495 (PARI) {a(n) = if( n<0, a(6-n), if( n<7, 1, (a(n-1) * a(n-6) + a(n-3) * a(n-4)) / a(n-7)))};
%o A078495 (PARI) {a(n) = 2^(b(n-9) + b(n-7)) * 3^b(n-8)}; {b(n) = (n^2 + 10*n + 1 - n%2*13) \ 60 + 1}; /* b(n) = A025795(n) */
%o A078495 (Haskell)
%o A078495 a078495 n = a078495_list !! n
%o A078495 a078495_list = [1, 1, 1, 1, 1, 1, 1] ++
%o A078495   zipWith div (foldr1 (zipWith (+)) (map b [1,3])) a078495_list
%o A078495   where b i = zipWith (*) (drop i a078495_list) (drop (7-i) a078495_list)
%o A078495 -- _Reinhard Zumkeller_, May 05 2013
%o A078495 (Magma) I:=[1,1,1,1,1,1,1]; [n le 7 select I[n] else (Self(n-1)*Self(n-6) + Self(n-3)*Self(n-4))/Self(n-7): n in [1..30]]; // _G. C. Greubel_, Feb 21 2018
%Y A078495 Cf. A000217, A006721, A006723, A025795, A060656, A078495.
%K A078495 nonn,easy,nice
%O A078495 0,8
%A A078495 _Michael Somos_, Nov 26 2002