A276529 - OEIS

%I #35 Oct 17 2021 13:30:34

%S 1,1,1,1,1,1,2,3,4,5,6,13,20,27,34,41,89,137,185,233,281,610,939,1268,

%T 1597,1926,4181,6436,8691,10946,13201,28657,44113,59569,75025,90481,

%U 196418,302355,408292,514229,620166,1346269,2072372,2798475,3524578,4250681,9227465

%N a(n) = (a(n-1) * a(n-5) + 1) / a(n-6), a(0) = a(1) = ... = a(5) = 1.

%C Thanks to the linear recurrence signature, we see that this is actually five separate linear recurrence sequences, each with signature (7,-1), interwoven together. - _Greg Dresden_, Oct 16 2021

%H Seiichi Manyama, <a href="/A276529/b276529.txt">Table of n, a(n) for n = 0..5985</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,7,0,0,0,0,-1).

%F a(n) + a(n+10) = 7*a(n+5).

%F a(5-n) = a(n).

%F G.f.: (1 +x +x^2 +x^3 +x^4 -6*x^5 -5*x^6 -4*x^7 -3*x^8 -2*x^9) / (1 -7*x^5 +x^10). - _Colin Barker_, Nov 16 2016

%F From _Greg Dresden_, Oct 16 2021: (Start)

%F a(5*n) = L(4*n-2)/3 = A049685(n-1),

%F a(5*n+1) = F(4*n-1) = A033891(n-1),

%F a(5*n+2) = L(4*n+2)/3 - F(4*n),

%F a(5*n+3) = L(4*n-2)/3 + F(4*n),

%F a(5*n+4) = F(4*n+1) = A033889(n). (End)

%t LinearRecurrence[{0,0,0,0,7,0,0,0,0,-1}, {1,1,1,1,1,1,2,3,4,5}, 50] (* _G. C. Greubel_, Nov 18 2016 *)

%t RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==a[4]==a[5]==1,a[n]==(a[n-1]a[n-5]+ 1)/a[n-6]},a,{n,50}] (* _Harvey P. Dale_, Oct 08 2020 *)

%t Flatten[Table[{LucasL[4 n - 2]/3, Fibonacci[4 n - 1], LucasL[4 n + 2]/3 - Fibonacci[4 n], LucasL[4 n - 2]/3 + Fibonacci[4 n], Fibonacci[4 n + 1]}, {n, 0, 10}]] (* _Greg Dresden_, Oct 16 2021 *)

%o (Ruby)

%o def A(k, m, n)

%o   a = Array.new(2 * k, 1)

%o   ary = [1]

%o   while ary.size < n + 1

%o     i = a[-1] * a[1] + a[k] ** m

%o     break if i % a[0] > 0

%o     a = *a[1..-1], i / a[0]

%o     ary << a[0]

%o   end

%o   ary

%o end

%o def A276529(n)

%o   A(3, 0, n)

%o end

%o (PARI) Vec((1 +x +x^2 +x^3 +x^4 -6*x^5 -5*x^6 -4*x^7 -3*x^8 -2*x^9)/(1 -7*x^5 +x^10) + O(x^50)) \\ _Colin Barker_, Nov 16 2016

%Y Cf. A275173, A102276, A276530.

%Y 5th-sections: A049685, A033891, A033889.

%K nonn,easy

%O 0,7

%A _Seiichi Manyama_, Nov 16 2016