A206282 - OEIS

%I #30 Mar 15 2024 12:49:08

%S 1,1,-1,-4,-5,1,9,11,-4,-25,-31,9,64,79,-25,-169,-209,64,441,545,-169,

%T -1156,-1429,441,3025,3739,-1156,-7921,-9791,3025,20736,25631,-7921,

%U -54289,-67105,20736,142129,175681,-54289,-372100,-459941,142129,974169,1204139

%N a(n) = ( a(n-1) * a(n-3) + a(n-2) ) / a(n-4), a(1) = a(2) = 1, a(3) = -1, a(4) = -4.

%C This satisfies the same recurrence as Dana Scott's sequence A048736.

%H Reinhard Zumkeller, <a href="/A206282/b206282.txt">Table of n, a(n) for n = 1..5000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,-2,0,0,2,0,0,1).

%F G.f.: x * (1 + x - x^2 - 2*x^3 - 3*x^4 - x^5 - x^6 - x^7) / (1 + 2*x^3 - 2*x^6 - x^9).

%F a(n) = a(-5 - n) = a(n+2) * a(n-2) - a(n+1) * a(n-1) for all n in Z.

%F a(3*n) = (-1)^n * F(n)^2, a(3*n + 1) = (-1)^n * F(n + 2)^2 where F = Fibonacci A000045.

%F a(6*n - 4) = - A110034(2*n), a(6*n - 1) = - A110035(2*n), a(3*n + 2) = (-1)^n * A126116(2*n + 3).

%e G.f. = x + x^2 - x^3 - 4*x^4 - 5*x^5 + x^6 + 9*x^7 + 11*x^8 - 4*x^9 - 25*x^10 + ...

%t CoefficientList[Series[x*(1+x)*(1-x^2)*(1+x^3)/(1-2*x^2-2*x^4-2*x^6+x^8 ), {x,0,50}], x] (* or *) RecurrenceTable[{a[n] == ( a[n-1]*a[n-3] + a[n-2] )/a[n-4], a[1] == a[2] == 1, a[3] == -1, a[4] == -4}, a, {n,1,50}] (* _G. C. Greubel_, Aug 12 2018 *)

%o (PARI) {a(n) = my(k = n\3); (-1)^k * if( n%3 == 0, fibonacci( k )^2, n%3 == 1, fibonacci( k+2 )^2, fibonacci( k ) * fibonacci( k+3 ) + fibonacci( k+1 ) * fibonacci( k+2 ))};

%o (PARI) x='x+O('x^30); Vec(x*(1+x)*(1-x^2)*(1+x^3)/(1-2*x^2-2*x^4 -2*x^6 +x^8 )) \\ _G. C. Greubel_, Aug 12 2018

%o (Haskell)

%o a206282 n = a206282_list !! (n-1)

%o a206282_list = 1 : 1 : -1 : -4 :

%o    zipWith div

%o      (zipWith (+)

%o        (zipWith (*) (drop 3 a206282_list)

%o                     (drop 1 a206282_list))

%o        (drop 2 a206282_list))

%o      a206282_list

%o -- Same program as in A048736, see comment.

%o -- _Reinhard Zumkeller_, Feb 08 2012

%o (Magma) I:=[1,1,-1,-4]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + Self(n-2))/Self(n-4): n in [1..30]]; // _G. C. Greubel_, Aug 12 2018

%Y Cf. A000045, A048736, A110034, A110035, A126116.

%K sign,easy

%O 1,4

%A _Michael Somos_, Feb 05 2012