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 A105073 Define a(1)=0, a(2)=2 then a(n) = 3*a(n-1) - a(n-2), a(n+1) = 3*a(n)-a(n-1) and a(n+2) = 3*a(n+1) - a(n) + 2. 1
 0, 2, 6, 16, 44, 116, 304, 798, 2090, 5472, 14328, 37512, 98208, 257114, 673134, 1762288, 4613732, 12078908, 31622992, 82790070, 216747218, 567451584, 1485607536, 3889371024, 10182505536, 26658145586, 69791931222, 182717648080, 478361013020, 1252365390980 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS From Jon E. Schoenfield, Jan 18 2019: (Start) Previously, the Name had included the comment, "This sequence is such that 20*(a(n)^2) + 20*a(n) + 1 = j^2 = a square." However, Anthony Hernandez observed that this statement is not true for all terms; e.g., at a(4)=16, 20*16^2 + 20*16 + 1 = 5441, a nonsquare. It is true that 20*a(n)^2 + 20*a(n) + 1 = A305315(n/3)^2 when n == 0 (mod 3) and A305316((n-2)/3)^2 when n == 2 (mod 3); however, for n == 1 (mod 3) with n > 1, sqrt(20*a(n)^2 + 20*a(n) + 1) is a noninteger number whose fractional part apparently approaches 3 - sqrt(5) as n increases, and Andrey Zabolotskiy observes that round(sqrt(20*a(n)^2 + 20*a(n) + 1) + sqrt(5)) appears to be equal to A002878(n). (End) LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (3,-1,1,-3,1). FORMULA a(n) = (1/6)*(Fibonacci(2n+4) - 2*Fibonacci(2n) - 2*cos((n+2)(2*Pi/3)) - 4). - Ralf Stephan, May 20 2007 From R. J. Mathar, Nov 13 2009: (Start) a(n) = 3*a(n-1) - a(n-2) + a(n-3) - 3*a(n-4) + a(n-5). G.f.: 2*x^2/((1-x) * (1+x+x^2) * (1-3*x+x^2)). a(n) = A061347(n+2)/6 + A001519(n+2)/2 - 2/3. (End) a(n) = floor(A027941(n)/2). - Anthony Hernandez, Jan 03 2019 MATHEMATICA a[n_]:=(1/6)*(Fibonacci[2*n+4] - 2*Fibonacci[2*n] - 2*Cos[(n+2)*(2*Pi/3)] - 4 ); Array[a, 50] (* Stefano Spezia, Jan 11 2019 *) RecurrenceTable[{a[1]==0, a[2]==2, a[3]==6, a[4]==16, a[5]==44, a[n]== 3 a[n-1] - a[n-2] + a[n-3] - 3 a[n-4] + a[n-5]}, a, {n, 35}] (* Vincenzo Librandi, Jan 13 2019 *) PROG (MAGMA) I:=[0, 2, 6, 16, 44]; [n le 5 select I[n] else 3*Self(n-1) - Self(n-2) + Self(n-3) - 3*Self(n-4) + Self(n-5): n in [1..35]]; // Vincenzo Librandi, Jan 13 2019 CROSSREFS Cf. A001519, A061347, A027941. Cf. A305315, A305316. Sequence in context: A003142 A335686 A118041 * A002605 A026134 A105696 Adjacent sequences:  A105070 A105071 A105072 * A105074 A105075 A105076 KEYWORD nonn,easy AUTHOR Pierre CAMI, Apr 06 2005 EXTENSIONS Extended by R. J. Mathar, Nov 13 2009 STATUS approved

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Last modified April 11 22:01 EDT 2021. Contains 342888 sequences. (Running on oeis4.)