A027994 - OEIS

%I #41 Jan 01 2025 23:31:03

%S 1,2,6,16,43,114,301,792,2080,5456,14301,37468,98137,256998,672946,

%T 1761984,4613239,12078110,31621701,82787980,216743836,567446112,

%U 1485598681,3889356696,10182482353,26658108074,69791870526,182717549872,478360854115,1252365133866,3278734743901,8583839415648,22472784017272

%N a(n) = (F(2n+3) - F(n))/2 where F() = Fibonacci numbers A000045.

%C Substituting x*(1-x)/(1-2x) into x^2/(1-x^2) yields x^2*(g.f. of sequence).

%C The number of (s(0), s(1), ..., s(n+1)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n+1, s(0) = 2, s(n+1) = 3. - _Herbert Kociemba_, Jun 02 2004

%C Diagonal sums of triangle in A125171. - _Philippe Deléham_, Jan 14 2014

%H Harvey P. Dale, <a href="/A027994/b027994.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,-2,1).

%F G.f.: (1-x)^2/((1-x-x^2)*(1-3*x+x^2)). - _Floor van Lamoen_ and _N. J. A. Sloane_, Jan 21 2001

%F a(n) = Sum_{k=0..n} T(n, k)*T(n, n+k), T given by A027926.

%F a(n) = 2*a(n-1) + Sum_{m < n-1} a(m) + F(n-1) = A059512(n+2) - F(n) where F(n) is the n-th Fibonacci number (A000045). - _Floor van Lamoen_, Jan 21 2001

%F a(n) = (2/5)*Sum_{k=1..4} sin(2*Pi*k/5)*sin(3*Pi*k/5)*(1+2*cos(Pi*k/5))^(n+1). - _Herbert Kociemba_, Jun 02 2004

%F a(-1-2n) = A056014(2n), a(-2n) = A005207(2n-1).

%F E.g.f.: exp(3*x/2)*cosh(sqrt(5)*x/2) + exp(x/2)*(2*exp(x) - 1)*sinh(sqrt(5)*x/2)/sqrt(5). - _Stefano Spezia_, Jan 01 2025

%t Table[(Fibonacci[2n+3]-Fibonacci[n])/2,{n,0,30}] (* or *) LinearRecurrence[{4,-3,-2,1},{1,2,6,16},30] (* _Harvey P. Dale_, Apr 28 2022 *)

%o (PARI) a(n)=(fibonacci(2*n+3)-fibonacci(n))/2

%o (Magma) [(Fibonacci(2*n+3)-Fibonacci(n))/2 : n in [0..40]]; // _Vincenzo Librandi_, Jan 01 2025

%Y Cf. A000667, A059216, A059219, A059502, A027926.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_