A027994 a(n) = (F(2n+3) - F(n))/2 where F() = Fibonacci numbers A000045.

1, 2, 6, 16, 43, 114, 301, 792, 2080, 5456, 14301, 37468, 98137, 256998, 672946, 1761984, 4613239, 12078110, 31621701, 82787980, 216743836, 567446112, 1485598681, 3889356696, 10182482353, 26658108074, 69791870526, 182717549872, 478360854115, 1252365133866, 3278734743901, 8583839415648, 22472784017272

Offset

0,2

Comments

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

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

Diagonal sums of triangle in A125171. - Philippe Deléham, Jan 14 2014

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).

Formula

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

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

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

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

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

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

Mathematica

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 *)

Prog

(PARI) a(n)=(fibonacci(2*n+3)-fibonacci(n))/2
(Magma) [(Fibonacci(2*n+3)-Fibonacci(n))/2 : n in [0..40]]; // Vincenzo Librandi, Jan 01 2025

Crossrefs

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

Sequence in context: A217661 A244399 A291142 * A319503 A295572 A372191

Adjacent sequences: A027991 A027992 A027993 * A027995 A027996 A027997

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved