A007574 Patterns in a dual ring. (Formerly M2653)

1, 3, 7, 15, 31, 60, 113, 207, 373, 663, 1167, 2038, 3537, 6107, 10499, 17983, 30703, 52272, 88769, 150407, 254321, 429223, 723167, 1216490, 2043361, 3427635, 5742463, 9609327, 16062463, 26821668, 44744657, 74576703, 124192237, 206650167, 343594479

Offset

1,2

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

C. A. Church, Jr., Lattice paths and Fibonacci and Lucas numbers, Fibonacci Quarterly 12(4) (1974) 336-338.

W. Dotson, F. Norwood and C. Taylor, Fiber optics and Fibonacci, Math. Mag., 66 (1993), 167-174.

Index entries for linear recurrences with constant coefficients, signature (4, -4, -2, 4, 0, -1).

Formula

a(n) = 4*a(n-1) -4*a(n-2) -2*a(n-3) +4*a(n-4) -a(n-6). G.f.: -x*(-1+x+x^2-x^3-x^4+2*x^5)/ ((x-1)^2 * (x^2+x-1)^2). [R. J. Mathar, Feb 06 2010]

Maple

with(combinat): A007574 := proc(n) local k; if n=1 then RETURN(1) fi; if n=2 then RETURN(3) fi; if n=3 then RETURN(7) fi; if n>3 then RETURN( fibonacci(n)+2*fibonacci(n-1)+n*sum(fibonacci(n-k), k=2..n-1)) fi; end;

Mathematica

Table[ Fibonacci[n] + 2 Fibonacci[n - 1] + n*Sum[Fibonacci[n - k], {k, 2, n - 1}], {n, 1, 35} ]
LinearRecurrence[{4, -4, -2, 4, 0, -1}, {1, 3, 7, 15, 31, 60}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)
CoefficientList[Series[-(- 1 + x + x^2 - x^3 - x^4 + 2 x^5) / ((x - 1)^2 (x^2 + x - 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)

Prog

(Magma) I:=[1, 3, 7, 15, 31, 60]; [n le 6 select I[n] else 4*Self(n-1)-4*Self(n-2)-2*Self(n-3)+4*Self(n-4)-Self(n-6): n in [1..50]]; // Vincenzo Librandi, Jun 09 2013

Crossrefs

Cf. A000045.

Sequence in context: A023424 A276647 A006778 * A034480 A218281 A057703

Adjacent sequences: A007571 A007572 A007573 * A007575 A007576 A007577

Keyword

nonn,easy

Author

Simon Plouffe and N. J. A. Sloane, Robert G. Wilson v

Status

approved