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 A007574 Patterns in a dual ring. (Formerly M2653) 1
 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 (list; graph; refs; listen; history; text; internal format)
 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 STATUS approved

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Last modified December 14 19:27 EST 2019. Contains 329987 sequences. (Running on oeis4.)