A001636 A Fielder sequence: a(n) = a(n-1) + a(n-2) - a(n-7), n >= 8.
0, 2, 3, 6, 10, 17, 21, 38, 57, 92, 143, 225, 351, 555, 868, 1366, 2142, 3365, 5282, 8296, 13023, 20451, 32108, 50417, 79160, 124295, 195159, 306431, 481139, 755462, 1186184, 1862486, 2924375, 4591702, 7209646, 11320209, 17774393, 27908418, 43820325
Offset: 1
Keywords
References
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- T. D. Noe, Table of n, a(n) for n = 1..1000
- Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
- Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, 0, 0, -1).
Crossrefs
Cf. A013983.
Programs
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Magma
I:=[0, 2, 3, 6, 10, 17, 21]; [n le 7 select I[n] else Self(n-1) + Self(n-2) - Self(n-7): n in [1..30]]; // G. C. Greubel, Jan 09 2018
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Maple
A001636:=-z*(2+3*z+4*z**2+5*z**3+6*z**4)/(z+1)/(z**5+z**3+z-1); # Simon Plouffe in his 1992 dissertation a:= n -> (Matrix([[6,-1$4,4,5]]). Matrix(7, (i,j)-> if (i=j-1) then 1 elif j=1 then [1$2,0$4,-1][i] else 0 fi)^n)[1,1]: seq(a(n), n=1..38); # Alois P. Heinz, Aug 01 2008
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Mathematica
LinearRecurrence[{1, 1, 0, 0, 0, 0, -1}, {0, 2, 3, 6, 10, 17, 21}, 50] (* T. D. Noe, Aug 09 2012 *) -
PARI
a(n)=if(n<0,0,polcoeff(x^2*(2+x+x^2+x^3+x^4-6*x^5)/(1-x-x^2+x^7)+x*O(x^n),n))
Formula
G.f.: x^2*(2+x+x^2+x^3+x^4-6*x^5)/(1-x-x^2+x^7).
a(n) = a(n-2) + a(n-3) + a(n-4) + a(n-5) + a(n-6), n >= 7.
Extensions
Edited by Michael Somos, Feb 17 2002