A001642 A Fielder sequence.
1, 3, 4, 11, 21, 36, 64, 115, 211, 383, 694, 1256, 2276, 4126, 7479, 13555, 24566, 44523, 80694, 146251, 265066, 480406, 870689, 1578040, 2860046, 5183558, 9394699, 17026986, 30859771, 55930361, 101368389, 183720435, 332975581, 603486148, 1093760479
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
- Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
- Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 1, 1).
Programs
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Magma
I:=[1,3,4,11,21]; [n le 5 select I[n] else Self(n-1) + Self(n-2) + Self(n-4) + Self(n-5): n in [1..30]]; // G. C. Greubel, Jan 09 2018
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Maple
A001642:=-(z+1)*(5*z**3-z**2+z+1)/(-1+z+z**2+z**4+z**5); # conjectured by Simon Plouffe in his 1992 dissertation
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Mathematica
LinearRecurrence[{1, 1, 0, 1, 1}, {1, 3, 4, 11, 21}, 50] (* T. D. Noe, Aug 09 2012 *) Rest[CoefficientList[Series[x (1+2x+4x^3+5x^4)/(1-x-x^2-x^4-x^5),{x,0,40}],x]] (* Harvey P. Dale, Feb 06 2025 *) -
PARI
a(n)=if(n<0,0,polcoeff(x*(1+2*x+4*x^3+5*x^4)/(1-x-x^2-x^4-x^5)+x*O(x^n),n))
Formula
G.f.: x(1+2x+4x^3+5x^4)/(1-x-x^2-x^4-x^5).