A001645 A Fielder sequence.
1, 3, 7, 11, 26, 45, 85, 163, 304, 578, 1090, 2057, 3888, 7339, 13862, 26179, 49437, 93366, 176321, 332986, 628852, 1187596, 2242800, 4235569, 7998951, 15106172, 28528288, 53876211, 101746240, 192149690, 362878313, 685302531, 1294206745, 2444133829
Offset: 1
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
- Wikipedia, Companion matrix.
- A. V. Zarelua, On Matrix Analogs of Fermat's Little Theorem, Mathematical Notes, vol. 79, no. 6, 2006, pp. 783-796. Translated from Matematicheskie Zametki, vol. 79, no. 6, 2006, pp. 840-855.
- Index entries for linear recurrences with constant coefficients, signature (1,1,1,0,1).
Programs
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Magma
I:=[1,3,7,11,26]; [n le 5 select I[n] else Self(n-1) + Self(n-2) + Self(n-3) + Self(n-5): n in [1..30]]; // G. C. Greubel, Dec 19 2017
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Maple
A001645:=-(1+2*z+3*z**2+5*z**4)/(-1+z+z**2+z**3+z**5); [Conjectured by Simon Plouffe in his 1992 dissertation.]
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Mathematica
LinearRecurrence[{1, 1, 1, 0, 1}, {1, 3, 7, 11, 26}, 50] (* T. D. Noe, Aug 09 2012 *) CoefficientList[Series[x*(1+2*x+3*x^2+5*x^4)/(1-x-x^2-x^3-x^5), {x, 0, 50}], x] (* G. C. Greubel, Dec 19 2017 *) -
PARI
a(n)=if(n<0,0,polcoeff(x*(1+2*x+3*x^2+5*x^4)/(1-x-x^2-x^3-x^5)+x*O(x^n),n))
Formula
G.f.: x*(1+2*x+3*x^2+5*x^4)/(1-x-x^2-x^3-x^5).
a(n) = trace(M^n), where M = [0, 0, 0, 0, 1; 1, 0, 0, 0, 0; 0, 1, 0, 0, 1; 0, 0, 1, 0, 1; 0, 0, 0, 1, 1] is the 5 x 5 companion matrix to the monic polynomial x^5 - x^4 - x^3 - x^2 - 1. It follows that the sequence satisfies the Gauss congruences: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for positive integers n and r and all primes p. See Zarelua. - Peter Bala, Jan 09 2023