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A001645
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A Fielder sequence.
(Formerly M2626 N1041)
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4
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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
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OFFSET
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1,2
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REFERENCES
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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).
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LINKS
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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.
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FORMULA
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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
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MAPLE
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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
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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 *)
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PROG
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(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))
(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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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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