%I M2626 N1041 #38 Jan 09 2023 09:31:24

%S 1,3,7,11,26,45,85,163,304,578,1090,2057,3888,7339,13862,26179,49437,

%T 93366,176321,332986,628852,1187596,2242800,4235569,7998951,15106172,

%U 28528288,53876211,101746240,192149690,362878313,685302531,1294206745,2444133829

%N A Fielder sequence.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001645/b001645.txt">Table of n, a(n) for n = 1..1000</a>

%H Daniel C. Fielder, <a href="http://www.fq.math.ca/Scanned/6-3/fielder.pdf">Special integer sequences controlled by three parameters</a>, Fibonacci Quarterly 6, 1968, 64-70.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Companion_matrix">Companion matrix</a>.

%H A. V. Zarelua, <a href="https://doi.org/10.1007/s11006-006-0090-y">On Matrix Analogs of Fermat's Little Theorem</a>, Mathematical Notes, vol. 79, no. 6, 2006, pp. 783-796. Translated from Matematicheskie Zametki, vol. 79, no. 6, 2006, pp. 840-855.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,0,1).

%F G.f.: x*(1+2*x+3*x^2+5*x^4)/(1-x-x^2-x^3-x^5).

%F 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

%p 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.]

%t LinearRecurrence[{1, 1, 1, 0, 1}, {1, 3, 7, 11, 26}, 50] (* _T. D. Noe_, Aug 09 2012 *)

%t 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 *)

%o (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))

%o (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

%Y Cf. A001609, A001634 - A001636, A001638 - A001645, A001648, A001649.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_