This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A001645 M2626 N1041 #40 Jan 05 2025 19:51:32
%S A001645 1,3,7,11,26,45,85,163,304,578,1090,2057,3888,7339,13862,26179,49437,
%T A001645 93366,176321,332986,628852,1187596,2242800,4235569,7998951,15106172,
%U A001645 28528288,53876211,101746240,192149690,362878313,685302531,1294206745,2444133829
%N A001645 A Fielder sequence.
%D A001645 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001645 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001645 T. D. Noe, <a href="/A001645/b001645.txt">Table of n, a(n) for n = 1..1000</a>
%H A001645 Daniel C. Fielder, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/6-3/fielder.pdf">Special integer sequences controlled by three parameters</a>, Fibonacci Quarterly 6, 1968, 64-70.
%H A001645 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 A001645 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A001645 Wikipedia, <a href="https://en.wikipedia.org/wiki/Companion_matrix">Companion matrix</a>.
%H A001645 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 A001645 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,0,1).
%F A001645 G.f.: x*(1+2*x+3*x^2+5*x^4)/(1-x-x^2-x^3-x^5).
%F A001645 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 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 A001645 LinearRecurrence[{1, 1, 1, 0, 1}, {1, 3, 7, 11, 26}, 50] (* _T. D. Noe_, Aug 09 2012 *)
%t A001645 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 A001645 (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 A001645 (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 A001645 Cf. A001609, A001634 - A001636, A001638 - A001645, A001648, A001649.
%K A001645 nonn,easy
%O A001645 1,2
%A A001645 _N. J. A. Sloane_