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%I M2648 N1055 #60 Sep 08 2022 08:44:29
%S 1,3,7,15,26,51,99,191,367,708,1365,2631,5071,9775,18842,36319,70007,
%T 134943,260111,501380,966441,1862875,3590807,6921503,13341626,
%U 25716811,49570747,95550687,184179871,355018116,684319421,1319068095,2542585503,4900991135
%N Tetranacci numbers A073817 without the leading term 4.
%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 Indranil Ghosh, <a href="/A001648/b001648.txt">Table of n, a(n) for n = 1..3501</a> (terms 1..200 from T. D. Noe)
%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 S. Saito, T. Tanaka, N. Wakabayashi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Saito/saito22.html">Combinatorial Remarks on the Cyclic Sum Formula for Multiple Zeta Values </a>, J. Int. Seq. 14 (2011) # 11.2.4, Table 3.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Lucasn-StepNumber.html">Lucas n-Step Number</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,1).
%F G.f.: x*(1+2*x+3*x^2+4*x^3)/(1-x-x^2-x^3-x^4).
%F a(n) = trace of M^n, where M = the 4 X 4 matrix [ 0 1 0 0 / 0 0 1 0 / 0 0 0 1 / 1 1 1 1]. E.g., the trace (sum of diagonal terms) of M^12 = a(12) = 2631 = (108 + 316 + 717 + 1490). - _Gary W. Adamson_, Feb 22 2004
%F a(n) = n*Sum_{k=ceiling(n/5)..n} Sum_{i=0..(n-k)/4} (-1)^i*binomial(k,k-i)*binomial(n-i*4-1,k-1))/k), n>0. - _Vladimir Kruchinin_, Jan 20 2012
%p A001648:=-(1+2*z+3*z**2+4*z**3)/(-1+z+z**2+z**3+z**4); # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t Rest@ CoefficientList[ Series[(4 - 3 x - 2 x^2 - x^3)/(1 - x - x^2 - x^3 - x^4), {x, 0, 40}], x] (* Or *)
%t a[0] = 4; a[1] = 1; a[2] = 3; a[3] = 7; a[4] = 15; a[n_] := 2*a[n - 1] - a[n - 5]; Array[a, 33] (* _Robert G. Wilson v_ *)
%t LinearRecurrence[{1, 1, 1, 1}, {1, 3, 7, 15}, 60] (* _Vladimir Joseph Stephan Orlovsky_, Feb 08 2012 *)
%o (PARI) a(n)=if(n<0,0,polcoeff(x*(1+2*x+3*x^2+4*x^3)/(1-x-x^2-x^3-x^4)+x*O(x^n),n))
%o (Maxima) a(n):=n*sum(sum((-1)^i*binomial(k,k-i)*binomial(n-i*4-1,k-1),i,0,((n-k)/4))/k,k,ceiling(n/5),n); /* _Vladimir Kruchinin_, Jan 20 2012 */
%o (Magma) I:=[1,3,7,15]; [n le 4 select I[n] else Self(n-1) + Self(n-2) + Self(n-3) + Self(n-4): n in [1..30]]; // _G. C. Greubel_, Dec 18 2017
%Y Cf. A000288, A073817.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
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