A003522 - OEIS

%I M1391 #44 Oct 19 2022 08:12:53

%S 1,1,1,1,2,5,11,21,37,64,113,205,377,693,1266,2301,4175,7581,13785,

%T 25088,45665,83097,151169,274969,500162,909845,1655187,3011157,

%U 5477917,9965312,18128529,32978725,59993817,109139117,198543154

%N a(n) = Sum_{k=0..n} C(n-k,3k).

%D A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 113.

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

%H Seiichi Manyama, <a href="/A003522/b003522.txt">Table of n, a(n) for n = 0..3850</a>

%H V. C. Harris, C. C. Styles, <a href="http://www.fq.math.ca/Scanned/2-4/harris.pdf">A generalization of Fibonacci numbers</a>, Fib. Quart. 2 (1964) 277-289, sequence u(n,1,3).

%H V. E. Hoggatt, Jr., <a href="/A005676/a005676.pdf">7-page typed letter to N. J. A. Sloane with suggestions for new sequences</a>, circa 1977.

%H Vedran Krcadinac, <a href="http://www.fq.math.ca/Papers1/44-4/quartkrcadinac04_2006.pdf">A new generalization of the golden ratio</a>, Fibonacci Quart. 44 (2006), no. 4, 335-340.

%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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3, -3, 1, 1).

%F G.f. : (1-x)^2/(1-3x+3x^2-x^3-x^4); a(n)=3a(n-1)-3a(n-2)+a(n-3)+a(n-4). - _Paul Barry_, Jul 07 2004

%p A003522:=-(z-1)**2/(-1+3*z-3*z**2+z**4+z**3); # conjectured by _Simon Plouffe_ in his 1992 dissertation

%t LinearRecurrence[{3, -3, 1, 1},{1, 1, 1, 1},35] (* _Ray Chandler_, Sep 23 2015 *)

%o (PARI) a(n)=if(n<0, 0, polcoeff((1-x)^2/(1-3*x+3*x^2-x^3-x^4)+x*O(x^n), n)) /* _Michael Somos_, Sep 20 2005 */

%K nonn,easy

%O 0,5

%A _N. J. A. Sloane_