A003522 a(n) = Sum_{k=0..n} C(n-k,3k). (Formerly M1391)

1, 1, 1, 1, 2, 5, 11, 21, 37, 64, 113, 205, 377, 693, 1266, 2301, 4175, 7581, 13785, 25088, 45665, 83097, 151169, 274969, 500162, 909845, 1655187, 3011157, 5477917, 9965312, 18128529, 32978725, 59993817, 109139117, 198543154

Offset

0,5

References

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

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..3850

V. C. Harris, C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,1,3).

V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.

Vedran Krcadinac, A new generalization of the golden ratio, Fibonacci Quart. 44 (2006), no. 4, 335-340.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Index entries for linear recurrences with constant coefficients, signature (3, -3, 1, 1).

Formula

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

Maple

A003522:=-(z-1)**2/(-1+3*z-3*z**2+z**4+z**3); # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

LinearRecurrence[{3, -3, 1, 1}, {1, 1, 1, 1}, 35] (* Ray Chandler, Sep 23 2015 *)

Prog

(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 */

Crossrefs

Sequence in context: A103198 A183929 A365252 * A112805 A119970 A326509

Adjacent sequences: A003519 A003520 A003521 * A003523 A003524 A003525

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved