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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Programs

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 *)

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 */

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