A004280 2 together with the odd numbers (essentially the result of the first stage of the sieve of Eratosthenes).

1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131

Offset: 1

Author

N. J. A. Sloane

Keywords

• easy
• nonn

Comments

Number of Fibonacci binary words of length n and having no subword 1011. A Fibonacci binary word is a binary word having no 00 subword. Example: a(5) = 9 because of the 13 Fibonacci binary words of length 5 the following do not qualify: 11011, 10110, 10111 and 01011. - Emeric Deutsch, May 13 2007

a(1) = 1; for n > 1, a(n) = least number > a(n-1) which is a unique sum of two earlier terms, not necessarily distinct. - Franklin T. Adams-Watters, Nov 01 2011

References

• F. S. Roberts, Applied Combinatorics, Prentice-Hall, 1984, p. 256.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Borys Kuca, Structures in Additive Sequences, arXiv:1804.09594 [math.NT], 2018. See V(1,2).
• Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 528.
• H. B. Meyer, Eratosthenes' sieve
• Index entries for linear recurrences with constant coefficients, signature (2,-1).
• Index entries for sequences generated by sieves

Crossrefs

Cf. A002858.

Programs

Maple

1,2,seq(2*n-1,n=2..70); # Emeric Deutsch, May 13 2007

Mathematica

Union[ Join[ 2Range[70] - 1, {2}]] (* Robert G. Wilson v *)

PARI

a(n)=2*n + 2\n - 3 \\ Charles R Greathouse IV, Nov 01 2011

Sage

[1,2]+[2*n-3 for n in (3..70)] # G. C. Greubel, Nov 25 2021

Formula

From Paul Barry, Mar 05 2007: (Start)

G.f.: x*(1+x^3)/(1-x)^2;

a(n) = 2*n - 3 + C(1, n-1) + C(0, n-1). (End)

a(n) = 2*n - 3 + floor(2/n). - Wesley Ivan Hurt, May 23 2013

E.g.f.: (1/2)*(6 + 4*x + x^2 - 2*(3 - 2*x)*exp(x)). - G. C. Greubel, Nov 25 2021

Extensions

Offset changed to 1 and formulas updated accordingly (at the suggestion of Michel Marcus) by Charles R Greathouse IV, Sep 03 2013