A046698 a(0) = 0, a(1) = 1, a(n) = a(a(n-1)) + a(a(n-2)) if n > 1.

0, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Author

N. J. A. Sloane, R. K. Guy

Keywords

• nonn
• easy

Comments

Partial sums are A004275. Binomial transform is A048492, starting with 0. - Paul Barry, Feb 28 2003

From Elmo R. Oliveira, Jul 25 2024: (Start)

Continued fraction expansion of 2 - sqrt(2) = A101465.

Decimal expansion of 101/9000. (End)

References

• Sequence proposed by Reg Allenby.

Links

• Ömür Deveci, Zafer Adıgüzel, and Taha Doğan, On the Generalized Fibonacci-circulant-Hurwitz numbers, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 1, 179-190.
• O. Deveci, Y. Akuzum, E. Karaduman, and O. Erdag, The Cyclic Groups via Bezout Matrices, Journal of Mathematics Research, Vol. 7, No. 2, 2015, pp. 34-41.
• Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
• Index entries for linear recurrences with constant coefficients, signature (1).

Crossrefs

Cf. A004275, A048492, A101465 (decimal expansion of 2 - sqrt(2)).

Cf. A033324, A114955, A343461.

Programs

Mathematica

CoefficientList[Series[x (1 + x^2)/(1 - x), {x, 0, 104}], x] (* or *)
Nest[Append[#, #[[#[[-1]] + 1]] + #[[#[[-2]] + 1 ]]] &, {0, 1}, 105] (* Michael De Vlieger, Jul 31 2020 *)

PARI

a(n)=(n>0)+(n>2)

Formula

G.f.: x*(1+x^2)/(1-x). - Paul Barry, Feb 28 2003

From Elmo R. Oliveira, Jul 25 2024: (Start)

E.g.f.: 2*exp(x) - x - 1.

a(n) = 2 for n > 2.

a(n) = 2 - A033324(n+2) = 4 - A343461(n+4) = A114955(n+6) - 6. (End)