A309492 a(1) = a(2) = 1, a(3) = 3, a(4) = 5, a(5) = 2; a(n) = a(n-a(n-2)) + a(n-a(n-3)) for n > 5.

1, 1, 3, 5, 2, 4, 3, 9, 6, 4, 3, 13, 10, 4, 3, 17, 14, 4, 3, 21, 18, 4, 3, 25, 22, 4, 3, 29, 26, 4, 3, 33, 30, 4, 3, 37, 34, 4, 3, 41, 38, 4, 3, 45, 42, 4, 3, 49, 46, 4, 3, 53, 50, 4, 3, 57, 54, 4, 3, 61, 58, 4, 3, 65, 62, 4, 3, 69, 66, 4, 3, 73, 70, 4, 3, 77, 74, 4, 3, 81, 78, 4, 3, 85, 82, 4, 3

Offset: 1

Author

Altug Alkan, Aug 04 2019

Keywords

• nonn
• easy

Comments

A well-defined solution sequence for recurrence a(n) = a(n-a(n-2)) + a(n-a(n-3)).

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,-1,1,1,-1,1,-1).

Crossrefs

Cf. A063892, A244477.

Programs

Magma

I:=[1,1,3,5,2];[n le 5 select I[n] else Self(n-Self(n-2))+Self(n-Self(n-3)): n in [1..90]]; // Marius A. Burtea, Aug 07 2019

Mathematica

a[n_] := a[n] = If[n < 6, {1, 1, 3, 5, 2}[[n]], a[n - a[n-2]] + a[n - a[n-3]]]; Array[a, 87] (* Giovanni Resta, Aug 07 2019 *)

PARI

q=vector(100); q[1]=1;q[2]=1;q[3]=3;q[4]=5;q[5]=2; for(n=6, #q, q[n]=q[n-q[n-2]]+q[n-q[n-3]]); q

PARI

Vec(x*(1 + 3*x^2 + 2*x^3 - 2*x^4 + 4*x^5 - 7*x^6 + 6*x^7 - 3*x^8) / ((1 - x)^2*(1 + x)*(1 + x^2)^2) + O(x^40)) \\ Colin Barker, Aug 15 2019

Formula

For k > 2:

a(4*k) = 4*k+1,

a(4*k+1) = 4*k-2,

a(4*k+2) = 4,

a(4*k+3) = 3.

From Colin Barker, Aug 04 2019: (Start)

G.f.: x*(1 + 3*x^2 + 2*x^3 - 2*x^4 + 4*x^5 - 7*x^6 + 6*x^7 - 3*x^8) / ((1 - x)^2*(1 + x)*(1 + x^2)^2).

a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-4) - a(n-5) + a(n-6) - a(n-7) for n > 9.

(End)