A358793 Lexicographically earliest sequence of positive and unique integers such that 2*Sum_{k = 1..n} a(k) = Sum_{k = 1..n} a(a(k)) for n > 1 and a(1) = 1.

1, 3, 7, 5, 10, 8, 14, 16, 11, 20, 22, 13, 26, 28, 17, 32, 34, 19, 38, 40, 23, 44, 46, 25, 50, 52, 29, 56, 58, 31, 62, 64, 35, 68, 70, 37, 74, 76, 41, 80, 82, 43, 86, 88, 47, 92, 94, 49, 98, 100, 53, 104, 106, 55, 110, 112, 59, 116, 118, 61, 122, 124, 65, 128

Offset: 1

Author

Thomas Scheuerle, Dec 01 2022

Keywords

• nonn
• easy

Comments

There is a second version of this sequence possible if we change the definition to a(1) = 2 and a(n) > 1, then the sequence will start 2, 4, 5, 8, 10, 7, 14, ... . It will after this continue in the same way as our actual sequence does (and would also extend the valid range of the recurrence formulas).

Start a(1) = 2 and value 1 allowed is A257794.

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,1,0,0,-1).

Crossrefs

Cf. A002516, A105753, A257794.

Programs

Mathematica

LinearRecurrence[{0, 0, 1, 0, 0, 1, 0, 0, -1}, {1, 3, 7, 5, 10, 8, 14, 16, 11, 20, 22, 13, 26, 28, 17}, 100] (* Paolo Xausa, Jan 23 2025 *)

PARI

a(n) = {my(v = [1, 3, 7, 5, 10, 8]);if(n < 7, v[n], n*(1+min(1, n%3))+(n%3 == 0)+(n%6 == 3))}

Formula

G.f.: x*(1 + 3*x + 7*x^2 + 4*x^3 + 7*x^4 + x^5 + 8*x^6 + 3*x^7 - 4*x^8 + 2*x^9 - x^10 + x^11 - 3*x^12 + x^14)/(1 - x^3 - x^6 + x^9).

a(n) = a(n-3) + a(n-6) - a(n-9) for n >= 16.

a((3*(2*n-1) - (-1)^n)/4) = (3*(2*n-1) - (-1)^n)/2, for n > 3.

a(6*n) = 6*n+1, for n > 1.

a(6*n+3) = 6*n+5, for n > 0.

a(n) = 30*n - 2*a(n-1) - 3*a(n-2) - 3*a(n-3) - 3*a(n-4) - 3*a(n-5) - 2*a(n-6) - a(n-7) - 96, for n > 13.