A003234 Numbers k such that A003231(A001950(k)) = A001950(A003231(k)) - 1.

3, 8, 11, 16, 19, 21, 24, 29, 32, 37, 42, 45, 50, 53, 55, 58, 63, 66, 71, 74, 76, 79, 84, 87, 92, 97, 100, 105, 108, 110, 113, 118, 121, 126, 129, 131, 134, 139, 142, 144, 147, 152, 155, 160, 163, 165, 168, 173, 176, 181, 186, 189, 194, 197, 199, 202, 207

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

See 3.3 p. 344 in Carlitz link. - Michel Marcus, Feb 02 2014

This is the function named s in [Carlitz]. - Eric M. Schmidt, Aug 14 2014

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386.

Crossrefs

Cf. A001950, A003231.

Programs

Haskell

a003234 n = a003234_list !! (n-1)
a003234_list = [x | x <- [1..],
                    a003231 (a001950 x) == a001950 (a003231 x) - 1]
-- Reinhard Zumkeller, Oct 03 2014

Maple

A003234 := proc(n)
    option remember;
    if n =1 then
        3;
    else
        for a from procname(n-1)+1 do
            if A003231(A001950(a)) = A001950(A003231(a))-1 then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A003234(n),n=1..80) ; # R. J. Mathar, Jul 16 2024

Mathematica

a3[n_] := Floor[n (Sqrt[5] + 3)/2];
a5[n_] := Floor[n (Sqrt[5] + 5)/2];
Select[Range[300], a5[a3[#]] == a3[a5[#]]-1&] (* Jean-François Alcover, Jul 31 2018 *)

PARI

A001950(n) = floor(n*(sqrt(5)+3)/2);
A003231(n) = floor(n*(sqrt(5)+5)/2);
isok(n) = A003231(A001950(n)) == A001950(A003231(n)) - 1; \\ Michel Marcus, Feb 02 2014

Python

from math import isqrt
from itertools import count, islice
def A003234_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:((m:=(n+isqrt(5*n**2)>>1)+n)+isqrt(5*m**2)>>1)+(m<<1)+1==((k:=(n+isqrt(5*n**2)>>1)+(n<<1))+isqrt(5*k**2)>>1)+k,count(max(1,startvalue)))
A003234_list = list(islice(A003234_gen(),30)) # Chai Wah Wu, Sep 02 2022

Extensions

More terms from Michel Marcus, Feb 02 2014

Definition from Michel Marcus moved from comment to name by Eric M. Schmidt, Aug 17 2014