A003234 Numbers k such that A003231(A001950(k)) = A001950(A003231(k)) - 1. (Formerly M2714)

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

1,1

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

Table of n, a(n) for n=1..57.

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

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 *)

Prog

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

Crossrefs

Cf. A001950, A003231.

Sequence in context: A145837 A111132 A188473 * A047470 A184401 A190251

Adjacent sequences: A003231 A003232 A003233 * A003235 A003236 A003237

Keyword

nonn

Author

N. J. A. Sloane

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

Status

approved