A003233 Numbers k such that A003231(A001950(k)) = A001950(A003231(k)). (Formerly M0944)

1, 2, 4, 5, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18, 20, 22, 23, 25, 26, 27, 28, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 43, 44, 46, 47, 48, 49, 51, 52, 54, 56, 57, 59, 60, 61, 62, 64, 65, 67, 68, 69, 70, 72, 73, 75, 77, 78, 80, 81, 82, 83, 85, 86, 88, 89, 90, 91

Offset

1,2

Comments

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

This is the function named r 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..67.

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

Mathematica

a3221[n_] := Floor[n(5 + Sqrt[5])/2];
a1950[n_] := Floor[n(1 + Sqrt[5])^2/4];
Select[Range[100], a3221[a1950[#]] == a1950[a3221[#]]&] (* Jean-François Alcover, Aug 04 2018 *)

Prog

(PARI) A001950(n) = floor(n*(sqrt(5)+3)/2);
A003231(n) = floor(n*(sqrt(5)+5)/2);
lista(nn) = { for(n=1, nn, if (A003231(A001950(n)) == A001950(A003231(n)), print1(n, ", "))); } \\ Michel Marcus, Feb 02 2014
(Haskell)
a003233 n = a003233_list !! (n-1)
a003233_list = [x | x <- [1..],
                    a003231 (a001950 x) == a001950 (a003231 x)]
-- Reinhard Zumkeller, Oct 03 2014
(Python)
from math import isqrt
from itertools import count, islice
def A003233_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)==((k:=(n+isqrt(5*n**2)>>1)+(n<<1))+isqrt(5*k**2)>>1)+k, count(max(1, startvalue)))
A003233_list = list(islice(A003233_gen(), 30)) # Chai Wah Wu, Sep 02 2022

Crossrefs

Cf. A001950, A003231, A003234.

Sequence in context: A187485 A039052 A047572 * A218544 A286045 A127032

Adjacent sequences: A003230 A003231 A003232 * A003234 A003235 A003236

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