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

%I A003233 M0944 #41 Jan 05 2025 19:51:33
%S A003233 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,
%T A003233 35,36,38,39,40,41,43,44,46,47,48,49,51,52,54,56,57,59,60,61,62,64,65,
%U A003233 67,68,69,70,72,73,75,77,78,80,81,82,83,85,86,88,89,90,91
%N A003233 Numbers k such that A003231(A001950(k)) = A001950(A003231(k)).
%C A003233 See 3.3 p. 344 in Carlitz link. - _Michel Marcus_, Feb 02 2014
%C A003233 This is the function named r in [Carlitz]. - _Eric M. Schmidt_, Aug 14 2014
%D A003233 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003233 L. Carlitz, R. Scoville and T. Vaughan, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/11-4/carlitz.pdf">Some arithmetic functions related to Fibonacci numbers</a>, Fib. Quart., 11 (1973), 337-386.
%t A003233 a3221[n_] := Floor[n(5 + Sqrt[5])/2];
%t A003233 a1950[n_] := Floor[n(1 + Sqrt[5])^2/4];
%t A003233 Select[Range[100], a3221[a1950[#]] == a1950[a3221[#]]&] (* _Jean-François Alcover_, Aug 04 2018 *)
%o A003233 (PARI) A001950(n) = floor(n*(sqrt(5)+3)/2);
%o A003233 A003231(n) = floor(n*(sqrt(5)+5)/2);
%o A003233 lista(nn) = { for(n=1, nn, if (A003231(A001950(n)) == A001950(A003231(n)), print1(n, ", ")));} \\ _Michel Marcus_, Feb 02 2014
%o A003233 (Haskell)
%o A003233 a003233 n = a003233_list !! (n-1)
%o A003233 a003233_list = [x | x <- [1..],
%o A003233                     a003231 (a001950 x) == a001950 (a003231 x)]
%o A003233 -- _Reinhard Zumkeller_, Oct 03 2014
%o A003233 (Python)
%o A003233 from math import isqrt
%o A003233 from itertools import count, islice
%o A003233 def A003233_gen(startvalue=1): # generator of terms >= startvalue
%o A003233     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)))
%o A003233 A003233_list = list(islice(A003233_gen(),30)) # _Chai Wah Wu_, Sep 02 2022
%Y A003233 Cf. A001950, A003231, A003234.
%K A003233 nonn
%O A003233 1,2
%A A003233 _N. J. A. Sloane_
%E A003233 More terms from _Michel Marcus_, Feb 02 2014
%E A003233 Definition from Michel Marcus moved from comment to name by _Eric M. Schmidt_, Aug 17 2014