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

%I A003234 M2714 #42 Jan 05 2025 19:51:33
%S A003234 3,8,11,16,19,21,24,29,32,37,42,45,50,53,55,58,63,66,71,74,76,79,84,
%T A003234 87,92,97,100,105,108,110,113,118,121,126,129,131,134,139,142,144,147,
%U A003234 152,155,160,163,165,168,173,176,181,186,189,194,197,199,202,207
%N A003234 Numbers k such that A003231(A001950(k)) = A001950(A003231(k)) - 1.
%C A003234 See 3.3 p. 344 in Carlitz link. - _Michel Marcus_, Feb 02 2014
%C A003234 This is the function named s in [Carlitz]. - _Eric M. Schmidt_, Aug 14 2014
%D A003234 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003234 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.
%p A003234 A003234 := proc(n)
%p A003234     option remember;
%p A003234     if n =1 then
%p A003234         3;
%p A003234     else
%p A003234         for a from procname(n-1)+1 do
%p A003234             if A003231(A001950(a)) = A001950(A003231(a))-1 then
%p A003234                 return a;
%p A003234             end if;
%p A003234         end do:
%p A003234     end if;
%p A003234 end proc:
%p A003234 seq(A003234(n),n=1..80) ; # _R. J. Mathar_, Jul 16 2024
%t A003234 a3[n_] := Floor[n (Sqrt[5] + 3)/2];
%t A003234 a5[n_] := Floor[n (Sqrt[5] + 5)/2];
%t A003234 Select[Range[300], a5[a3[#]] == a3[a5[#]]-1&] (* _Jean-François Alcover_, Jul 31 2018 *)
%o A003234 (PARI) A001950(n) = floor(n*(sqrt(5)+3)/2);
%o A003234 A003231(n) = floor(n*(sqrt(5)+5)/2);
%o A003234 isok(n) = A003231(A001950(n)) == A001950(A003231(n)) - 1; \\ _Michel Marcus_, Feb 02 2014
%o A003234 (Haskell)
%o A003234 a003234 n = a003234_list !! (n-1)
%o A003234 a003234_list = [x | x <- [1..],
%o A003234                     a003231 (a001950 x) == a001950 (a003231 x) - 1]
%o A003234 -- _Reinhard Zumkeller_, Oct 03 2014
%o A003234 (Python)
%o A003234 from math import isqrt
%o A003234 from itertools import count, islice
%o A003234 def A003234_gen(startvalue=1): # generator of terms >= startvalue
%o A003234     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)))
%o A003234 A003234_list = list(islice(A003234_gen(),30)) # _Chai Wah Wu_, Sep 02 2022
%Y A003234 Cf. A001950, A003231.
%K A003234 nonn
%O A003234 1,1
%A A003234 _N. J. A. Sloane_
%E A003234 More terms from _Michel Marcus_, Feb 02 2014
%E A003234 Definition from _Michel Marcus_ moved from comment to name by _Eric M. Schmidt_, Aug 17 2014