A003234 - OEIS

%I M2714 #40 Jul 16 2024 13:06:57

%S 3,8,11,16,19,21,24,29,32,37,42,45,50,53,55,58,63,66,71,74,76,79,84,

%T 87,92,97,100,105,108,110,113,118,121,126,129,131,134,139,142,144,147,

%U 152,155,160,163,165,168,173,176,181,186,189,194,197,199,202,207

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

%C See 3.3 p. 344 in Carlitz link. - _Michel Marcus_, Feb 02 2014

%C This is the function named s in [Carlitz]. - _Eric M. Schmidt_, Aug 14 2014

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

%H L. Carlitz, R. Scoville and T. Vaughan, <a href="http://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 := proc(n)

%p     option remember;

%p     if n =1 then

%p         3;

%p     else

%p         for a from procname(n-1)+1 do

%p             if A003231(A001950(a)) = A001950(A003231(a))-1 then

%p                 return a;

%p             end if;

%p         end do:

%p     end if;

%p end proc:

%p seq(A003234(n),n=1..80) ; # _R. J. Mathar_, Jul 16 2024

%t a3[n_] := Floor[n (Sqrt[5] + 3)/2];

%t a5[n_] := Floor[n (Sqrt[5] + 5)/2];

%t Select[Range[300], a5[a3[#]] == a3[a5[#]]-1&] (* _Jean-François Alcover_, Jul 31 2018 *)

%o (PARI) A001950(n) = floor(n*(sqrt(5)+3)/2);

%o A003231(n) = floor(n*(sqrt(5)+5)/2);

%o isok(n) = A003231(A001950(n)) == A001950(A003231(n)) - 1; \\ _Michel Marcus_, Feb 02 2014

%o (Haskell)

%o a003234 n = a003234_list !! (n-1)

%o a003234_list = [x | x <- [1..],

%o                     a003231 (a001950 x) == a001950 (a003231 x) - 1]

%o -- _Reinhard Zumkeller_, Oct 03 2014

%o (Python)

%o from math import isqrt

%o from itertools import count, islice

%o def A003234_gen(startvalue=1): # generator of terms >= startvalue

%o     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_list = list(islice(A003234_gen(),30)) # _Chai Wah Wu_, Sep 02 2022

%Y Cf. A001950, A003231.

%K nonn

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _Michel Marcus_, Feb 02 2014

%E Definition from _Michel Marcus_ moved from comment to name by _Eric M. Schmidt_, Aug 17 2014