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A034587 Fibonacci iteration starting with (1, a(n)) leads to a "nine digits anagram".

%I A034587 #57 Feb 19 2024 01:49:30
%S A034587 718,1790,1993,2061,2259,3888,3960,4004,4396,5093,5832,7031,7310,7712,
%T A034587 8039,8955,9236,11598,11742,12312,13295,15095,15432,16044,16355,16472,
%U A034587 18109,18559,19144,19950,19968,20116,20180,20494,21170,21376,21998
%N A034587 Fibonacci iteration starting with (1, a(n)) leads to a "nine digits anagram".
%C A034587 By "nine digits anagram" the author means a number whose digits are a permutation of {1, ..., 9}. These are more commonly known as restricted zeroless pandigital numbers and form the first 9! terms of A050289.
%C A034587 The largest term is a(750767) = 987654320.
%C A034587 More generally, the last N = 9! - 158323 = 204557 (> 56% of 9!) terms are given as A050289(k)-1 with indices k = 9!-N+1, ..., 9!. Indeed, a number > (987654321-1)/2 = 493827160 is a term if and only if it equals a "9-digit anagram" minus 1, since all results beyond the first iteration (1 + n = n+1) will be too large. Since 493827165 = A050289(158324) > 493827160, starting with a(546211) = 493827164 the terms are given by A050289(158324 .. 9!) - 1, for a total of 546211 + N - 1 = 750767 terms. (The term 493827164 is preceded by 493827160 (which yields 987654321 but is not in A050289 - 1) and 493827155 = A050289(158323) - 1.) - M. F. Hasler, Jan 07 2020
%C A034587 The ratio between consecutive terms in a Fibonacci sequence x(n+1) = x(n) + x(n-1) tends quickly to the golden ratio Phi = (sqrt(5)+1)/2 = A001622. We can tell whether a starting value N is in this sequence or not from the terms between 123456789 and 987654321 ~ 1e9. From N*Phi^k = 1e9 we get k = log(1e9/N)/log(Phi) ~ 43 - 2*log(N) for the maximum (and 3 less for the minimum) number of required iterations. - _M. F. Hasler_, Jan 06 2020
%H A034587 M. F. Hasler, <a href="/A034587/b034587.txt">Table of n, a(n) for n = 1..10000</a>, Jan 06 2020. (Full list of 750767 terms is available on request.)
%H A034587 Patrick De Geest, <a href="http://www.worldofnumbers.com/ninedigits.htm">Nine Digits Digressions</a>
%H A034587 M. F. Hasler, <a href="/A034587/a034587.png">Graph of A034587, n = 1..750767 (full sequence)</a>, Jan 10 2020
%H A034587 M. F. Hasler, <a href="/A034587/a034587_1.png">Slope of A034587, averaged over n +- 10 sqrt(n)</a>, Jan 10 2020
%F A034587 a(n) = A050289(m) with n = 387887 + m for 158324 <= m <= 9! or 546211 <= n <= 750767 = total number of terms in this sequence. - _M. F. Hasler_, Jan 07 2020
%e A034587 Denote by F(a,b) the Fibonacci-type sequence x(n+1) = x(n) + x(n-1) starting with x(0) = a, x(1) = b.
%e A034587 Then F(1,21998) = (1, 21998, 21999, 43997, 65996, 109993, 175989, 285982, 461971, 747953, 1209924, 1957877, 3167801, 5125678, 8293479, 13419157, 21712636, 35131793, 56844429, 91976222, 148820651, 240796873, 389617524, ...) where a nine-digits anagram has been reached.
%e A034587 The growth is roughly linear in three parts, with a slope of 700 up to a(292967) = 206993812, then an average slope of 1130 before it rises to (9.87e8 - 4.94e8)/2.05e5 ~ 2400 for 546211 <= n <= 750767 (cf. formula & comments): a(100) = 71960, a(200) = 149540, a(500) = 351868, a(1000) = 649921, a(2000) = 1400539, a(5000) = 3209798, a(10^4) = 6595301, a(2e4) = 13351498, a(5e4) = 32441506, a(10^5) = 67090523, a(2e5) = 134759627, a(3e5) = 214973567, a(4e5) = 327136594, a(5e5) = 439256717. - _M. F. Hasler_, Jan 07 2020
%o A034587 (PARI)
%o A034587 A034587=select( {is_A034587(n,s=1,L=[1..9])=while( 123456789 > n=s+s=n,); n<1e9 && until( 987654321 < n=s+s=n, Set(digits(n))==L&&return(n))}, [1..22222]) \\ Function is_A034587 returns the 9-digit anagram if one is reached; null == false == 0 else.
%o A034587 nxt_A034587(n)={until(is_A034587(n+=1),);n} \\ Returns next larger term
%o A034587 A034587(n)={if(n>546210, A050289(n-387887)-1, #A034587>=n, A034587[n], A034587=concat( A034587, vector(n-#A034587,i, n=nxt_A034587(if(i>1,n,A034587[#A034587])))); n)} \\ Uses the two functions above. Could use Vecsmall(...) in definition of A034587 and vectorsmall in A034587(n) to reduce memory.
%o A034587 \\ _M. F. Hasler_, Jan 06 2020 and Jan 07 2020
%o A034587 (Python)
%o A034587 def ok(n):
%o A034587     f, g = n, n+1
%o A034587     while g < 10**9:
%o A034587         if g > 123456788 and "".join(sorted(str(g))) == "123456789":
%o A034587             return True
%o A034587         f, g = g, f+g
%o A034587     return False
%o A034587 print([k for k in range(10**4) if ok(k)]) # _Michael S. Branicky_, Feb 18 2024
%Y A034587 Subsequences: A034588 (primes), A034589 (lucky numbers), A034306 (palindromes).
%K A034587 nonn,base,fini
%O A034587 1,1
%A A034587 _Patrick De Geest_, Oct 15 1998
%E A034587 Edited and offset changed to 1 by _M. F. Hasler_, Jan 06 2020
%E A034587 Results confirmed by _Giovanni Resta_, Jan 07 2020