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A265905 a(1) = 1; for n > 1, a(n) = a(n-1) + A153880(a(n-1)).

%I A265905 #35 Feb 14 2024 01:05:10
%S A265905 1,3,11,49,291,1979,15217,136659,1349627,14561425,174637707,
%T A265905 2254758155,31206959833,467925825795,7453435202483,125743951819681,
%U A265905 2262941842058883,42863071603162571,852618666050008129,17902734514975521891,392964858422866610699,9001537965557375522737,216015564123360144707139,5390978540058458090266187
%N A265905 a(1) = 1; for n > 1, a(n) = a(n-1) + A153880(a(n-1)).
%C A265905 In factorial base (A007623) these numbers look as:
%C A265905   1, 11, 121, 2001, 22011, 242121, 3004001, 33044011, 363524121, 4011111001, 44122221011, 485344431121, 5018801043001, <the first term with digit-value "10">, ...
%C A265905 This sequence is obtained by setting a(1) = 1, and then adding to each previous term a(n-1) the same factorial-base representation, but shifted by one factorial digit left. Only when a term does not contain any adjacent nonzero digits, as is the case with a(4) = "2001" or a(7) = "3004001", does the next term a(5) = "22011" (or respectively a(8) = "33044011") show the uncorrupted "double vision pattern". In other cases, for example, when going from a(2) to a(3), "11" to "121", two nonzero digits are summed up and there is possibly also a carry digit propagating to the left.
%C A265905 Note that the sequence is computed in such a way that factorial-base digits larger than 9 are also correctly summed together. That is, the eventual decimal corruption present in sequences like A007623 does not affect the actual values of this sequence. (See the implementation of A153880.)
%H A265905 Antti Karttunen, <a href="/A265905/b265905.txt">Table of n, a(n) for n = 1..120</a>
%H A265905 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>.
%F A265905 a(1) = 1; for n > 1, a(n) = a(n-1) + A153880(a(n-1)).
%F A265905 Other identities. For all n >= 1:
%F A265905 A084558(a(n)) = n. [The length of the factorial-base representation of the n-th term is always n.]
%t A265905 f[n_] := Module[{k = n, m = 2, r, s = {0}}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, AppendTo[s, r]; m++]; FromDigits[Reverse[s], MixedRadix[Reverse@ Range[2, Length[s] + 1]]]]; NestList[f[#] + # &, 1, 23] (* _Amiram Eldar_, Feb 14 2024 *)
%o A265905 (Scheme, with memoization-macro definec)
%o A265905 (definec (A265905 n) (if (= 1 n) n (+  (A265905 (- n 1)) (A153880 (A265905 (- n 1))))))
%Y A265905 Row 1 of A275950.
%Y A265905 Binomial transform of A275955 (when both are considered as offset-0 sequences).
%Y A265905 Cf. A084558 (left inverse), A153880.
%Y A265905 Cf. A001710, A265906 (first differences), A265907 (variant).
%K A265905 nonn,base
%O A265905 1,2
%A A265905 _Antti Karttunen_, Dec 20 2015
%E A265905 Comment and the note about binomial transform corrected - _Antti Karttunen_, Sep 20 2016