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A265905 a(1) = 1; for n > 1, a(n) = a(n-1) + A153880(a(n-1)). 6
1, 3, 11, 49, 291, 1979, 15217, 136659, 1349627, 14561425, 174637707, 2254758155, 31206959833, 467925825795, 7453435202483, 125743951819681, 2262941842058883, 42863071603162571, 852618666050008129, 17902734514975521891, 392964858422866610699, 9001537965557375522737, 216015564123360144707139, 5390978540058458090266187 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
In factorial base (A007623) these numbers look as:
1, 11, 121, 2001, 22011, 242121, 3004001, 33044011, 363524121, 4011111001, 44122221011, 485344431121, 5018801043001, <the first term with digit-value "10">, ...
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.
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.)
LINKS
FORMULA
a(1) = 1; for n > 1, a(n) = a(n-1) + A153880(a(n-1)).
Other identities. For all n >= 1:
A084558(a(n)) = n. [The length of the factorial-base representation of the n-th term is always n.]
MATHEMATICA
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 *)
PROG
(Scheme, with memoization-macro definec)
(definec (A265905 n) (if (= 1 n) n (+ (A265905 (- n 1)) (A153880 (A265905 (- n 1))))))
CROSSREFS
Row 1 of A275950.
Binomial transform of A275955 (when both are considered as offset-0 sequences).
Cf. A084558 (left inverse), A153880.
Cf. A001710, A265906 (first differences), A265907 (variant).
Sequence in context: A261600 A331617 A193319 * A058733 A203163 A024333
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Dec 20 2015
EXTENSIONS
Comment and the note about binomial transform corrected - Antti Karttunen, Sep 20 2016
STATUS
approved

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Last modified March 29 09:59 EDT 2024. Contains 371268 sequences. (Running on oeis4.)