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

1, 3, 11, 49, 291, 1979, 15217, 136659, 1349627, 14561425, 174637707, 2254758155, 31206959833, 467925825795, 7453435202483, 125743951819681, 2262941842058883, 42863071603162571, 852618666050008129, 17902734514975521891, 392964858422866610699, 9001537965557375522737, 216015564123360144707139, 5390978540058458090266187

Offset: 1

Author

Antti Karttunen, Dec 20 2015

Keywords

• nonn
• base

Comments

In factorial base (A007623) these numbers look as:

1, 11, 121, 2001, 22011, 242121, 3004001, 33044011, 363524121, 4011111001, 44122221011, 485344431121, 5018801043001, , ...

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

• Antti Karttunen, Table of n, a(n) for n = 1..120
• Index entries for sequences related to factorial base representation.

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).

Programs

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 *)

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.]

Extensions

Comment and the note about binomial transform corrected - Antti Karttunen, Sep 20 2016