A265905 - OEIS

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

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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

Antti Karttunen, Table of n, a(n) for n = 1..120

Index entries for sequences related to factorial base representation.

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

Adjacent sequences: A265902 A265903 A265904 * A265906 A265907 A265908

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