A265906 -id:A265906 - cp's OEIS frontend

A275953 First differences of A265906; second differences of A265905.

6, 30, 204, 1446, 11550, 108204, 1091526, 11998830, 146864484, 1920044166, 26872081230, 407766664284, 6548790510726, 111305007240510, 2018907373622004, 38462931870864486, 769155464685741870, 16240360254478668204, 358012008058965575046, 8233510983226617823230, 198405453050668260272364, 4967948949777295176374646, 129158062469199137001844830

Offset: 1

Antti Karttunen, Aug 13 2016

Antti Karttunen, Table of n, a(n) for n = 1..52
Index entries for sequences related to factorial base representation

Row 3 of A275950.

Cf. A265905, A265906.

(define (A275953 n) (- (A265906 (+ 1 n)) (A265906 n)))

a(n) = A265906(1+n) - A265906(n).

A275950 Square array A(1,k) = A265905(k), A(n>1,k) = A(n-1, k+1) - A(n-1, k); successive differences of A265905 read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 3, 2, 11, 8, 6, 49, 38, 30, 24, 291, 242, 204, 174, 150, 1979, 1688, 1446, 1242, 1068, 918, 15217, 13238, 11550, 10104, 8862, 7794, 6876, 136659, 121442, 108204, 96654, 86550, 77688, 69894, 63018, 1349627, 1212968, 1091526, 983322, 886668, 800118, 722430, 652536, 589518, 14561425, 13211798, 11998830, 10907304, 9923982, 9037314, 8237196, 7514766, 6862230, 6272712

Offset: 1

Antti Karttunen, Aug 13 2016

			The top left 7 x 8 corner of the array:
      1, 3, 11, 49, 291, 1979, 15217
      2, 8, 38, 242, 1688, 13238, 121442
      6, 30, 204, 1446, 11550, 108204, 1091526
     24, 174, 1242, 10104, 96654, 983322, 10907304
    150, 1068, 8862, 86550, 886668, 9923982, 123958350
    918, 7794, 77688, 800118, 9037314, 114034368, 1514355678
   6876, 69894, 722430, 8237196, 104997054, 1400321310, 20026187676
  63018, 652536, 7514766, 96759858, 1295324256, 18625866366, 291196957578

Transpose: A275951.
Row 1: A265905, row 2: A265906, row 3: A275953.
Column 1: A275955.
Cf. A153880.
Variant: A275960.

(define (A275950 n) (A275950bi (A002260 n) (A004736 n)))
(define (A275950bi row col) (if (= 1 row) (A265905 col) (- (A275950bi (- row 1) (+ 1 col)) (A275950bi (- row 1) col))))

A(1,k) = A265905(k); and for n > 1, A(n,k) = A(n-1, k+1) - A(n-1, k).

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

Original entry on oeis.org

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

Antti Karttunen, Dec 20 2015

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.)

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

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).

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 *)

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.]

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

A265908 a(n) = A255411(A265907(n)); also the first differences of A265907.

Original entry on oeis.org

4, 22, 256, 2500, 24598, 262192, 3005356, 36562174, 478487968, 12927533332, 280630789030, 5778343352464, 118583043104764, 2476606038823342, 53484469903211776, 1188931280602126420, 27430026590262346558, 653821165282804596712, 16172901278558141600116, 413537682797697142621894, 10959122779052635897843288, 300659352550430117464479652

Offset: 1

Antti Karttunen, Dec 20 2015

In factorial base (A007623) these numbers are almost just like those of A265906, but shifted once left, with an extra zero appended, and then each nonzero digit incremented by one:

20, 320, 20220, 324020, 4604320, 64004220, 824203020, , ...

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

Row 2 of A275960.
Cf. A007623, A255411.
First differences of A265907.

a(n) = A255411(A265907(n))

a(n) = A265907(n+1) - A265907(n).

cp's OEIS Frontend

A275953 First differences of A265906; second differences of A265905.

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A275950 Square array A(1,k) = A265905(k), A(n>1,k) = A(n-1, k+1) - A(n-1, k); successive differences of A265905 read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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

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A265908 a(n) = A255411(A265907(n)); also the first differences of A265907.

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