A265907 -id:A265907 - cp's OEIS frontend

A275960 Square array A(1,k) = A265907(k), A(n>1,k) = A(n-1, k+1) - A(n-1, k); successive differences of A265907 read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

1, 5, 4, 27, 22, 18, 283, 256, 234, 216, 2783, 2500, 2244, 2010, 1794, 27381, 24598, 22098, 19854, 17844, 16050, 289573, 262192, 237594, 215496, 195642, 177798, 161748, 3294929, 3005356, 2743164, 2505570, 2290074, 2094432, 1916634, 1754886, 39857103, 36562174, 33556818, 30813654, 28308084, 26018010, 23923578, 22006944, 20252058

Offset: 1

Antti Karttunen, Aug 13 2016

			The top left corner of the array:
       1, 5, 27, 283, 2783, 27381, 289573
       4, 22, 256, 2500, 24598, 262192, 3005356
      18, 234, 2244, 22098, 237594, 2743164, 33556818
     216, 2010, 19854, 215496, 2505570, 30813654, 408368976
    1794, 17844, 195642, 2290074, 28308084, 377555322, 11598750594
   16050, 177798, 2094432, 26018010, 349247238, 11221195272, 231648340170
  161748, 1916634, 23923578, 323229228, 10871948034, 220427144898, ...

Transpose: A275961.
Row 1: A265907, row 2: A265908, row 3: A275963.
Column 1: A275965.
Cf. A255411.
Variant: A275950.

(define (A275960 n) (A275960bi (A002260 n) (A004736 n)))
(define (A275960bi row col) (if (= 1 row) (A265907 col) (- (A275960bi (- row 1) (+ 1 col)) (A275960bi (- row 1) col))))

A(1,k) = A265907(k); and for n > 1, A(n,k) = A(n-1, k+1) - A(n-1, k).

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).

A275963 First differences of A265908; second differences of A265907.

Original entry on oeis.org

18, 234, 2244, 22098, 237594, 2743164, 33556818, 441925794, 12449045364, 267703255698, 5497712563434, 112804699752300, 2358022995718578, 51007863864388434, 1135446810698914644, 26241095309660220138, 626391138692542250154, 15519080113275337003404, 397364781519139001021778, 10545585096254938755221394, 289700229771377481566636364

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 A275960.

Cf. A265907, A265908.

(define (A275963 n) (- (A265908 (+ 1 n)) (A265908 n)))

a(n) = A265908(1+n) - A265908(n).

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

A275965 Leftmost column of array A275960.

Original entry on oeis.org

1, 4, 18, 216, 1794, 16050, 161748, 1754886, 20252058, 257046648, 9715067802, 169069882650, 3334258948644, 65130972856014, 1314162130657122, 27756075741907992, 599859130159322082, 13631049605170919850, 318062816759187443364, 7775218520916136276470, 195524294317191054759258, 5132224614520884668600904, 139264442985628101392954898

Offset: 1

Antti Karttunen, Aug 13 2016

nonn

Inverse binomial transform of A265907 (when both are considered as offset-0 sequences).

Antti Karttunen, Table of n, a(n) for n = 1..32

Cf. A265907, A275960.

Cf. also A275955.

(definec (A275965 n) (A275960bi n 1)) ;; A275960bi given in A275960.

cp's OEIS Frontend

A275960 Square array A(1,k) = A265907(k), A(n>1,k) = A(n-1, k+1) - A(n-1, k); successive differences of A265907 read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Scheme

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A275963 First differences of A265908; second differences of A265907.

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A275965 Leftmost column of array A275960.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme