A001731 -id:A001731

Search: a001731 -id:a001731

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A063432

Triangle read by rows in which k-th entry in row n is representation of n in base k, for 1 <= k <= n.

+10
6

1, 11, 10, 111, 11, 10, 1111, 100, 11, 10, 11111, 101, 12, 11, 10, 111111, 110, 20, 12, 11, 10, 1111111, 111, 21, 13, 12, 11, 10, 11111111, 1000, 22, 20, 13, 12, 11, 10, 111111111, 1001, 100, 21, 14, 13, 12, 11, 10, 1111111111, 1010, 101, 22, 20, 14, 13

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Representation of n in base 1 is defined to be a concatenation of n 1's.

It is difficult to write twenty-one in base 11 using decimal digits.

Representation in bases greater than 10 are written in base 10. This is really nasty! - N. J. A. Sloane, Dec 06 2002

LINKS

Table of n, a(n) for n=1..52.

EXAMPLE

Rows start (1), (11, 10), (111, 11, 10), (1111, 100, 11, 10), etc.

MATHEMATICA

f[n_] := Flatten[ Append[ {FromDigits[ Table[1, {n}]] }, Table[ FromDigits[ IntegerDigits[n, i]], {i, 2, n}]]]; Flatten[ Table[ f[n], {n, 1, 10}]] (* Robert G. Wilson v *)

CROSSREFS

Cf. A063431.

Rows are effectively the reverse of A001731, A001732, A001733, A001734, A001735, A001736, A008707, A008708, A008709, A008710, A008711, A008712, A008713, A008714, A008715, A008716, A008717, etc.

Columns are truncated versions of A000042, A007088, A007089, A007090, A007091, A007092, A007093, A007094, A007095, A000027 and perhaps A055649, etc.

Without the 1st column becomes A004053.

KEYWORD

base,nonn,tabl

AUTHOR

Henry Bottomley, Jul 20 2001

STATUS

approved

A094327

a(n) = 9 written in base n.

+10
3

111111111, 1001, 100, 21, 14, 13, 12, 11, 10, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The term 111....1111 should officially be called the "unary expansion", since in base 1 only the digit 0 may appear.

LINKS

Table of n, a(n) for n=1..90.

Index entries for linear recurrences with constant coefficients, signature (1).

FORMULA

G.f.: x*(111111111 - 111110110*x - 901*x^2 - 79*x^3 - 7*x^4 - x^5 - x^6 - x^7 - x^8 - x^9)/(1 - x). - Stefano Spezia, Nov 10 2023

MATHEMATICA

Join[{111111111}, Table[FromDigits[IntegerDigits[9, n]], {n, 2, 90}]] (* Stefano Spezia, Nov 10 2023 *)

CROSSREFS

Cf. A001731, A063432, A035613, A094324, A094325, A094326, A094232, etc.

KEYWORD

nonn,base,easy

AUTHOR

N. J. A. Sloane, Jun 04 2004

STATUS

approved

A063431

Square array read by antidiagonals of n written in base k (n,k>0).

+10
1

1, 1, 11, 1, 10, 111, 1, 2, 11, 1111, 1, 2, 10, 100, 11111, 1, 2, 3, 11, 101, 111111, 1, 2, 3, 10, 12, 110, 1111111, 1, 2, 3, 4, 11, 20, 111, 11111111, 1, 2, 3, 4, 10, 12, 21, 1000, 111111111, 1, 2, 3, 4, 5, 11, 13, 22, 1001, 1111111111, 1, 2, 3, 4, 5, 10, 12, 20, 100

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

It is difficult to write ten in base 11 using decimal digits.

LINKS

Table of n, a(n) for n=1..64.

EXAMPLE

Rows start (1, 1, 1, 1, 1,...), (11, 10, 2, 2, 2,...), (111, 11, 10, 3, 3,...), (1111, 100, 11, 10, 4,...), etc.

CROSSREFS

Cf. A063432. Rows start effectively as the reverse of A001731, A001732, A001733, A001734, A001735, A001736, A008707, A008708, A008709, A008710, A008711, A008712, A008713, A008714, A008715, A008716, A008717 etc. Columns are A000042, A007088, A007089, A007090, A007091, A007092, A007093, A007094, A007095, A000027 and perhaps A055649 etc.

KEYWORD

base,nonn,tabl

AUTHOR

Henry Bottomley, Jul 20 2001

STATUS

approved

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