cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A340156 Square array read by upward antidiagonals: T(n, k) is the number of n-ary strings of length k containing 00.

Original entry on oeis.org

1, 1, 3, 1, 5, 8, 1, 7, 21, 19, 1, 9, 40, 79, 43, 1, 11, 65, 205, 281, 94, 1, 13, 96, 421, 991, 963, 201, 1, 15, 133, 751, 2569, 4612, 3217, 423, 1, 17, 176, 1219, 5531, 15085, 20905, 10547, 880, 1, 19, 225, 1849, 10513, 39186, 86241, 92935, 34089, 1815
Offset: 2

Views

Author

Robert P. P. McKone, Dec 29 2020

Keywords

Examples

			For n = 3 and k = 4, there are 21 strings: {0000, 0001, 0002, 0010, 0011, 0012, 0020, 0021, 0022, 0100, 0200, 1000, 1001, 1002, 1100, 1200, 2000, 2001, 2002, 2100, 2200}.
Square table T(n,k):
     k=2:  k=3:  k=4:   k=5:    k=6:     k=7:
n=2:   1     3     8     19      43       94
n=3:   1     5    21     79     281      963
n=4:   1     7    40    205     991     4612
n=5:   1     9    65    421    2569    15085
n=6:   1    11    96    751    5531    39186
n=7:   1    13   133   1219   10513    87199
n=8:   1    15   176   1849   18271   173608
n=9:   1    17   225   2665   29681   317817

Links

Crossrefs

Cf. A008466 (row 2), A186244 (row 3), A000567 (column 4).
Cf. A180165 (not containing 00), A340242 (containing 000).
Cf. A000045, A028859, A125145, A086347, A180033, A180167, A322054.
Cf. A005563, A033445.

Programs

  • Mathematica
    m[r_] := Normal[With[{p = 1/n}, SparseArray[{Band[{1, 2}] -> p, {i_, 1} /; i <= r -> 1 - p, {r + 1, r + 1} -> 1}]]];
T[n_, k_, r_] := MatrixPower[m[r], k][[1, r + 1]]*n^k;
Reverse[Table[T[n, k - n + 2, 2], {k, 2, 11}, {n, 2, k}], 2] // Flatten (* Robert P. P. McKone, Jan 26 2021 *)

Formula

T(n, k) = n^k - A180165(n+1,k-1), where A180165 in the number of strings not containing 00.
m(2) = [1 - 1/n, 1/n, 0; 1 - 1/n, 0, 1/n; 0, 0, 1], is the probability/transition matrix for two consecutive "0" -> "containing 00".

A322052 Number of decimal strings of length n that contain a specific string xy where x and y are distinct digits.

Original entry on oeis.org

0, 1, 20, 299, 3970, 49401, 590040, 6850999, 77919950, 872348501, 9645565060, 105583302099, 1146187455930, 12356291257201, 132416725116080, 1411810959903599, 14985692873919910, 158445117779295501, 1669465484919035100, 17536209731411055499, 183692631829191519890, 1919390108560504143401
Offset: 1

Views

Author

N. J. A. Sloane, Dec 21 2018

Keywords

Comments

See A004189 for the number that do not contain the specified string.

Examples

			Suppose the desired string is 03. At length 2 that is the only possibility. At length 3 there are 20 strings that contain it: 03d and d03, where d is any digit.

Links

Crossrefs

Cf. A004189, A322053, A322054.
Partial sums of A322628.

Programs

  • Maple
    f:= gfun:-rectoproc({10*a(n) - 101*a(n + 1) + 20*a(n + 2) - a(n + 3), a(0) = 0, a(1) = 0, a(2) = 1},a(n),remember):
map(f, [$1..40]); # Robert Israel, Mar 27 2020

Formula

G.f.: x^2/((1-10*x)*(1-10*x+x^2)).

A322053 Number of decimal strings of length n that contain a specific string xx (where x is a single digit).

Original entry on oeis.org

0, 1, 19, 280, 3691, 45739, 544870, 6315481, 71743159, 802527760, 8868438271, 97038694279, 1053164192950, 11351825985061, 121644911602099, 1296970638284440, 13767539948978851, 145580595285369619, 1534133217109136230, 16117424311550552641
Offset: 1

Views

Author

N. J. A. Sloane, Dec 21 2018

Keywords

Comments

See A322054 for the number that do not contain the specified string.

Examples

			Suppose the desired string is 00. At length 2 that is the only possibility. At length 3 there are 19 strings that contain it: 000, 00d, and d00, where d is any nonzero digit.

Crossrefs

Cf. A322052, A004189, A322054.
Suggested by A322628.

Formula

G.f. = x^2/((1-10*x)*(1-9*x-9*x^2)).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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