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.

User: Harrie Grondijs

Harrie Grondijs's wiki page.

Harrie Grondijs has authored 4 sequences.

A114972 Array read by antidiagonals: consider a doubly infinite chessboard with squares labeled (i,j), i in Z, j in Z; T(i,j) = number of king-paths of length max{i,j} from (0,0) to (i,j).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 7, 2, 2, 7, 19, 6, 1, 6, 19, 51, 16, 3, 3, 16, 51, 141, 45, 10, 1, 10, 45, 141, 393, 126, 30, 4, 4, 30, 126, 393, 1107, 357, 90, 15, 1, 15, 90, 357, 1107, 3139, 1016, 266, 50, 5, 5, 50, 266, 1016, 3139, 8953, 2907, 784, 161, 21, 1, 21, 161, 784, 2907
Offset: 0

Views

Author

N. J. A. Sloane, based on May 27 2005 email from Harrie Grondijs, Feb 27 2006

Keywords

Examples

			Array begins:
1 1 3 7 19 ...
1 1 2 6 ...
3 2 1 3 ...
7 6 3 1 ...
...

References

  • Harrie Grondijs, Neverending Quest of Type C, Volume B - the endgame study-as-struggle.

Crossrefs

Cf. A111808, A026300, A114929.

Formula

Equals triangle A111808 next to same triangle reflected in mirror. See A111808 for obvious recurrence.

Extensions

More terms from Joshua Zucker, May 20 2006
T(0,0) corrected by Johannes W. Meijer, Oct 05 2010

A115229 Number of ways in chess to force mate in n moves for a single knight against a single pawn, without duals, without short mates and excluding rotations, mirroring and color reversing.

Original entry on oeis.org

3, 12, 16, 31, 21, 44, 3, 6
Offset: 0

Views

Author

N. J. A. Sloane, based on email from Harrie Grondijs, Mar 06 2006

Keywords

Comments

The mate is only possible against a pawn on a border line (a-line or h-line).
No non-dual (single solution) mates are possible beyond n = 7 (cf. Springerzauber, page 178).
Obviously the positions with the higher number of moves lead via a unique chain of positions with the lower numbers.

Examples

			Example: The three positions with 0 moves (checkmate) are:
a) White: Kc2, Nb3, Black: Ka1, Pawn a2
b) White: Kc1, Nb3, Black: Ka1, Pawn a2
c) White: Kc1, Nc2, Black: Ka1, pawn a2

References

  • John Selman and Harrie Grondijs, Springerzauber, 1998, chapter 16.

A114929 Array read by antidiagonals: consider a semi-infinite chessboard with squares labeled (i,j), i >= 0, j >= 0; T(i,j) = number of king-paths of length max{i,j} from (0,0) to (i,j).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 4, 2, 2, 4, 9, 5, 1, 5, 9, 21, 12, 3, 3, 12, 21, 51, 30, 9, 1, 9, 30, 51, 127, 76, 25, 4, 4, 25, 76, 127, 323, 196, 69, 14, 1, 14, 69, 196, 323, 835, 512, 189, 44, 5, 5, 44, 189, 512, 835, 2188, 1353, 518, 133, 20, 1, 20, 133, 518, 1353, 2188, 5798, 3610, 1422
Offset: 0

Views

Author

N. J. A. Sloane, based on May 27 2005 email from Harrie Grondijs, Feb 27 2006

Keywords

Examples

			Array begins:
1 1 2 4 9 21 51 ...
1 1 2 5 12 30 ...
2 2 1 3 9 25 ...
4 5 3 1 4 14 ...
...

References

  • Harrie Grondijs, Neverending Quest of Type C, Volume B - the endgame study-as-struggle.

Crossrefs

Cf. A026300, A020474, A111808, A114972.

Formula

Equals Motzkin triangle (A026300) next to same triangle reflected in mirror. See A026300 for the obvious recurrence.

Extensions

More terms from Joshua Zucker, May 20 2006
T(0,0) corrected by Johannes W. Meijer, Oct 07 2010

A115218 Triangle read by rows: zeroth row is 0; to get row n >= 1, append next 2^n numbers to end of previous row.

Original entry on oeis.org

0, 0, 1, 2, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22
Offset: 0

Views

Author

N. J. A. Sloane, based on a suggestion from Harrie Grondijs, Mar 04 2006

Keywords

Examples

			Triangle begins:
0
0 1 2
0 1 2 3 4 5 6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
...

Links

Crossrefs

Cf. A126646 (length of n-th row).

Programs

  • Maple
    seq($0..2^n-2, n=0..5); # Robert Israel, Jan 02 2018
  • Mathematica
    Range[0,#-1]&/@Accumulate[2^Range[0,5]]//Flatten (* Harvey P. Dale, Jan 20 2021 *)

Formula

From Robert Israel, Jan 02 2018: (Start)
G.f.: x^2/(1-x)^2 - (1-x)^(-1)*Sum_{n>=2} (2^n-1)*x^(2^(n+1)-n-2).
a(n) = k if n = 2^m - m + k - 1, 0 <= k <= 2^m-2.
G.f. as triangle: (1-y)^(-2)*Sum_{n>=1} x^n*(y + (1-2^n)*y^(2^n-1)+(2^n-2)*y^(2^n)). (End)

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

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