A133090 -id:A133090 - cp's OEIS frontend

A133081 An interpolation operator, companion to A133080.

1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Gary W. Adamson, Sep 09 2007

A133081 * [1,2,3,...] = A133090: (1, 1, 5, 3, 9, 5, 13, 7, 17, ...).

A133080: diagonal and subdiagonal are switched.

			First few rows of the triangle:
  1;
  1, 0;
  0, 1, 1;
  0, 0, 1, 0;
  0, 0, 0, 1, 1;
  ...

Cf. A133080, A133090, A040001 (row sums).

row(n) = vector(n, k, if (k==n-1, 1, if (k==n, n%2)));
lista(nn) = my(list=List()); for (n=1, nn, my(v=row(n)); for (k=1, #v, listput(list, v[k]))); Vec(list); \\ Michel Marcus, Mar 06 2022

Infinite lower triangular matrix, (1,0,1,0,...) in the main diagonal and (1,1,1,...) in the subdiagonal.

More terms from Michel Marcus, Mar 06 2022

A309805 Maximum number of nonattacking kings placeable on a hexagonal board with edge-length n in Glinski's hexagonal chess.

Original entry on oeis.org

1, 2, 7, 10, 19, 24, 37, 44, 61, 70, 91, 102, 127, 140, 169, 184, 217, 234, 271, 290, 331, 352, 397, 420, 469, 494, 547, 574, 631, 660, 721, 752, 817, 850, 919, 954, 1027, 1064, 1141, 1180, 1261, 1302, 1387, 1430, 1519, 1564, 1657, 1704, 1801, 1850, 1951, 2002

Offset: 1

Sangeet Paul, Aug 17 2019

			a(1) = 1
.
  o
.
a(2) = 2
.
   . .
  o . o
   . .
.
a(3) = 7
.
    o . o
   . . . .
  o . o . o
   . . . .
    o . o
.
a(4) = 10
.
     . . . .
    o . o . o
   . . . . . .
  o . o . o . o
   . . . . . .
    o . o . o
     . . . .
.

Chess variants, Glinski's Hexagonal Chess
Wikipedia, Hexagonal chess - Gliński's hexagonal chess
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A003215, A049450.

Partial sums of A133090.

nn:=51; CoefficientList[Series[- (1 + x + 3*x^2 + x^3)/((- 1 + x)^3*(1 + x)^2),{x, 0, nn}], x] (* Georg Fischer, May 10 2020 *)

a(n) = n^2 - (n\2) - (n\2)^2; \\ Andrew Howroyd, Aug 17 2019

def A309805(n): return n**2-(m:=n>>1)*(m+1) # Chai Wah Wu, Apr 04 2024

a(n) = n^2 - floor(n/2) - floor(n/2)^2.
From Stefano Spezia, Aug 18 2019 (Start)
G.f.: - (1 + x + 3*x^2 + x^3)/((- 1 + x)^3*(1 + x)^2).
E.g.f.: (1/8)*exp(-x)*(-1 + 2*x + exp(2*x)*(1 + 4*x + 6*x^2)).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5.
a(n) = (1/16)*(3 + (-1)^(1+2*n) - 4*n + 12*n^2 - 2*(-1)^n*(1 + 2*n)).
a(2*n-1) = A003215(n).
a(2*n) = A049450(n).
(End)

A009661 Smallest number m such that m^m+1 is divisible by n.

Original entry on oeis.org

0, 0, 5, 3, 2, 5, 3, 7, 17, 9, 21, 11, 6, 3, 29, 15, 24, 17, 27, 19, 41, 21, 11, 23, 18, 25, 53, 3, 14, 29, 15, 31, 35, 33, 69, 35, 6, 27, 77, 39, 25, 41, 63, 35, 89, 11, 23, 47, 97, 49, 101, 51, 26, 53, 109, 55, 113, 35, 117, 59, 30, 15, 125, 63, 18, 35, 99, 67, 11, 69, 35, 71, 9, 27

Offset: 1

David W. Wilson

nonn

If n is odd, then a(n) <= 2*n - 1. If n is even, then a(n) <= n - 1. - Thomas Ordowski, Dec 03 2023

Cf. A014566, A133090, A268466.

a(n) = my(m=0); while ((1+Mod(m, n)^m) != 0, m++); m; \\ Michel Marcus, Dec 03 2023

cp's OEIS Frontend

A133081 An interpolation operator, companion to A133080.

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A309805 Maximum number of nonattacking kings placeable on a hexagonal board with edge-length n in Glinski's hexagonal chess.

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A009661 Smallest number m such that m^m+1 is divisible by n.

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Author

Keywords

Comments

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PARI