A371967 -id:A371967 - cp's OEIS frontend

A064225 a(n) = (9n^2 + 5n + 2)/2.

1, 8, 24, 49, 83, 126, 178, 239, 309, 388, 476, 573, 679, 794, 918, 1051, 1193, 1344, 1504, 1673, 1851, 2038, 2234, 2439, 2653, 2876, 3108, 3349, 3599, 3858, 4126, 4403, 4689, 4984, 5288, 5601, 5923, 6254, 6594, 6943, 7301, 7668, 8044, 8429, 8823, 9226

Offset: 0

N. J. A. Sloane, Sep 22 2001

Diagonal of triangular spiral in A051682. - Michael Somos, Jul 22 2006
Ehrhart polynomial of closed quadrilateral with vertices (0,2),(2,3),(3,1),(2,0). - Michael Somos, Jul 22 2006
In the natural number array A000027 this sequence is the first knight moves diagonal (A081267 is the second, A001844 is the main diagonal). It can be used to define this diagonal for any array: A007318(A064225-1)=A005809 (Subtraction by 1 because A007318 is defined with offset 0.) - Tilman Piesk, Mar 24 2012
Or positions of pentagonal numbers, such that p(a(n)) = p(a(n)-1) + p(3*n+1), where p=A000326. - Vladimir Shevelev, Jan 21 2014

			Illustration of initial terms:
.
.                                    o
.                                 o o
.                      o       o o o o
.                   o o     o o o o o
.           o    o o o o     o o o o o
.        o o      o o o     o o o o o
.   o     o o    o o o o     o o o o o
.        o o      o o o     o o o o o
.           o    o o o o     o o o o o
.                   o o     o o o o o
.                      o       o o o o
.                                 o o
.                                    o
.
.   1     8        24           49
- _Aaron David Fairbanks_, Feb 23 2025

Harry J. Smith, Table of n, a(n) for n = 0..1000
National Security Agency, Intrigued? (advertisement), Notices of the Amer. Math. Soc., vol. 49 (2002), p. 216.
J. A. Siehler, Selections without adjacency on a rectangular grid, arXiv:1409.3869, Table 3, k=2 (different offset)
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000027, A000326, A001844, A005809, A007318, A051682, A064226, A081267, A235332.

Column w=2 of A371967.

Table[(9n^2+5n+2)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,8,24},51] (* Harvey P. Dale, Sep 13 2011 *)

{a(n) = 1 + n * (9*n + 5) / 2}; /* Michael Somos, Jul 22 2006 */

(define (A064225 n) (/ (+ (* 9 n n) (* 5 n) 2) 2))

a(n) = 9*n+a(n-1)-2, with n>0, a(0) = 1. - Vincenzo Librandi, Aug 07 2010
a(0)=1, a(1)=8, a(2)=24, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Sep 13 2011
G.f.: (1+5*x+3*x^2)/(1-x)^3. - Colin Barker, Feb 23 2012
A064226(n) = a(-1-n). - Michael Somos, Jul 22 2006 (While the sequence itself is only one-way infinite, this identity works, as the defining formula (in the Name-field) produces integers also for the negative values of n, -1, -2, -3, etc.) - Antti Karttunen, Mar 24 2012
E.g.f.: exp(x)*(2 + 14*x + 9*x^2)/2. - Stefano Spezia, Dec 25 2022

A051736 Number of 3 X n (0,1)-matrices with no consecutive 1's in any row or column.

Original entry on oeis.org

1, 5, 17, 63, 227, 827, 2999, 10897, 39561, 143677, 521721, 1894607, 6879979, 24983923, 90725999, 329460929, 1196397873, 4344577397, 15776816033, 57291635519, 208047769363, 755500774443, 2743511349031, 9962735709201, 36178491743225, 131377896967213, 477083233044745

Offset: 0

Stephen G Penrice, Dec 06 1999

Also the number of independent vertex sets and vertex covers in the 3 X n grid graph. - Eric W. Weisstein, Sep 21 2017

			There are five 3 X 1 (0,1)-matrices with no consecutive 1's:
  0 0 0
  0 0 1
  0 1 0
  1 0 0
  1 0 1
There are 17 3 X 2 (0,1)-matrices with no consecutive 1's:
0 0, 0 1, 0 0, 0 0, 0 1, 1 0, 1 0, 1 0, 0 0, 0 1, 0 0, 0 1, 0 0, 0 1, 0 0, 1 0, 1 0
0 0, 0 0, 0 1, 0 0, 0 0, 0 0, 0 1, 0 0, 1 0, 1 0, 1 0, 1 0, 0 0, 0 0, 0 1, 0 0, 0 1
0 0, 0 0, 0 0, 0 1, 0 1, 0 0, 0 0, 0 1, 0 0, 0 0, 0 1, 0 1, 1 0, 1 0, 1 0, 1 0, 1 0

Reinhard Zumkeller, Table of n, a(n) for n = 0..999
N. J. Calkin and H. S. Wilf, The number of independent sets in a grid graph, preprint, SIAM J. Discrete Math., 11(1), 54-60.
N. J. Calkin and H. S. Wilf, The number of independent sets in a grid graph, SIAM J. Discrete Math, 11 (1998) 54-60.
Reinhardt Euler, The Fibonacci Number of a Grid Graph and a New Class of Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.6.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover
Index entries for linear recurrences with constant coefficients, signature (2,6,0,-1).

Row 3 of A089934. Row sums of A371967.

Cf. A051737.

a051736 n = a051736_list !! (n-1)
a051736_list = 1 : 5 : 17 : 63 : zipWith (-) (map (* 2) $ drop 2 $
   zipWith (+) (map (* 3) a051736_list) (tail a051736_list)) a051736_list
-- Reinhard Zumkeller, Apr 02 2012

LinearRecurrence[{2, 6, 0, -1}, {1, 5, 17, 63}, 40] (* Harvey P. Dale, Mar 05 2013 *)
CoefficientList[Series[(1 + 3 x + x^2 - x^3)/(1 - 2 x - 6 x^2 + x^4), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 21 2017 *)
Table[-RootSum[1 - 6 #1^2 - 2 #1^3 + #1^4 &, 263 #1^n - 657 #1^(n + 1) - 331 #1^(n + 2) + 81 #1^(n + 3) &]/1994, {n, 0, 20}] (* Eric W. Weisstein, Sep 21 2017 *)

Vec((1+3*x+x^2-x^3)/(1-2*x-6*x^2+x^4)+O(x^50)) \\ Michel Marcus, Sep 17 2014

a(n) = 2*a(n-1) + 6*a(n-2) - a(n-4).

G.f.: (1+x)*(1+2*x-x^2)/(1-2*x-6*x^2+x^4). - Philippe Deléham, Sep 07 2006

More terms from James Sellers, Dec 08 1999
More terms from Michel Marcus, Sep 17 2014
Offset fixed by Eric W. Weisstein, Sep 21 2017

A172229 Number of ways to place 3 nonattacking wazirs on a 3 X n board.

Original entry on oeis.org

0, 2, 22, 84, 215, 442, 792, 1292, 1969, 2850, 3962, 5332, 6987, 8954, 11260, 13932, 16997, 20482, 24414, 28820, 33727, 39162, 45152, 51724, 58905, 66722, 75202, 84372, 94259, 104890, 116292, 128492, 141517, 155394, 170150, 185812, 202407, 219962, 238504

Offset: 1

Vaclav Kotesovec, Jan 29 2010

A wazir is a (fairy chess) leaper [0,1].

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
R. J. Mathar, Bivariate generating functions for non-attacking wazirs on rectangular boards, viXra:2404.0122 (2024) Section 3.
Eric Weisstein's World of Mathematics, Grid Graph
Wikipedia, Wazir (chess)

Cf. A172226, A061989.

Column w=3 of A371967.

CoefficientList[Series[x (3 x^3 + 8 x^2 + 14 x + 2) / (x - 1)^4, {x, 0, 50}], x] (* Vincenzo Librandi, May 28 2013 *)

a(n) = (3*n - 5)*(3*n^2 - 8*n + 8)/2, n>=2.

G.f.: x^2*(3*x^3+8*x^2+14*x+2)/(x-1)^4. - Vaclav Kotesovec, Mar 25 2010

A371978 Number of ways of placing n non-attacking wazirs on a 3 X n board.

Original entry on oeis.org

1, 3, 8, 22, 61, 174, 504, 1478, 4374, 13035, 39062, 117585, 355279, 1076845, 3272692, 9969385, 30430982, 93055869, 285013326, 874193006, 2684778104, 8254967674, 25408703236, 78283452265, 241403160254, 745024894092, 2301051484006, 7111897305089, 21995136183906

Offset: 0

Alois P. Heinz, Apr 14 2024

nonn

			a(2) = 8:
  +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+
  | W . | | W . | | W . | | . W | | . W | | . W | | . . | | . . |
  | . W | | . . | | . . | | W . | | . . | | . . | | W . | | . W |
  | . . | | W . | | . W | | . . | | W . | | . W | | . W | | W . |
  +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ .

Alois P. Heinz, Table of n, a(n) for n = 0..2011
Wikipedia, Wazir (chess)

Main diagonal of A371967.

Cf. A201511, A371979.

b:= proc(n, l) option remember; `if`(n=0, 1,
      add(`if`(Bits[And](j, l)>0, 0, expand(b(n-1, j)*
      x^add(i, i=Bits[Split](j)))), j=[0, 1, 2, 4, 5]))
    end:
a:= n-> coeff(b(n, 0), x, n):
seq(a(n), n=0..30);

a(n) = A371967(n,n).
From Vaclav Kotesovec, Apr 16 2024: (Start)
Recurrence: (n+1)*(72*n^4 - 700*n^3 + 2288*n^2 - 2803*n + 796)*a(n) = 2*(144*n^5 - 1328*n^4 + 3814*n^3 - 3083*n^2 - 1479*n + 1194)*a(n-1) - 2*(72*n^5 - 700*n^4 + 2050*n^3 - 1979*n^2 + 409*n + 16)*a(n-2) - 4*(36*n^5 - 368*n^4 + 1437*n^3 - 2421*n^2 + 1398*n + 95)*a(n-3) - (72*n^5 - 772*n^4 + 2404*n^3 - 1365*n^2 - 4749*n + 5704)*a(n-4) + 2*(72*n^5 - 808*n^4 + 2858*n^3 - 3067*n^2 - 1494*n + 2666)*a(n-5) - (n-6)*(72*n^4 - 412*n^3 + 620*n^2 - 39*n - 347)*a(n-6).
a(n) ~ sqrt(c) * d^n / sqrt(Pi*n), where d = (188 + 12*sqrt(93))^(1/3)/6 + 14/(3*(188 + 12*sqrt(93))^(1/3)) + 4/3 and c = 11/6 + (1465336244224 - 5597165568*sqrt(93))^(1/3)/5952 + ((23080523 + 88161*sqrt(93))/2)^(1/3) / (12*31^(2/3)). (End)

A371979 Number of ways of placing k non-attacking wazirs on a 3 X n board, where k is chosen so as to maximize this number.

Original entry on oeis.org

1, 3, 8, 24, 84, 276, 880, 3063, 10692, 36257, 121580, 436847, 1530534, 5259906, 18389910, 65748491, 230935493, 799429185, 2860613606, 10203350814, 35899202776, 125660232367, 453360413253, 1614905346286, 5688690345179, 20241845359246, 72805688610204

Offset: 0

Alois P. Heinz, Apr 14 2024

nonn

			a(3) = 24 = A371967(3,2):
  +-------+ +-------+ +-------+ +-------+ +-------+ +-------+
  | W . W | | W . . | | W . . | | W . . | | W . . | | W . . |
  | . . . | | . W . | | . . W | | . . . | | . . . | | . . . |
  | . . . | | . . . | | . . . | | W . . | | . W . | | . . W |
  +-------+ +-------+ +-------+ +-------+ +-------+ +-------+
  +-------+ +-------+ +-------+ +-------+ +-------+ +-------+
  | . W . | | . W . | | . W . | | . W . | | . W . | | . . W |
  | W . . | | . . W | | . . . | | . . . | | . . . | | W . . |
  | . . . | | . . . | | W . . | | . W . | | . . W | | . . . |
  +-------+ +-------+ +-------+ +-------+ +-------+ +-------+
  +-------+ +-------+ +-------+ +-------+ +-------+ +-------+
  | . . W | | . . W | | . . W | | . . W | | . . . | | . . . |
  | . W . | | . . . | | . . . | | . . . | | W . W | | W . . |
  | . . . | | W . . | | . W . | | . . W | | . . . | | . W . |
  +-------+ +-------+ +-------+ +-------+ +-------+ +-------+
  +-------+ +-------+ +-------+ +-------+ +-------+ +-------+
  | . . . | | . . . | | . . . | | . . . | | . . . | | . . . |
  | W . . | | . W . | | . W . | | . . W | | . . W | | . . . |
  | . . W | | W . . | | . . W | | W . . | | . W . | | W . W |
  +-------+ +-------+ +-------+ +-------+ +-------+ +-------+ .

Alois P. Heinz, Table of n, a(n) for n = 0..1788
Wikipedia, Wazir (chess)

Row maxima of A371967.

Cf. A371978.

b:= proc(n, l) option remember; `if`(n=0, 1,
      add(`if`(Bits[And](j, l)>0, 0, expand(b(n-1, j)*
      x^add(i, i=Bits[Split](j)))), j=[0, 1, 2, 4, 5]))
    end:
a:= n-> max(coeffs(b(n, 0))):
seq(a(n), n=0..30);

cp's OEIS Frontend

A064225 a(n) = (9*n^2 + 5*n + 2)/2.

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A051736 Number of 3 X n (0,1)-matrices with no consecutive 1's in any row or column.

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A172229 Number of ways to place 3 nonattacking wazirs on a 3 X n board.

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A371978 Number of ways of placing n non-attacking wazirs on a 3 X n board.

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A371979 Number of ways of placing k non-attacking wazirs on a 3 X n board, where k is chosen so as to maximize this number.

Views

Author

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Examples

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Programs

Maple

A064225 a(n) = (9n^2 + 5n + 2)/2.