A030511 -id:A030511 - cp's OEIS frontend

A069813 Maximum number of triangles in polyiamond with perimeter n.

1, 2, 3, 6, 7, 10, 13, 16, 19, 24, 27, 32, 37, 42, 47, 54, 59, 66, 73, 80, 87, 96, 103, 112, 121, 130, 139, 150, 159, 170, 181, 192, 203, 216, 227, 240, 253, 266, 279, 294, 307, 322, 337, 352, 367, 384, 399, 416, 433, 450, 467, 486, 503, 522, 541, 560, 579

Offset: 3

Winston C. Yang (winston(AT)cs.wisc.edu), Apr 30 2002

			a(10) = 16 because the maximum number of triangles in a polyiamond of perimeter 10 is 16.

Colin Barker, Table of n, a(n) for n = 3..1000
W. C. Yang, R. R. Meyer, Maximal and minimal polyiamonds, 2002.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A000105, A000577, A027709, A030511 (bisection), A057729, A067628.

R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( x^3*(x^2-x-1)*(x^2+1)/((x-1)^3*(x+1)*(x^2+x+1)))); // Marius A. Burtea, Jan 03 2020

A069813 := proc(n)
    round(n^2/6) ;
    if modp(n,6) <> 0 then
        %-1 ;
    else
        % ;
    end if;
end proc: # R. J. Mathar, Jul 14 2015

LinearRecurrence[{1, 1, 0, -1, -1, 1}, {1, 2, 3, 6, 7, 10}, 60] (* Jean-François Alcover, Jan 03 2020 *)

a(n) = round(n^2/6) - (n % 6 != 0) \\ Michel Marcus, Jul 17 2013

Vec(x^3*(x^2-x-1)*(x^2+1)/((x-1)^3*(x+1)*(x^2+x+1)) + O(x^60)) \\ Colin Barker, Jan 19 2015

a(n) = round(n^2/6) - (0 if n = 0 mod 6, 1 else) = A056829(n)-A097325(n).
From Colin Barker, Jan 18 2015: (Start)
a(n) = round((-25 + 9*(-1)^n + 8*exp(-2/3*i*n*Pi) + 8*exp((2*i*n*Pi)/3) + 6*n^2)/36), where i=sqrt(-1).
G.f.: x^3*(1+x-x^2)*(1+x^2) / ((1-x)^3*(1+x)*(1+x+x^2)). (End)
a(n) = A001399(n-3) + A001399(n-4) + A001399(n-6) - A001399(n-7). - R. J. Mathar, Jul 14 2015

A194069 1+floor((2/3)*n^2).

Original entry on oeis.org

1, 3, 7, 11, 17, 25, 33, 43, 55, 67, 81, 97, 113, 131, 151, 171, 193, 217, 241, 267, 295, 323, 353, 385, 417, 451, 487, 523, 561, 601, 641, 683, 727, 771, 817, 865, 913, 963, 1015, 1067, 1121, 1177, 1233, 1291, 1351, 1411

Offset: 1

Clark Kimberling, Aug 14 2011

nonn

Cf. A194070 (natural fractal sequence),

A194071 (natural fractal interspersion).

c[k_]:=1+Floor[(2/3)k^2]; c=Table[c[k],{k,1,90}]

a(n)=1+floor(2*n^2/3).

a(n) = 1 +A030511(n+1). - R. J. Mathar, Aug 25 2011

A358163 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = ij - ceiling(ij/3).

Original entry on oeis.org

1, 0, 1, 30, 1272, 113224, 18615680, 4299553536, 1507609286784, 781464165813504, 525599814806986752, 473934337123421786112, 567876971785035135320064, 837723761443461191423754240, 1549608938859438129393893376000, 3582000047767392376356107059200000, 9838495669776145718724862743674880000

Offset: 0

Stefano Spezia, Nov 01 2022

nonn

The matrix M(n) is the n-th principal submatrix of the rectangular array A143979.

det(M(0)) = 1, det(M(2)) = -1, det(M(3)) = 2, and otherwise det(M(n)) = 0.

			a(5) = 113224:
    0  1   2   2   3
    1  2   4   5   6
    2  4   6   8  10
    2  5   8  10  13
    3  6  10  13  16

Cf. A143979.

Cf. A030511 (matrix element M[n-1,n-1]), A358164 (hafnian of M(2*n)).

a[n_]:=Permanent[Table[i*j-Ceiling[i*j/3],{i,n},{j,n}]]; Join[{1},Array[a,16]]

from fractions import Fraction
from sympy import Matrix
def A358163(n): return Matrix(n,n,[i*j-Fraction(i*j,3)._ceil_() for i in range(1,n+1) for j in range(1,n+1)]).per() if n else 1 # Chai Wah Wu, Nov 02 2022

A071408 a(n+1) - 2a(n) + a(n-1) = (2/3)(1 + w^(n+1) + w^(2*n+2)) with a(1)=0, a(2)=1, and where w is the imaginary cubic root of unity.

Original entry on oeis.org

0, 1, 4, 7, 10, 15, 20, 25, 32, 39, 46, 55, 64, 73, 84, 95, 106, 119, 132, 145, 160, 175, 190, 207, 224, 241, 260, 279, 298, 319, 340, 361, 384, 407, 430, 455, 480, 505, 532, 559, 586, 615, 644, 673, 704, 735, 766, 799, 832, 865, 900, 935, 970, 1007, 1044

Offset: 1

Robert G. Wilson v, Jun 24 2002

w = exp(2*Pi*i/3)= (-1 - sqrt(-3))/2. Beginning with a(2) the first differences are 3,3,3,5,5,5,7,7,7,9,9,9,11, etc.

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A071618.

a[1] = 0; a[2] = 1; w = Exp[2Pi*I/3]; a[n_] := a[n] = Simplify[(2/3)(1 + w^n + w^(2n)) + 2a[n - 1] - a[n - 2]]; Table[ a[n], {n, 1, 60}]
Table[If[n<3,n-1,Floor[((n+1)^2-4)/3]],{n,1,100}] (*  Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)
LinearRecurrence[{2,-1,1,-2,1},{0,1,4,7,10},60] (* Harvey P. Dale, Jun 10 2016 *)

a(n)=n*(n+2)\3 - 1 \\ Charles R Greathouse IV, Mar 02 2017

a(n) = A032765(n)-1.
a(n) = floor((n-1)*(n+1)*(n+3)/(3*n+3)). - Gary Detlefs, Jul 13 2010
a(n) = (n-1)^2 - A030511(n-1). - Wesley Ivan Hurt, Jun 19 2013
G.f.: x^2*(1+x)*(x^2-x-1) / ( (1+x+x^2)*(x-1)^3 ). - R. J. Mathar, Jun 23 2013
a(n) = n + floor(n*(n-1)/3) - 1. - Bruno Berselli, Mar 02 2017

A358164 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = ij - ceiling(ij/3).

Original entry on oeis.org

1, 1, 26, 2704, 698568, 384890688, 378771904512, 597991783196160, 1450380828625459200, 5077825865646165964800, 24487520383436615392204800

Offset: 0

Stefano Spezia, Nov 01 2022

The matrix M(n) is the n-th principal submatrix of the rectangular array A143979.

			a(2) = 26:
    0  1   2   2
    1  2   4   5
    2  4   6   8
    2  5   8  10

Wikipedia, Hafnian
Wikipedia, Symmetric matrix

Cf. A143979.

Cf. A030511 (matrix element M[n-1,n-1]), A358163 (permanent of M(n)).

M[i_, j_, n_]:=Part[Part[Table[r*c-Ceiling[r*c/3], {r, n}, {c, n}], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]

tm(n) = matrix(n, n, i, j, i*j - ceil((i*j)/3));
a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ Michel Marcus, May 02 2023

a(6) from Michel Marcus, May 02 2023

a(7)-a(10) from Pontus von Brömssen, Oct 15 2023

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A069813 Maximum number of triangles in polyiamond with perimeter n.

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A194069 1+floor((2/3)*n^2).

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A358163 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = i*j - ceiling(i*j/3).

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A071408 a(n+1) - 2*a(n) + a(n-1) = (2/3)*(1 + w^(n+1) + w^(2*n+2)) with a(1)=0, a(2)=1, and where w is the imaginary cubic root of unity.

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A358164 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = i*j - ceiling(i*j/3).

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A358163 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = ij - ceiling(ij/3).

A071408 a(n+1) - 2a(n) + a(n-1) = (2/3)(1 + w^(n+1) + w^(2*n+2)) with a(1)=0, a(2)=1, and where w is the imaginary cubic root of unity.

A358164 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = ij - ceiling(ij/3).