A087508 -id:A087508 - cp's OEIS frontend

A157897 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-3,k-3) + delta(n,0)*delta(k,0), T(n,k<0) = T(n

1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 3, 1, 2, 0, 1, 4, 3, 3, 2, 0, 1, 5, 6, 5, 6, 0, 1, 1, 6, 10, 9, 12, 3, 3, 0, 1, 7, 15, 16, 21, 12, 6, 3, 0, 1, 8, 21, 27, 35, 30, 14, 12, 0, 1, 1, 9, 28, 43, 57, 61, 35, 30, 6, 4, 0, 1, 10, 36, 65, 91, 111, 81, 65, 30, 10, 4, 0, 1, 11, 45, 94, 142, 189, 169, 135, 90, 30, 20, 0, 1

Offset: 0

Gary W. Adamson, Mar 08 2009

T(n, k) is the number of tilings of an n-board that use k (1/2, 1)-fences and n-k squares. A (1/2, 1)-fence is a tile composed of two pieces of width 1/2 separated by a gap of width 1. (Result proved in paper by K. Edwards - see the links section.) - Michael A. Allen, Apr 28 2019

T(n, k) is the (n, n-k)-th entry in the (1/(1-x^3), x*(1+x)/(1-x^3)) Riordan array. - Michael A. Allen, Mar 11 2021

			First few rows of the triangle are:
  1;
  1,  0;
  1,  1,  0;
  1,  2,  0,  1;
  1,  3,  1,  2,  0;
  1,  4,  3,  3,  2,  0;
  1,  5,  6,  5,  6,  0,  1;
  1,  6, 10,  9, 12,  3,  3,  0;
  1,  7, 15, 16, 21, 12,  6,  3,  0;
  1,  8, 21, 27, 35, 30, 14, 12,  0,  1;
  ...
T(9,3) = 27 = T(8,3) + T(7,2) + T(6,0) = 16 + 10 + 1.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
K. Edwards, A Pascal-like triangle related to the tribonacci numbers, Fib. Q., 46/47 (2008/2009), 18-25.
Kenneth Edwards and Michael A. Allen, New combinatorial interpretations of the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers using two types of tile, J. Int. Seq. 24 (2021) Article 21.3.8.

Cf. A000073 (row sums), A006498, A120415.
Cf. A001477, A079978, A087508, A120415, A161680.
Other triangles related to tiling using fences: A059259, A123521, A335964.

function T(n,k) // T = A157897
  if k lt 0 or k gt n then return 0;
  elif k eq 0 then return 1;
  else return T(n-1, k) + T(n-2, k-1) + T(n-3, k-3);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..14]]; // G. C. Greubel, Sep 01 2022

T[n_,k_]:= If[nMichael A. Allen, Apr 28 2019 *)

def T(n,k): # T = A157897
    if (k<0 or k>n): return 0
    elif (k==0): return 1
    else: return T(n-1, k) + T(n-2, k-1) + T(n-3, k-3)
flatten([[T(n,k) for k in (0..n)] for n in (0..14)]) # G. C. Greubel, Sep 01 2022

T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-3,k-3) + delta(n,0)*delta(k,0), T(n,k<0) = T(n

Sum_{k=0..n} T(n, k) = A000073(n+2). - Reinhard Zumkeller, Jun 25 2009

From G. C. Greubel, Sep 01 2022: (Start)

T(n, k) = T(n-1, k) + T(n-2, k-1) + T(n-3, k-3), with T(n, 0) = 1.

T(n, n) = A079978(n).

T(n, n-1) = A087508(n), n >= 1.

T(n, 1) = A001477(n-1).

T(n, 2) = A161680(n-2).

Sum_{k=0..floor(n/2)} T(n-k, k) = A120415(n). (End)

Name clarified by Michael A. Allen, Apr 28 2019

Definition improved by Michael A. Allen, Mar 11 2021

A087509 Number of k such that (k*n) == 2 (mod 3) for 0 <= k <= n.

Original entry on oeis.org

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

Offset: 0

Paul Barry, Sep 11 2003

			a(8) = #{1,4,7} = 3.

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A087507, A087508.

{#-1,1+#,0}[[Mod[#,3,1]]]/3&/@Range[0, 99] (* Federico Provvedi, Jun 15 2021 *)
LinearRecurrence[{0,0,2,0,0,-1},{0,0,1,0,1,2},100] (* Harvey P. Dale, May 04 2023 *)

a(n) = sum(k=0, n, (k*n % 3)==2); \\ Michel Marcus, Sep 25 2017

a(n) = Sum_{k=0..n} [(k*n) == 2 (mod 3)];
a(n) = n - 2*(floor(n/3) + 1)*(1 - cos(2*Pi*n/3))/3 - floor(n/3)*(5 + 4*cos(2*Pi*n/3))/3.
a(n) = n - A087507(n) - A087508(n).
G.f.: x^2*(x^2+1) / ((x-1)^2*(x^2+x+1)^2). - Colin Barker, Mar 31 2013
a(n) = 2*sin(n*Pi/3)*(sqrt(3)*cos(n*Pi) + 2*n*sin(n*Pi/3))/9. - Wesley Ivan Hurt, Sep 24 2017

A087507 #{0<=k<=n: k*n is divisible by 3}.

Original entry on oeis.org

1, 1, 1, 4, 2, 2, 7, 3, 3, 10, 4, 4, 13, 5, 5, 16, 6, 6, 19, 7, 7, 22, 8, 8, 25, 9, 9, 28, 10, 10, 31, 11, 11, 34, 12, 12, 37, 13, 13, 40, 14, 14, 43, 15, 15, 46, 16, 16, 49, 17, 17, 52, 18, 18, 55, 19, 19, 58, 20, 20, 61, 21, 21, 64, 22, 22, 67, 23, 23, 70, 24, 24, 73, 25, 25, 76

Offset: 0

Paul Barry, Sep 11 2003

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A016777 (trisection).

Vec((2*x^3+x^2+x+1)/((x-1)^2*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, May 02 2015

a(n) = sum{k=0..n, if (mod(kn, 3)=0, 1, 0) }.
a(n) = floor(n/3)(5/3+4/3cos(2Pi*/3))+1.
a(n)+A087508(n)+A087509(n) = n.
a(n) = 2*a(n-3)-a(n-6) for n>5. - Colin Barker, May 02 2015
G.f.: (2*x^3+x^2+x+1) / ((x-1)^2*(x^2+x+1)^2). - Colin Barker, May 02 2015

A115265 Correlation triangle for floor((n+3)/3).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 4, 4, 4, 4, 2, 3, 4, 5, 7, 5, 4, 3, 3, 5, 6, 8, 8, 6, 5, 3, 3, 6, 7, 9, 11, 9, 7, 6, 3, 4, 6, 8, 12, 12, 12, 12, 8, 6, 4, 4, 7, 9, 13, 15, 15, 15, 13, 9, 7, 4

Offset: 0

Paul Barry, Jan 18 2006

Row sums are A115266. Diagonal sums are A115267.

T(2n,n) is A092353. T(2n,n)-T(2n,n+1)=A087508(n+1).

			Triangle begins
1;
1,1;
1,2,1;
2,2,2,2;
2,3,3,3,2;
2,4,4,4,4,2;
3,4,5,7,5,4,3;
3,5,6,8,8,6,5,3;
3,6,7,9,11,9,7,6,3;

T[n_, k_] := Sum[Boole[j <= k] * Floor[(k - j + 3)/3] * Boole[j <= n-k] * Floor[(n - k - j + 3)/3], {j, 0, n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 15 2017 *)

G.f.: (1+x+x^2)(1+xy+x^2*y^2)/((1-x^3)^2*(1-x^3*y^3)^2*(1-x^2*y)).

T(n, k) = sum{j=0..n, [j<=k]*floor((k-j+3)/3)*[j<=n-k]*floor((n-k-j+3)/3)}.

A128615 Expansion of x/(1 + x + x^2 - x^3 - x^4 - x^5).

Original entry on oeis.org

0, 1, -1, 0, 2, -2, 0, 3, -3, 0, 4, -4, 0, 5, -5, 0, 6, -6, 0, 7, -7, 0, 8, -8, 0, 9, -9, 0, 10, -10, 0, 11, -11, 0, 12, -12, 0, 13, -13, 0, 14, -14, 0, 15, -15, 0, 16, -16, 0, 17, -17, 0, 18, -18, 0, 19, -19

Offset: 0

Paul Barry, Mar 13 2007

Partial sums are 0,1,0,0,2,0,0,3,0,0,4,...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,-1,1,1,1).

Cf. A049347, A087508.

[Floor((n+3)/3)*((n+1) mod 3 -1): n in [0..40]]; // G. C. Greubel, Mar 26 2024

CoefficientList[Series[x/(1+x+x^2-x^3-x^4-x^5),{x,0,60}],x] (* or *) LinearRecurrence[{-1,-1,1,1,1},{0,1,-1,0,2},60] (* or *) Table[{0,n,-n},{n,20}]//Flatten (* Harvey P. Dale, Jul 15 2017 *)
Table[Floor[(n+3)/3]*(Mod[n+1,3] -1), {n,0,40}] (* G. C. Greubel, Mar 26 2024 *)

[((n+3)//3)*((n+1)%3 -1) for n in range(41)] # G. C. Greubel, Mar 26 2024

G.f.: x/((1-x)*(1+x+x^2)^2) = x*(1-x)/(1-x^3)^2.
a(n) = (1/9)*(1 - cos(2*Pi*n/3) + sqrt(3)*(2*n + 3)*sin(2*Pi*n/3)).
a(n) = floor((n+3)/3)*A049347(n+2). - G. C. Greubel, Mar 26 2024

Showing 1-5 of 5 results.

cp's OEIS Frontend

A157897 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-3,k-3) + delta(n,0)*delta(k,0), T(n,k<0) = T(n

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A087509 Number of k such that (k*n) == 2 (mod 3) for 0 <= k <= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A087507 #{0<=k<=n: k*n is divisible by 3}.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A115265 Correlation triangle for floor((n+3)/3).

Views

Author

Keywords

Comments

Examples

Programs

Mathematica

Formula

A128615 Expansion of x/(1 + x + x^2 - x^3 - x^4 - x^5).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula