Dimitri Boscainos - cp's OEIS frontend

A273220 a(n) = 8n^2 - 12n + 1.

9, 37, 81, 141, 217, 309, 417, 541, 681, 837, 1009, 1197, 1401, 1621, 1857, 2109, 2377, 2661, 2961, 3277, 3609, 3957, 4321, 4701, 5097, 5509, 5937, 6381, 6841, 7317, 7809, 8317, 8841, 9381, 9937, 10509, 11097, 11701, 12321, 12957, 13609, 14277, 14961, 15661

Offset: 2

Dimitri Boscainos, May 18 2016

Sequence may be obtained by starting with the segment (9, 37) followed by the line from 37 in the direction 37, 81,... in the square spiral whose vertices are the generalized hexagonal numbers (A000217). - Omar E. Pol, Jun 26 2016

Colin Barker, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Table[8 n^2 - 12 n + 1, {n, 2, 45}] (* or *)
Drop[#, 2] &@ CoefficientList[Series[x^2 (9 + 10 x - 3 x^2)/(1 - x)^3, {x, 0, 45}], x] (* Michael De Vlieger, Jun 26 2016 *)

Vec(x^2*(9+10*x-3*x^2)/(1-x)^3 + O(x^50)) \\ Colin Barker, May 18 2016

From Colin Barker, May 18 2016: (Start)
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>4.
G.f.: x^2*(9+10*x-3*x^2) / (1-x)^3.
(End)

A261360 Pentatope of coefficients in expansion of (1 + 2x + 2y + 2*z)^n.

Original entry on oeis.org

1, 1, 2, 2, 2, 1, 4, 4, 4, 4, 8, 8, 4, 8, 4, 1, 6, 6, 6, 12, 24, 24, 12, 24, 12, 8, 24, 24, 24, 48, 24, 8, 24, 24, 8, 1, 8, 8, 8, 24, 48, 48, 24, 48, 24, 32, 96, 96, 96, 192, 96, 32, 96, 96, 32, 16, 64, 64, 96, 192, 96, 64, 192, 192, 64, 16, 64, 96, 64, 96, 1, 10, 10, 10, 40, 80, 80, 40, 80, 40, 80, 240, 240, 240, 480, 240, 80, 240, 240, 80, 80, 320, 320, 480, 960, 480, 320, 960, 960, 320, 80, 320, 480, 320, 80, 32, 160, 160, 320, 640, 320, 320, 960, 960, 320, 160, 640, 960, 640, 160, 32, 160, 320, 320, 160, 32

Offset: 0

Dimitri Boscainos, Aug 16 2015

T(n,i,j,k) is the number of lattice paths from (0,0,0,0) to (n,i,j,k) with steps (1,0,0,0) and two kinds of steps (1,1,0,0), (1,1,1,0) and (1,1,1,1).

The sum of the numbers in each cell of the pentatope is 7^n (A000420).

			The 5th slice (n=4) of this 4D simplex starts at a(35). It comprises a 3D tetrahedron of 35 terms whose sum is 2401. It is organized as follows:
.
.               1
.
.               8
.            8     8
.
.              24
.           48    48
.        24    48    24
.
.              32
.           96    96
.        96   192    96
.     32    96    96    32
.
.              16
.           64    64
.        96   192    96
.     64   192   192    64
.  16    64    96    64    16

Cf. A000420, A189225, A261358, A261359.

p:= proc(i, j, k, l) option remember;
      if l<0 or j<0 or i<0 or i>l or j>i or k<0 or k>j then 0
    elif {i, j, k, l}={0} then 1
    else p(i, j, k, l-1) +2*p(i-1, j, k, l-1) +2*p(i-1, j-1, k, l-1)+2*p(i-1, j-1, k-1, l-1)
      fi
    end:
seq(seq(seq(seq(p(i, j, k, l), k=0..j), j=0..i), i=0..l), l=0..5);
# Adapted from Alois P. Heinz's Maple program for A261356

lista(nn) = {for (n=0, nn, for (i=0, n, for (j=0, i, for (k=0, j, print1(2^i*binomial(n,i)*binomial(i,j)*binomial(j,k), ", ")););););} \\ Michel Marcus, Oct 07 2015

T(i+1,j,k,l) = 2*T(i,j-1,k-1,l-1) + 2*T(i,j-1,k-1,l) + 2*T(i,j-1,k,l) + T(i,j,k,l); T(i,j,k,-1)=0, ...; T(0,0,0,0)=1.

T(n,i,j,k) = 2^i*binomial(n,i)*binomial(i,j)*binomial(j,k). - Dimitri Boscainos, Aug 21 2015

A261358 Pentatope of coefficients in expansion of (1 + x + y + 2*z)^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 4, 1, 4, 4, 1, 3, 3, 6, 3, 6, 12, 3, 12, 12, 1, 3, 6, 3, 12, 12, 1, 6, 12, 8, 1, 4, 4, 8, 6, 12, 24, 6, 24, 24, 4, 12, 24, 12, 48, 48, 4, 24, 48, 32, 1, 4, 8, 6, 24, 24, 4, 24, 48, 32, 1, 8, 24, 32, 16, 1, 5, 5, 10, 10, 20, 40, 10, 40, 40, 10, 30, 60, 30, 120, 120, 30, 120, 240, 160, 5, 40, 120, 160, 80, 1, 5, 10, 10, 40, 40, 10, 60, 120, 80, 5, 40, 120, 160, 80, 1, 10, 40, 80, 80, 32

Offset: 0

Dimitri Boscainos, Aug 16 2015

T(n,i,j,k) is the number of lattice paths from (0,0,0,0) to (n,i,j,k) with steps (1,0,0,0),(1,1,0,0),(1,1,1,0) and two kinds of steps (1,1,1,1).
The sum of the numbers in each cell of the pentatope is 5^n (A000351).
The sum of the antidiagonals of each triangle in each slice gives A261357. - Dimitri Boscainos, Aug 21 2015

			The 5th slice (n=4) of this 4D simplex starts at a(35). It comprises a 3D tetrahedron of 35 terms whose sum is 625. It is organized as follows:
.           1
.
.           4
.         4   8
.
.           6
.         12  24
.        6  24  24
.
.           4
.        12  24
.      12  48  48
.     4  24  48  32
.
.          1
.        4   8
.      6   24  24
.    4  24   48   32
.  1   8   24   32   16

Cf. A000351, A189225, A261357, A261359, A261360.

p:= proc(i, j, k, l) option remember;
      if l<0 or j<0 or i<0 or i>l or j>i or k<0 or k>j then 0
    elif {i, j, k, l}={0} then 1
    else p(i, j, k, l-1) +p(i-1, j, k, l-1) +p(i-1, j-1, k, l-1)+2*p(i-1, j-1, k-1, l-1)
      fi
    end:
seq(seq(seq(seq(p(i, j, k, l), k=0..j), j=0..i), i=0..l), l=0..5);
# Adapted from Alois P. Heinz's Maple program for A261356

T(i+1,j,k,l) = 2*T(i,j-1,k-1,l-1) + T(i,j-1,k-1,l) + T(i,j-1,k,l) + T(i,j,k,l); a(i,j,k,-1)=0,...; a(0,0,0,0)=1.

T(n,i,j,k) = 2^k*binomial(n,i)*binomial(i,j)*binomial(j,k). - Dimitri Boscainos, Aug 21 2015

A261359 Pentatope of coefficients in expansion of (1 + x + 2y + 2z)^n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 4, 4, 1, 4, 4, 4, 8, 4, 1, 3, 6, 6, 3, 12, 12, 12, 24, 12, 1, 6, 6, 12, 24, 12, 8, 24, 24, 8, 1, 4, 8, 8, 6, 24, 24, 24, 48, 24, 4, 24, 24, 48, 96, 48, 32, 96, 96, 32, 1, 8, 8, 24, 48, 24, 32, 96, 96, 32, 16, 64, 96, 64, 16, 1, 5, 10, 10, 10, 40, 40, 40, 80, 40, 10, 60, 60, 120, 240, 120, 80, 240, 240, 80, 5, 40, 40, 120, 240, 120, 160, 480, 480, 160, 80, 320, 480, 320, 80, 1, 10, 10, 40, 80, 40, 80, 240, 240, 80, 80, 320, 480, 320, 80, 32, 160, 320, 320, 160, 32

Offset: 0

Dimitri Boscainos, Aug 16 2015

T(n,i,j,k) is the number of lattice paths from (0,0,0,0) to (n,i,j,k) with steps (1,0,0,0), (1,1,0,0) and two kinds of steps (1,1,1,0) and (1,1,1,1).

The sum of the numbers in each cell of the pentatope is 6^n (A000400).

			The 5th slice (n=4) of this 4D simplex starts at a(35). It comprises a 3D tetrahedron of 35 terms whose sum is 1296. It is organized as follows:
.
.           1
.
.           4
.         8   8
.
.           6
.        24  24
.      24  48  24
.
.           4
.        24  24
.      48  96  48
.    32  96  96  32
.
.           1
.         8   8
.      24  48  24
.    32  96  96  32
.  16  64  96  64  16

Cf. A000400, A189225, A261358, A261360.

p:= proc(i, j, k, l) option remember;
      if l<0 or j<0 or i<0 or i>l or j>i or k<0 or k>j then 0
    elif {i, j, k, l}={0} then 1
    else p(i, j, k, l-1) +p(i-1, j, k, l-1) +2*p(i-1, j-1, k, l-1)+2*p(i-1, j-1, k-1, l-1)
      fi
    end:
seq(seq(seq(seq(p(i, j, k, l), k=0..j), j=0..i), i=0..l), l=0..5);
# Adapted from Alois P. Heinz's Maple program for A261356

lista(nn) = {for (n=0, nn, for (i=0, n, for (j=0, i, for (k=0, j, print1(2^j*binomial(n,i)*binomial(i,j)*binomial(j,k), ", ")););););} \\ Michel Marcus, Oct 07 2015

T(i+1,j,k,l) = 2*T(i,j-1,k-1,l-1) + 2*T(i,j-1,k-1,l) + T(i,j-1,k,l) + T(i,j,k,l); T(i,j,k,-1)=0,...; T(0,0,0,0)=1.

T(n,i,j,k) = 2^j*binomial(n,i)*binomial(i,j)*binomial(j,k). - Dimitri Boscainos, Aug 21 2015

A261357 Pyramid of coefficients in expansion of (1 + 2x + 2y)^n.

Original entry on oeis.org

1, 1, 2, 2, 1, 4, 4, 4, 8, 4, 1, 6, 6, 12, 24, 12, 8, 24, 24, 8, 1, 8, 8, 24, 48, 24, 32, 96, 96, 32, 16, 64, 96, 64, 16, 1, 10, 10, 40, 80, 40, 80, 240, 240, 80, 80, 320, 480, 320, 80, 32, 160, 320, 320, 160, 32

Offset: 0

Dimitri Boscainos, Aug 16 2015

T(n,j,k) is the number of lattice paths from (0,0,0) to (n,j,k) with steps (1,0,0), two kinds of steps (1,1,0) and two kinds of steps (1,1,1).
The sum of the numbers in each slice of the pyramid is 5^n.
The terms of the j-th row of the n-th slice of this pyramid are the sum of the terms in each antidiagonal of the j-th triangle of the n-th slice of A261358. - Dimitri Boscainos, Aug 21 2015

			Here is the fourth (n=3) slice of the pyramid:
        1
      6   6
   12  24  12
  8  24  24   8

Alois P. Heinz, Rows n = 0..38, flattened

Cf. A000079, A046816, A059304, A261356, A261358.

p:= proc(i, j, k) option remember;
      if k<0 or i<0 or i>k or j<0 or j>i then 0
    elif {i, j, k}={0} then 1
    else p(i, j, k-1) +2*p(i-1, j, k-1) +2*p(i-1, j-1, k-1)
      fi
    end:
seq(seq(seq(p(i, j, k), j=0..i), i=0..k), k=0..5);
# Adapted from Alois P. Heinz's Maple program for A261356

p[i_, j_, k_] := p[i, j, k] = If[k < 0 || i < 0 || i > k || j < 0 || j > i, 0, If[Union@{i, j, k} == {0}, 1, p[i, j, k - 1] + 2*p[i - 1, j, k - 1] + 2*p[i - 1, j - 1, k - 1]]];
Table[Table[Table[p[i, j, k], {j, 0, i}], {i, 0, k}], {k, 0, 5}] // Flatten (* Jean-François Alcover, Mar 17 2025, after Alois P. Heinz *)

tabf(nn) = {for (n=0, nn, for (j=0, n, for (k=0, j, print1(2^j*binomial(n,j)*binomial(j,k), ", ")); print();); print(););} \\ Michel Marcus, Oct 07 2015

T(i+1,j,k) = 2*T(i,j-1,k-1)+ 2*T(i,j-1,k) + T(i,j,k); T(i,j,-1) = 0, ...; T(0,0,0) = 1.

T(n,j,k) = 2^j*binomial(n,j)*binomial(j,k). - Dimitri Boscainos, Aug 21 2015

A261356 Pyramid of coefficients in expansion of (1+x+2*y)^n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 4, 1, 4, 4, 1, 3, 6, 3, 12, 12, 1, 6, 12, 8, 1, 4, 8, 6, 24, 24, 4, 24, 48, 32, 1, 8, 24, 32, 16, 1, 5, 10, 10, 40, 40, 10, 60, 120, 80, 5, 40, 120, 160, 80, 1, 10, 40, 80, 80, 32, 1, 6, 12, 15, 60, 60, 20, 120, 240, 160, 15, 120, 360, 480

Offset: 0

Dimitri Boscainos, Aug 16 2015

T(n,j,k) is the number of lattice paths from (0,0,0) to (n,j,k) with steps (1,0,0), (1,1,0) and two kinds of steps (1,1,1).
The sum of the terms in each slice of the pyramid is 4^n (A000302).
The terms of the j-th row of the n-th slice of this pyramid are the sum of the terms in each row of the j-th triangle of the n-th slice of A189225. - Dimitri Boscainos, Aug 21 2015

			Here is the fourth (n=3) slice of the pyramid:
.....1......
...3   6....
..3  12  12.
.1  6  12  8
As an irregular triangle, rows begin:
1;
1, 1, 2;
1, 2, 4, 1, 4, 4;
1, 3, 6, 3, 12, 12, 1, 6, 12, 8;
...

Alois P. Heinz, Rows n = 0..38, flattened

Cf. A046816, A261357.

p:= proc(i, j, k) option remember;
      if k<0 or i<0 or i>k or j<0 or j>i then 0
    elif {i, j, k}={0} then 1
    else p(i, j, k-1) +p(i-1, j, k-1) +2*p(i-1, j-1, k-1)
      fi
    end:
seq(seq(seq(p(i, j, k), j=0..i), i=0..k), k=0..5);
# Alois P. Heinz, Aug 20 2015

p[i_, j_, k_] := p[i, j, k] = If[k < 0 || i < 0 || i > k || j < 0 || j > i, 0, If[Union@{i, j, k} == {0}, 1, p[i, j, k - 1] + p[i - 1, j, k - 1] + 2* p[i - 1, j - 1, k - 1]]];
Table[Table[Table[p[i, j, k], {j, 0, i}], {i, 0, k}], {k, 0, 5}] // Flatten (* Jean-François Alcover, Sep 16 2023, after Alois P. Heinz *)

T(i+1,j,k) = 2*T(i,j-1,k-1)+T(i,j-1,k)+T(i,j,k); T(i,j,-1)=0,...; T(0,0,0)=1.

T(n,j,k) = 2^k*binomial(n,j)*binomial(j,k). - Dimitri Boscainos, Aug 21 2015

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User: Dimitri Boscainos

A273220 a(n) = 8n^2 - 12n + 1.

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A261360 Pentatope of coefficients in expansion of (1 + 2*x + 2*y + 2*z)^n.

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A261358 Pentatope of coefficients in expansion of (1 + x + y + 2*z)^n.

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Programs

Maple

Formula

A261359 Pentatope of coefficients in expansion of (1 + x + 2*y + 2*z)^n.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

PARI

Formula

A261357 Pyramid of coefficients in expansion of (1 + 2*x + 2*y)^n.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A261356 Pyramid of coefficients in expansion of (1+x+2*y)^n.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A261360 Pentatope of coefficients in expansion of (1 + 2x + 2y + 2*z)^n.

A261359 Pentatope of coefficients in expansion of (1 + x + 2y + 2z)^n.

A261357 Pyramid of coefficients in expansion of (1 + 2x + 2y)^n.