A060747 -id:A060747 - cp's OEIS frontend

A343559 Irregular triangle read by rows: the n-th row gives the column indices of the consecutive elements of the spiral of the n X n matrix defined in A126224.

1, 1, 2, 2, 1, 1, 2, 3, 3, 3, 2, 1, 1, 2, 1, 2, 3, 4, 4, 4, 4, 3, 2, 1, 1, 1, 2, 3, 3, 2, 1, 2, 3, 4, 5, 5, 5, 5, 5, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 4, 4, 3, 2, 2, 3, 1, 2, 3, 4, 5, 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1, 1, 1, 1, 1, 2, 3, 4, 5, 5, 5, 5, 4, 3, 2, 2, 2, 3, 4, 4, 3

Offset: 1

Stefano Spezia, Apr 19 2021

			The triangle begins
1
1   2   2   1
1   2   3   3   3   2   1   1   2
1   2   3   4   4   4   4   3   2   1   1   1   2   3   3   2
...

Stefano Spezia, First 30 rows of the triangle, flattened

Cf. A000290 (row length), A002265, A002411 (row sums), A010873, A060747, A126224, A343558 (row indices).

a:={};nmax:=6;For[n=1,n<=nmax,n++,For[s=1,s<=2n-1,s++,If[OddQ[s]&&Mod[s,4]==1 ,k=Floor[s/4];For[i=1,i<=Ceiling[n-s/2],i++,AppendTo[a,k+i]];k=+Floor[s/4]+Ceiling[n-s/2],If[EvenQ[s],For[i=1,i<=Ceiling[n-s/2],i++,AppendTo[a,k]],For[i=1,i<=Ceiling[n-s/2],i++,AppendTo[a,k-i]];k=k-i+1]]]]; a

A343853 Irregular triangle read by rows: the n-th row gives the row indices of the matrix of 1..n^2 filled successively back and forth along antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 2, 3, 3, 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 3, 4, 4, 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1, 2, 3, 4, 5, 5, 4, 3, 4, 5, 5, 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 3, 4, 5, 6, 6, 5, 4, 5, 6, 6

Offset: 1

Stefano Spezia, May 01 2021

			The triangle begins:
1
1   1   2   2
1   1   2   3   2   1   2   3   3
1   1   2   3   2   1   1   2   3   4   4   3   2   3   4   4
...

Stefano Spezia, First 30 rows of the triangle, flattened

Cf. A000290 (row length), A002411 (row sums), A060747 (number of antidiagonals), A078475, A319573, A343854 (column indices).

a={};For[n=1,n<=6,n++,For[d=1,d<=n,d++, If[OddQ[d],i=d;For [k=1,k<=d,k++, AppendTo[a,i-k+1]],i=1;For[k=1,k<=d,k++, AppendTo[a,i+k-1]]]];For[d=n+1,d<=2n-1,d++, If[OddQ[d],i= n; For[k=1,k<=2n-d,k++,AppendTo[a,i-k+1]],If[EvenQ[d],i=d-n+1;For[k=1,k<=2n-d,k++, AppendTo[a,i+k-1]]]]]]; a

A343854 Irregular triangle read by rows: the n-th row gives the column indices of the matrix of 1..n^2 filled successively back and forth along antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 3, 2, 3, 1, 2, 1, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 4, 3, 4, 1, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 3, 4, 5, 5, 4, 3, 2, 3, 4, 5, 5, 4, 5, 1, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 4, 5, 6, 6, 5, 6

Offset: 1

Stefano Spezia, May 01 2021

			The triangle begins:
1
1   2   1   2
1   2   1   1   2   3   3   2   3
1   2   1   1   2   3   4   3   2   1   2   3   4   4   3   4
...

Stefano Spezia, First 30 rows of the triangle, flattened

Cf. A000290 (row length), A002411 (row sums), A060747 (number of antidiagonals), A078475, A319572, A343853 (row indices).

a={};For[n=1,n<=6,n++,For[d=1,d<=n,d++, If[EvenQ[d],i=d;For [k=1,k<=d,k++, AppendTo[a,i-k+1]],i=1;For[k=1,k<=d,k++, AppendTo[a,i+k-1]]]];For[d=n+1,d<=2n-1,d++, If[EvenQ[d],i= n; For[k=1,k<=2n-d,k++,AppendTo[a,i-k+1]],If[OddQ[d],i=d-n+1;For[k=1,k<=2n-d,k++, AppendTo[a,i+k-1]]]]]]; a

A350549 a(n) is the permanent of a square matrix M(n) whose general element M_{i,j} is defined by floor((j - i + 1)/2).

Original entry on oeis.org

1, 0, 0, -1, 2, 20, -120, -4608, 41952, 2325024, -34876800, -3133087200, 66120252480, 8258565859200, -239533775631360, -40631838221721600, 1532513262269767680, 335620705700380262400, -16054693916748370329600, -4428138916386119015424000, 261291002534430572648448000

Offset: 0

Stefano Spezia, Jan 04 2022

sign

The matrix M(n) is the n-th principal submatrix of the array A010751.
In the n X n matrix M(n): the zero element appears 2*n - 1 times; the positive integers k appears iff 0 < k < floor(n/2), 2*n - 1 - A040002(k-1) times; the negative integer k appears iff -k < ceiling(n/2),  2*n - 5 + 4*(k + 1) times.
det(M(n)) = 0, except for n = 3 for which det(M(3)) = -1.
The trace and the subdiagonal sum of the matrix M(n) are zero.
The antitrace of the matrix M(n) is A142150(n+1).
The superdiagonal sum of the matrix M(n) is equal to n - 1.
The sum of the elements of the matrix M(n) is A002620(n).

			For n = 3 the matrix M(3) is
     0, 1, 1
     0, 0, 1
    -1, 0, 0
with permanent a(3) = -1.
For n = 4 the matrix M(4) is
    0,  1,  1,  2
    0,  0,  1,  1
   -1,  0,  0,  1
   -1, -1,  0,  0
with permanent a(4) = 2.

Cf. A002620, A004526, A010751, A040002, A060747, A110654, A142150.

a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](
         Matrix(n, (i, j)-> floor((j-i+1)/2)))):
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 19 2022

Join[{1},Table[Permanent[Table[Floor[(j-i+1)/2],{i,n},{j,n}]],{n,20}]]

a(n) = matpermanent(matrix(n, n, i, j, (j - i + 1)\2)); \\ Michel Marcus, Jan 04 2022

from sympy import Matrix
def A350549(n): return 1 if n == 0 else Matrix(n,n,lambda i,j:(j-i+1)//2).per() # Chai Wah Wu, Jan 12 2022

A193251 Small rhombicosidodecahedron with faces of centered polygons.

Original entry on oeis.org

1, 123, 605, 1687, 3609, 6611, 10933, 16815, 24497, 34219, 46221, 60743, 78025, 98307, 121829, 148831, 179553, 214235, 253117, 296439, 344441, 397363, 455445, 518927, 588049, 663051, 744173, 831655, 925737, 1026659, 1134661, 1249983, 1372865, 1503547, 1642269

Offset: 1

Craig Ferguson, Jul 19 2011

The sequence starts with a central dot and expands outward with (n-1) centered polygonal pyramids producing a small rhombicosidodecahedron. Each iteration requires the addition of (n-2) edges and (n-1) vertices to complete the centered polygon of each face. [centered triangles (A005448), centered squares (A001844) and centered pentagons (A005891)]

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
OEIS Wiki, (Centered_polygons) pyramidal numbers.
Wikipedia, Tetrahedral number.
Wikipedia, Triangular number.
Wikipedia, Centered polygonal number.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A001844, A005448, A005891, A060747, A195317.

[40*n^3-60*n^2+22*n-1: n in [1..50]]; // Vincenzo Librandi, Aug 30 2011

A193251[n_] := (2*n - 1)*(20*(n - 1)*n + 1); Array[A193251, 50] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {1, 123, 605, 1687}, 50] (* Paolo Xausa, Aug 25 2025 *)

a(n)=40*n^3-60*n^2+22*n-1 \\ Charles R Greathouse IV, Oct 19 2022

a(n) = 40*n^3 - 60*n^2 + 22*n - 1.
G.f.: x*(1+x)*(x^2 + 118*x + 1)/(x-1)^4. - R. J. Mathar, Aug 26 2011
From Elmo R. Oliveira, Aug 22 2025: (Start)
E.g.f.: 1 + exp(x)*(-1 + 2*x + 60*x^2 + 40*x^3).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4.
a(n) = A060747(n)*A195317(n). (End)

A195662 Triangle T(n,k) read by rows: T(0,0)= -3, T(1,0)= 2, T(1,1) = 0 and T(n,k) = T(n-1,k) -T(n-2,k-2) otherwise.

Original entry on oeis.org

-3, 2, 0, 2, 0, 3, 2, 0, 1, 0, 2, 0, -1, 0, -3, 2, 0, -3, 0, -4, 0, 2, 0, -5, 0, -3, 0, 3, 2, 0, -7, 0, 0, 0, 7, 0, 2, 0, -9, 0, 5, 0, 10, 0, -3, 2, 0, -11, 0, 12, 0, 10, 0, -10, 0, 2, 0, -13, 0, 21, 0, 5, 0, -20, 0, 3, 2, 0, -15, 0, 32, 0, -7, 0, -30, 0, 13, 0

Offset: 0

Paul Curtz, Sep 22 2011

In the notation of A195673, this defines polynomials P(n,x,p=-3,q=2), where p and q are the values of the constant and linear order for n=0 and 1.
Row sums -- the value P(n,1,-3,2) of the polynomial -- are A130848(n+5).
For general seed values in the two top rows of the triangle, the recurrence T(n,k) = T(n-1,k) - T(n-2,k-2) defines the triangle
p;
q,   0;
q,   0,      -p;
q,   0,    -p-q, 0;
q,   0,   -p-2q, 0,     p;
q,   0,   -p-3q, 0,  2p+q,  0;
and a companion triangle by adding 1 to both seed values:
p+1;
q+1, 0;
q+1, 0,    -p-1;
q+1, 0,  -p-q-2, 0;
q+1, 0, -p-2q-3, 0,    p+1;
q+1, 0, -p-3q-4, 0, 2p+q+3, 0;
The point-by-point difference between two companions is P(n,x,p+1,q+1) - P(n,x,p,q) = S(n,x) as represented (with increasing exponents) by A053119.
Examples of such triangles are A053119 (p=q=1), A192575 (p=1, q=2),
A162514 (p=2, q=1, up to a sign factor), A192011 (p=-1, q=2), A135929 (p=-2, q=1, apart from a irregular leading T(0,0)).

			The first few rows are
-3;
2, 0;
2, 0,   3;
2, 0,   1, 0;
2, 0,  -1, 0, -3;
2, 0,  -3, 0, -4, 0;
2, 0,  -5, 0, -3, 0,  3;
2, 0,  -7, 0,  0, 0,  7, 0;
2, 0,  -9, 0,  5, 0, 10, 0,  -3;
2, 0, -11, 0, 12, 0, 10, 0, -10, 0;
2, 0, -13, 0, 21, 0,  5, 0, -20, 0, 3;

Cf. A135929, A192011.

p = -3; q = 2; t[0, 0] = p; t[, 0] = q; t[, ?OddQ] = 0; t[n, k_] /; k > n = 0; t[n_ /; n >= 0, k_ /; k >= 0] := t[n, k] = t[n-1, k] - t[n-2, k-2]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 27 2012 *)

T(n,0) = 2 (n>0).
T(n,2) = -A060747(n-3), n>2.
T(n,4) = A028347(n-5), n>6.
T(2n,2n) = -3*(-1)^n ; T(n, 2k-1) = 0 ; T(2n+1,2n) = -(3n-2)*(-1)^n.  - M. F. Hasler, Sep 28 2011

A215415 a(2n) = n, a(4n+1) = 2n-1, a(4n+3) = 2*n+3.

Original entry on oeis.org

0, -1, 1, 3, 2, 1, 3, 5, 4, 3, 5, 7, 6, 5, 7, 9, 8, 7, 9, 11, 10, 9, 11, 13, 12, 11, 13, 15, 14, 13, 15, 17, 16, 15, 17, 19, 18, 17, 19, 21, 20, 19, 21, 23, 22, 21, 23, 25, 24, 23, 25, 27, 26, 25, 27, 29, 28, 27, 29, 31, 30, 29, 31, 33, 32, 31, 33, 35, 34, 33, 35, 37

Offset: 0

Paul Curtz, Aug 09 2012

a(n) and higher order differences in further rows:
0,  -1,  1,  3,  2,  1,
-1,  2,  2, -1, -1, -2,    A134430(n).
3,   0, -3,  0,  3,  0,
-3, -3,  3,  3, -3, -3,
0,   6,  0, -6,  0,  6,
6,  -6, -6,  6,  6, -6.
a(n) is the binomial transform of 0, -1, 3, -3, 0, 6, -12, 12, 0, -24, 48, -48, 0, 96..., essentially negated A134813.
By definition, all differences a(n+k)-a(n) are periodic sequences with period length 4. For k=1, 3 and 4 these are A134430, A021307 and A007395, for example.

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

Quadrisections: A005843, A060747, A005408, A144396.

Flatten[Table[{2n, 2n - 1, 2n + 1, 2n + 3}, {n, 0, 19}]] (* Alonso del Arte, Aug 09 2012 *)

a(n) = ((-3*I)*((-I)^n-I^n)+2*n)/4 \\ Colin Barker, Oct 19 2015

concat(0, Vec(-x*(1-3*x+x^2)/((x^2+1)*(x-1)^2) + O(x^100))) \\ Colin Barker, Oct 19 2015

a(2*n) = n, a(2*n+1) = A097062(n+1).
a(n) = (A214297(n+1) - A214297(n-1))/2.
a(3*n) =3*A004525(n).
a(n) = +2*a(n-1) -2*a(n-2) +2*a(n-3) -a(n-4).
G.f. -x*(1-3*x+x^2) / ( (x^2+1)*(x-1)^2 ). - R. J. Mathar, Aug 11 2012
a(n) = ((-3*I)*((-I)^n-I^n)+2*n)/4. - Colin Barker, Oct 19 2015

A380747 Array read by ascending antidiagonals: A(n,k) = [x^n] (1 - x)/(1 - k*x)^2.

Original entry on oeis.org

1, -1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 8, 5, 1, 0, 1, 20, 21, 7, 1, 0, 1, 48, 81, 40, 9, 1, 0, 1, 112, 297, 208, 65, 11, 1, 0, 1, 256, 1053, 1024, 425, 96, 13, 1, 0, 1, 576, 3645, 4864, 2625, 756, 133, 15, 1, 0, 1, 1280, 12393, 22528, 15625, 5616, 1225, 176, 17, 1

Offset: 0

Stefano Spezia, Jan 31 2025

			The array begins as:
   1, 1,   1,    1,     1,     1, ...
  -1, 1,   3,    5,     7,     9, ...
   0, 1,   8,   21,    40,    65, ...
   0, 1,  20,   81,   208,   425, ...
   0, 1,  48,  297,  1024,  2625, ...
   0, 1, 112, 1053,  4864, 15625, ...
   0, 1, 256, 3645, 22528, 90625, ...
   ...

Cf. A000012 (k=1 or n=0), A000567 (n=2), A001792 (k=2), A007778, A060747 (n=1), A081038 (k=3), A081039 (k=4), A081040 (k=5), A081041 (k=6), A081042 (k=7), A081043 (k=8), A081044 (k=9), A081045 (k=10), A103532, A154955, A380748 (antidiagonal sums).

A[0,0]:=1; A[1,0]:=-1; A[n_,k_]:=((k-1)*n+k)k^(n-1); Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=SeriesCoefficient[(1-x)/(1-k*x)^2,{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=n!SeriesCoefficient[Exp[k*x](1+(k-1)*x),{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

A(n,k) = ((k - 1)*n + k)*k^(n-1) with A(0,0) = 1.
A(n,k) = n! * [x^n] exp(k*x)*(1 + (k - 1)*x).
A(n,0) = A154955(n+1).
A(3,n) = A103532(n-1) for n > 0.
A(n,n) = A007778(n) for n > 0.

A381059 Array read by ascending antidiagonals: A(n,k) = numerator(binomial(n-1/2,k)) with k >=0.

Original entry on oeis.org

1, 1, -1, 1, 1, 3, 1, 3, -1, -5, 1, 5, 3, 1, 35, 1, 7, 15, -1, -5, -63, 1, 9, 35, 5, 3, 7, 231, 1, 11, 63, 35, -5, -3, -21, -429, 1, 13, 99, 105, 35, 3, 7, 33, 6435, 1, 15, 143, 231, 315, -7, -5, -9, -429, -12155, 1, 17, 195, 429, 1155, 63, 7, 5, 99, 715, 46189

Offset: 0

Stefano Spezia, Feb 12 2025

Numerators of the binomial coefficients for half-integers. The denominators are given by the absolute values of A173755.

			The array of the binomial coefficients for half-integers begins as:
  1, -1/2,  3/8,  -5/16,   35/128, -63/256, ...
  1,  1/2, -1/8,   1/16,   -5/128,   7/256, ...
  1,  3/2,  3/8,  -1/16,    3/128,  -3/256, ...
  1,  5/2, 15/8,   5/16,   -5/128,   3/256, ...
  1,  7/2, 35/8,  35/16,   35/128,  -7/256, ...
  1,  9/2, 63/8, 105/16,  315/128,  63/256, ...
  1, 11/2, 99/8, 231/16, 1155/128, 693/256, ...
  ...

Stefano Spezia, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals of the array)
Gábor Hegedüs, Sho Suda, and Ziqing Xiang, Codes with symmetric distances, arXiv:2501.11461 [math.CO], 2025. See p. 10.

Cf. A000466, A161200, A161202, A162540, A173755.
Columns k=0..1 give A000012, A060747.
Row n=1 gives A002596.
Main diagonal gives A001790.

A[n_,k_]:=Numerator[Binomial[n-1/2,k]]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=Numerator[(2n-1)!!/((2(n-k)-1)!!2^k k!)]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

A(n,k) = numerator((2*n - 1)!!/((2*(n - k) - 1)!!*2^k*k!)).
A(n,2) = A000466(n-1) for n > 0.
A(n,3) = A162540(n-3) for n > 3.
A(0,n) = (-1)^n*A001790(n).
abs(A(2,n)) = abs(A161200(n)).
abs(A(3,n)) = abs(A161202(n)).

A084847 a(n) = 23^n+2^(2n-1)(n-2).

Original entry on oeis.org

1, 4, 18, 86, 418, 2022, 9650, 45334, 209730, 956870, 4312402, 19228662, 84948962, 372287398, 1620178674, 7008019670, 30150864514, 129107299206, 550530654866, 2338786731958, 9902578218786, 41802362561894, 175984622563378

Offset: 0

Paul Barry, Jun 09 2003

Binomial transform of A084643. Third binomial transform of abs(A060747).

Index entries for linear recurrences with constant coefficients, signature (11,-40,48).

Cf. A060747, A084643.

G.f.: (1-7x+14x^2)/((1-3x)(1-4x)^2).

cp's OEIS Frontend

A343559 Irregular triangle read by rows: the n-th row gives the column indices of the consecutive elements of the spiral of the n X n matrix defined in A126224.

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A343853 Irregular triangle read by rows: the n-th row gives the row indices of the matrix of 1..n^2 filled successively back and forth along antidiagonals.

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A343854 Irregular triangle read by rows: the n-th row gives the column indices of the matrix of 1..n^2 filled successively back and forth along antidiagonals.

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A350549 a(n) is the permanent of a square matrix M(n) whose general element M_{i,j} is defined by floor((j - i + 1)/2).

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A193251 Small rhombicosidodecahedron with faces of centered polygons.

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A195662 Triangle T(n,k) read by rows: T(0,0)= -3, T(1,0)= 2, T(1,1) = 0 and T(n,k) = T(n-1,k) -T(n-2,k-2) otherwise.

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A215415 a(2*n) = n, a(4*n+1) = 2*n-1, a(4*n+3) = 2*n+3.

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A380747 Array read by ascending antidiagonals: A(n,k) = [x^n] (1 - x)/(1 - k*x)^2.

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A215415 a(2n) = n, a(4n+1) = 2n-1, a(4n+3) = 2*n+3.

A084847 a(n) = 23^n+2^(2n-1)(n-2).