A202453 -id:A202453 - cp's OEIS frontend

A115255 "Correlation triangle" of central binomial coefficients A000984.

1, 2, 2, 6, 5, 6, 20, 14, 14, 20, 70, 46, 41, 46, 70, 252, 160, 134, 134, 160, 252, 924, 574, 466, 441, 466, 574, 924, 3432, 2100, 1672, 1534, 1534, 1672, 2100, 3432, 12870, 7788, 6118, 5506, 5341, 5506, 6118, 7788, 12870, 48620, 29172, 22692, 20152, 19174

Offset: 0

Paul Barry, Jan 18 2006

Row sums are A033114. Diagonal sums are A115256. T(2n,n) is A115257. Corresponds to the triangle of antidiagonals of the correlation matrix of the sequence array for C(2n,n).

Let s=(1,2,6,20,...), (central binomial coefficients), and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A115255 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203005 for characteristic polynomials of principal submatrices of M, with interlacing zeros. - Clark Kimberling, Dec 27 2011

			Triangle begins:
  1;
  2, 2;
  6, 5, 6;
  20, 14, 14, 20;
  70, 46, 41, 46, 70;
  252, 160, 134, 134, 160, 252;
Northwest corner (square format):
  1    2    6    20    70
  2    5    14   46    160
  6    14   41   134   466
  20   46   134  441   1534

Cf. A203004, A203001, A202453.

s[k_] := Binomial[2 k - 2, k - 1];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]]; (* A115255 in square format *)
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]; Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]  (* A006134 *)
Table[m[1, j], {j, 1, 12}]     (* A000984 *)
Table[m[j, j], {j, 1, 12}]     (* A115257 *)
Table[m[j, j + 1], {j, 1, 12}] (* 2*A082578 *)
(* Clark Kimberling, Dec 27 2011 *)

G.f.: 1/(sqrt(1-4*x)*sqrt(1-4*x*y)*(1-x^2*y)) (format due to Christian G. Bower).

T(n, k) = Sum_{j=0..n} [j<=k]*C(2*k-2*j, k-j)*[j<=n-k]*C(2*n-2*k-2*j, n-k-j).

A203001 Symmetric matrix based on A007598, by antidiagonals.

Original entry on oeis.org

1, 1, 1, 4, 2, 4, 9, 5, 5, 9, 25, 13, 18, 13, 25, 64, 34, 41, 41, 34, 64, 169, 89, 113, 99, 113, 89, 169, 441, 233, 290, 266, 266, 290, 233, 441, 1156, 610, 765, 689, 724, 689, 765, 610, 1156, 3025, 1597, 1997, 1811, 1866, 1866, 1811, 1997, 1597, 3025

Offset: 1

Clark Kimberling, Dec 27 2011

Let s=A007598 (squared Fibonacci numbers), and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A203001 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203002 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1...1...4....9....25....64
1...2...5....13...34....89
4...5...18...41...113...290
9...13..41...99...266...724

Cf. A203002, A202453.

s[k_] := Fibonacci[k]^2;
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]   (* A001654 *)
Table[m[1, j], {j, 1, 12}]      (* A007598 *)
Table[m[2, j], {j, 1, 12}]      (* A001519 *)
Table[m[j, j], {j, 1, 12}]      (* A005969 *)

A204004 Symmetric matrix based on f(i,j) = max{2i+j-2,i+2j-2}, by antidiagonals.

Original entry on oeis.org

1, 3, 3, 5, 4, 5, 7, 6, 6, 7, 9, 8, 7, 8, 9, 11, 10, 9, 9, 10, 11, 13, 12, 11, 10, 11, 12, 13, 15, 14, 13, 12, 12, 13, 14, 15, 17, 16, 15, 14, 13, 14, 15, 16, 17, 19, 18, 17, 16, 15, 15, 16, 17, 18, 19, 21, 20, 19, 18, 17, 16, 17, 18, 19, 20, 21, 23, 22, 21, 20, 19, 18

Offset: 1

Clark Kimberling, Jan 09 2012

A204004 represents the matrix M given by f(i,j)=max{2i+j,i+2j}for i>=1 and j>=1. See A204005 for characteristic polynomials of principal submatrices of M, with interlacing zeros.
General case A206772. Let m be natural number. Table T(n,k)=max{m*n+k-m,n+m*k-m} read by antidiagonals.
  For m=1 the result is A002024,
  for m=2 the result is A204004,
  for m=3 the result is A204008,
  for m=4 the result is A206772. - Boris Putievskiy, Jan 24 2013

			Northwest corner:
  1,  3,  5,  7,  9
  3,  4,  6,  8, 10
  5,  6,  7,  9, 11
  7,  8,  9, 10, 12

Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.

Cf. A204005, A202453, A002024, A204008, A002260, A004736, A206772.

f[i_, j_] := Max[2 i + j - 2, 2 j + i - 2];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 12}, {i, 1, n}]]  (* A204004 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]   (* A204005 *)
TableForm[Table[c[n], {n, 1, 10}]]

From Boris Putievskiy, Jan 24 2013: (Start)
For the general case, a(n) = m*A002024(n) + (m-1)*max{-A002260(n),-A004736(n)}.
a(n) = m*(t+1) + (m-1)*max{t*(t+1)/2-n,n-(t*t+3*t+4)/2}, where t=floor((-1+sqrt(8*n-7))/2).
For m=2, a(n) = 2*(t+1) + max{t*(t+1)/2-n,n-(t*t+3*t+4)/2}, where t=floor((-1+sqrt(8*n-7))/2). (End)

A204158 Symmetric matrix based on f(i,j)=max(3i-2j, 3j-2i), by antidiagonals.

Original entry on oeis.org

1, 4, 4, 7, 2, 7, 10, 5, 5, 10, 13, 8, 3, 8, 13, 16, 11, 6, 6, 11, 16, 19, 14, 9, 4, 9, 14, 19, 22, 17, 12, 7, 7, 12, 17, 22, 25, 20, 15, 10, 5, 10, 15, 20, 25, 28, 23, 18, 13, 8, 8, 13, 18, 23, 28, 31, 26, 21, 16, 11, 6, 11, 16, 21, 26, 31, 34, 29, 24, 19, 14, 9, 9, 14

Offset: 1

Clark Kimberling, Jan 12 2012

A204158 represents the matrix M given by f(i,j)=max(3i-2j, 3j-2i) for i>=1 and j>=1. See A204159 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

			Northwest corner:
1....4....7....10...13
4....2....5....8....11
7....5....3....6....9
10...8....6....4....7

Cf. A204159, A204016, A202453.

f[i_, j_] := Max[3 i - 2 j, 3 j - 2 i];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]   (* A204158 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]                  (* A204159 *)
TableForm[Table[c[n], {n, 1, 10}]]

A202452 Lower triangular Fibonacci matrix, by SW antidiagonals.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 3, 1, 0, 0, 5, 2, 1, 0, 0, 8, 3, 1, 0, 0, 0, 13, 5, 2, 1, 0, 0, 0, 21, 8, 3, 1, 0, 0, 0, 0, 34, 13, 5, 2, 1, 0, 0, 0, 0, 55, 21, 8, 3, 1, 0, 0, 0, 0, 0, 89, 34, 13, 5, 2, 1, 0, 0, 0, 0, 0, 144, 55, 21, 8, 3, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Clark Kimberling, Dec 19 2011

			Northwest corner:
1...0...0...0...0...0...0...0...0
1...1...0...0...0...0...0...0...0
2...1...1...0...0...0...0...0...0
3...2...1...1...0...0...0...0...0
5...3...2...1...1...0...0...0...0

Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.

Cf. A202451, A202453, A202462, A188516, A000045.

n = 12;
Q = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[Fibonacci[k], {k, 1, n}]];
P = Transpose[Q]; F = P.Q;
Flatten[Table[P[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202451 as a sequence *)
Flatten[Table[Q[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202452 as a sequence *)
Flatten[Table[F[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202453 as a sequence *)
TableForm[Q]  (* A202451, upper triangular Fibonacci array *)
TableForm[P]  (* A202452, lower triangular Fibonacci array *)
TableForm[F]  (* A202453, Fibonacci self-fusion matrix *)
TableForm[FactorInteger[F]]

Column n consists of n-1 zeros followed by the Fibonacci sequence (1,1,2,3,5,8,...).

A203949 Symmetric matrix based on (1,1,0,1,1,0,1,1,0,...), by antidiagonals.

Original entry on oeis.org

1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 2, 1, 1, 2, 0, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 0, 2, 1, 1, 4, 1, 1, 2, 0, 1, 1, 1, 3, 2, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 4, 2, 2, 2, 1, 1, 0, 2, 1, 1, 4, 2, 2, 4, 1, 1, 2, 0, 1, 1, 1, 3, 2, 2, 5, 2, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 4, 3, 3

Offset: 1

Clark Kimberling, Jan 08 2012

Let s be the periodic sequence (1,1,0,1,1,0,...) and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A203949 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203950 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1 1 0 1 1 0 1 1 0 1
1 2 1 1 2 1 1 2 1 1
0 1 2 1 1 2 1 1 2 1
1 1 1 3 2 1 3 2 1 3
1 2 1 2 4 2 2 4 2 2
0 1 2 1 2 4 2 2 4 2
1 1 1 3 2 2 5 3 2 5

Cf. A203950, A202453.

t = {1, 1, 0}; t1 = Flatten[{t, t, t, t, t, t, t, t, t}];
s[k_] := t1[[k]];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[
   Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M] (* A203949 *)
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]

A203951 Symmetric matrix based on (1,0,0,0,1,0,0,0,...), by antidiagonals.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Clark Kimberling, Jan 08 2012

Let s be the periodic sequence (1,0,0,0,1,0,0,0,...) and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A203951 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203952 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1 0 0 0 1 0 0 0 1 0
0 1 0 0 0 1 0 0 0 1
0 0 1 0 0 0 1 0 0 0
0 0 0 1 0 0 0 1 0 0
1 0 0 0 2 0 0 0 2 0
0 1 0 0 0 2 0 0 0 2
0 0 1 0 0 0 2 0 0 0
0 0 0 1 0 0 0 2 0 0
1 0 0 0 2 0 0 0 3 0

Cf. A203951, A202453.

t = {1, 0, 0, 0}; t1 = Flatten[{t, t, t, t, t, t, t, t}];
f[k_] := t1[[k]];
U[n_] :=
  NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[
   Table[f[k], {k, 1, n}]];
L[n_] := Transpose[U[n]];
p[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]  (* A203952 *)

A203990 Symmetric matrix based on f(i,j) = (i+j)*min(i,j), by antidiagonals.

Original entry on oeis.org

2, 3, 3, 4, 8, 4, 5, 10, 10, 5, 6, 12, 18, 12, 6, 7, 14, 21, 21, 14, 7, 8, 16, 24, 32, 24, 16, 8, 9, 18, 27, 36, 36, 27, 18, 9, 10, 20, 30, 40, 50, 40, 30, 20, 10, 11, 22, 33, 44, 55, 55, 44, 33, 22, 11, 12, 24, 36, 48, 60, 72, 60, 48, 36, 24, 12, 13, 26, 39, 52, 65, 78, 78, 65, 52, 39, 26, 13

Offset: 1

Clark Kimberling, Jan 09 2012

This sequence represents the matrix M given by f(i,j) = (i+j)*min{i,j} for i >= 1 and j >= 1.

See A203991 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
  2,  3,  4,  5,  6,  7
  3,  8, 10, 12, 14, 16
  4, 10, 18, 21, 24, 27
  5, 12, 21, 32, 36, 40

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A203991, A202453.

Flat(List([1..15], n-> List([1..n], k-> (n+1)*Minimum(n-k+1,k) ))); # G. C. Greubel, Jul 23 2019

[(n+1)*Min(n-k+1,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 23 2019

(* First program *)
f[i_, j_] := (i + j) Min[i, j];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]  (* A203990 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]              (* A203991 *)
TableForm[Table[c[n], {n, 1, 10}]]
(* Second program *)
Table[(n+1)*Min[n-k+1, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jul 23 2019 *)

for(n=1,15, for(k=1,n, print1((n+1)*min(n-k+1,k), ", "))) \\ G. C. Greubel, Jul 23 2019

[[(n+1)*min(n-k+1,k) for n in (1..n)] for n in (1..15)] # G. C. Greubel, Jul 23 2019

A204022 Symmetric matrix based on f(i,j) = max(2i-1, 2j-1), by antidiagonals.

Original entry on oeis.org

1, 3, 3, 5, 3, 5, 7, 5, 5, 7, 9, 7, 5, 7, 9, 11, 9, 7, 7, 9, 11, 13, 11, 9, 7, 9, 11, 13, 15, 13, 11, 9, 9, 11, 13, 15, 17, 15, 13, 11, 9, 11, 13, 15, 17, 19, 17, 15, 13, 11, 11, 13, 15, 17, 19, 21, 19, 17, 15, 13, 11, 13, 15, 17, 19, 21, 23, 21, 19, 17, 15, 13, 13, 15, 17, 19, 21, 23

Offset: 1

Clark Kimberling, Jan 11 2012

This sequence represents the matrix M given by f(i,j) = max(2i-1, 2j-1) for i >= 1 and j >= 1. See A204023 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

			Northwest corner:
  1 3 5 7 9
  3 3 5 7 9
  5 5 5 7 9
  7 7 7 7 9
  9 9 9 9 9

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A202453, A204016, A204022, A209302.

Flat(List([1..12], n-> List([1..n], k-> Maximum(2*k-1, 2*(n-k)+1) ))); # G. C. Greubel, Jul 23 2019

[[Max(2*k-1, 2*(n-k)+1): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 23 2019

(* First program *)
f[i_, j_] := Max[2 i - 1, 2 j - 1];
m[n_] := Table[f[i, j], {i, n}, {j, n}]
TableForm[m[6]] (* 6 X 6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 15}, {i, n}]]                (* A204022 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 10}]]]
Table[c[n], {n, 12}]
Flatten[%]                         (* A204023 *)
TableForm[Table[c[n], {n, 10}]]
(* Second program *)
Table[Max[2*k-1, 2*(n-k)+1], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 23 2019 *)

{T(n, k) = max(2*k-1, 2*(n-k)+1)};
for(n=1, 12, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jul 23 2019

from math import isqrt
def A204022(n): return (m:=isqrt(n<<3)+1>>1)+abs(m**2-(n<<1)+1) # Chai Wah Wu, Jun 08 2025

[[max(2*k-1, 2*(n-k)+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 23 2019

From Ridouane Oudra, May 27 2019: (Start)
a(n) = t + |t^2-2n+1|, where t = floor(sqrt(2n-1)+1/2).
a(n) = A209302(2n-1).
a(n) = A002024(n) + |A002024(n)^2-2n+1|.
a(n) = t + |t^2-2n+1|, where t = floor(sqrt(2n)+1/2). (End)

A204026 Symmetric matrix based on f(i,j)=min(F(i+1),F(j+1)), where F=A000045 (Fibonacci numbers), by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 3, 2, 1, 1, 2, 3, 5, 3, 2, 1, 1, 2, 3, 5, 5, 3, 2, 1, 1, 2, 3, 5, 8, 5, 3, 2, 1, 1, 2, 3, 5, 8, 8, 5, 3, 2, 1, 1, 2, 3, 5, 8, 13, 8, 5, 3, 2, 1, 1, 2, 3, 5, 8, 13, 13, 8, 5, 3, 2, 1, 1, 2, 3, 5, 8, 13, 21, 13, 8, 5, 3, 2, 1, 1, 2, 3, 5, 8

Offset: 1

Clark Kimberling, Jan 11 2012

A204026 represents the matrix M given by f(i,j)=min(F(i+1),F(j+1)) for i>=1 and j>=1. See A204027 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

			Northwest corner:
1 1 1 1 1 1
1 2 2 2 2 2
1 2 3 3 3 3
1 2 3 5 5 5
1 2 3 5 8 8
1 2 3 5 8 13

Cf. A204026, A204016, A202453.

f[i_, j_] := Min[Fibonacci[i + 1], Fibonacci[j + 1]]
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]  (* A204026 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]                 (* A204027 *)
TableForm[Table[c[n], {n, 1, 10}]]

cp's OEIS Frontend

A115255 "Correlation triangle" of central binomial coefficients A000984.

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A203001 Symmetric matrix based on A007598, by antidiagonals.

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A204004 Symmetric matrix based on f(i,j) = max{2i+j-2,i+2j-2}, by antidiagonals.

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A204158 Symmetric matrix based on f(i,j)=max(3i-2j, 3j-2i), by antidiagonals.

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A202452 Lower triangular Fibonacci matrix, by SW antidiagonals.

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A203949 Symmetric matrix based on (1,1,0,1,1,0,1,1,0,...), by antidiagonals.

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A203951 Symmetric matrix based on (1,0,0,0,1,0,0,0,...), by antidiagonals.

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A203990 Symmetric matrix based on f(i,j) = (i+j)*min(i,j), by antidiagonals.

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