cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A217536 Square array read by antidiagonals, where the top row is the nonnegative integers and the other numbers are the sum of the neighbors in the preceding row.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 3, 6, 10, 14, 4, 9, 18, 32, 46, 5, 12, 27, 55, 101, 147, 6, 15, 36, 81, 168, 315, 462, 7, 18, 45, 108, 244, 513, 975, 1437, 8, 21, 54, 135, 324, 736, 1564, 3001, 4438, 9, 24, 63, 162, 405, 973, 2222, 4761, 9199, 13637, 10, 27, 72, 189, 486, 1215, 2924, 6710, 14472, 28109, 41746
Offset: 0

Views

Author

WG Zeist, Oct 06 2012

Keywords

Comments

Each number in the top row of the array is determined by the pre-defined sequence (in this case, the nonnegative integers). Each number in lower rows is the sum of the numbers vertically or diagonally above it (so, the number at the left end of each row is the sum of two numbers, and all other numbers the sum of three).
Replacing the top row with A000012 (the all 1's sequence) and constructing the rest of the array the same way produces A062105. Similarly, replacing the top row with A000007 (a(n) = 0^n) produces A020474. - WG Zeist, Aug 24 2024
For any array constructed with this method, regardless of the sequence chosen for the top row, the sequence in the first column of the array can be computed from the sequence in the top row as follows: let a(0), a(1), a(2), ... be the terms in the top row, and b(0), b(1), b(2), ... the terms in the first column. Then b(n) = Sum_{k=0..n} A064189(n,k) * a(k). The inverse operation, to compute the top row from the first column, is given by a(n) = Sum_{k=0..n} A104562(n,k) * b(k). - WG Zeist, Aug 26 2024

Examples

			The array starts:
  0  1  2  3
  1  3  6  9
  4  10 18 27
  14 32 55 81

Links

Crossrefs

Main diagonal gives A036290. First column gives A330796.
Cf. A020474, A062105, A064189, A104562.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(k<0, 0,
     `if`(n=0, k, add(A(n-1, k+i), i=-1..1)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 24 2024

Formula

T(m+1,n) = sum(T(m,k), |k-n| <= 1) (and T(0,n)=n), m, n >= 0. - M. F. Hasler, Oct 09 2012

Extensions

Offset 0 from Alois P. Heinz, Aug 24 2024

A125090 Triangle read by rows: T(0,0)=1; for 0<=k<=n, n>=1, T(n,k) is the coefficient of x^(n-k) in the monic characteristic polynomial of the tridiagonal n X n matrix with diagonal (0,1,1,...) and super- and subdiagonals (1,1,1,...).

Original entry on oeis.org

1, 1, 0, 1, -1, -1, 1, -2, -1, 1, 1, -3, 0, 3, 0, 1, -4, 2, 5, -2, -1, 1, -5, 5, 6, -7, -2, 1, 1, -6, 9, 5, -15, 0, 5, 0, 1, -7, 14, 1, -25, 9, 12, -3, -1, 1, -8, 20, -7, -35, 29, 18, -15, -3, 1, 1, -9, 27, -20, -42, 63, 14, -42, 0, 7, 0, 1, -10, 35, -39, -42, 112, -14, -85, 24, 22, -4, -1, 1, -11, 44, -65, -30, 174, -84, -134, 95, 40, -26, -4
Offset: 1

Views

Author

Gary W. Adamson, Nov 19 2006

Keywords

Comments

The characteristic polynomial of the n X n matrix has a root = 1+2*cos(2*Pi/(2n+1)).

Examples

			Triangle starts:
1;
1, 0;
1, -1, -1;
1, -2, -1, 1;
1, -3, 0, 3, 0;
1, -4, 2, 5, -2, -1;
1, -5, 5, 6, -7, -2, 1;
1, -6, 9, 5, -15, 0, 5, 0;

Crossrefs

Cf. A104562.

Programs

  • Maple
    with(linalg): m:=proc(i,j): if i=1 and j=1 then 0 elif i=j then 1 elif abs(i-j)=1 then 1 else 0 fi end: T:=proc(n,k) if n=0 and k=0 then 1 else coeff(charpoly(matrix(n,n,m),x),x,n-k) fi end: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
  • Mathematica
    T[n_, k_] := T[n, k] = Which[n < 0, 0, n == 0, If[k == 0, 1, 0], True, T[n-1, k-1] - T[n-2, k] - If[n == 1, 0, T[n-1, k]]];
A[n_, k_] := T[n, n-k];
Table[A[n, k], {n, 0, 12}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019, after Peter Luschny *)
  • Sage
    @CachedFunction
def T(n, k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 0 if n==1 else T(n-1, k)
    return T(n-1,k-1) - T(n-2, k) - h
A125090 = lambda n,k: T(n, n-k)
for n in (0..9): [A125090(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

Formula

f(n,x)=(x-1)f(n-1,x)-f(n-2,x), where f(n,x) is the monic characteristic polynomial of the n X n matrix from the definition and f(0,x)=1.

Extensions

Edited by N. J. A. Sloane, Nov 29 2006

A124025 Duplicate of A123965.

Original entry on oeis.org

1, 3, -1, 8, -6, 1, 21, -25, 9, -1, 55, -90, 51, -12, 1, 144, -300, 234, -86, 15, -1, 377, -954, 951, -480, 130, -18, 1, 987, -2939, 3573, -2305, 855, -183, 21, -1, 2584, -8850, 12707, -10008, 4740, -1386, 245, -24, 1, 6765, -26195, 43398, -40426, 23373, -8715, 2100, -316, 27, -1, 17711, -76500, 143682
Offset: 1

Views

Author

Roger L. Bagula, Oct 31 2006

Keywords

References

  • Joanne Dombrowski, Tridiagonal matrix representations of cyclic selfadjoint operators, Pacific J. Math. 114, no. 2 (1984), 325-334

Crossrefs

Cf. A101950, A104562, A123965.

Programs

  • Mathematica
    b[k_] = 3; a[k_] = -1; p[0, x] = 1; p[1, x] = (x - b[1])/a[1]; p[k_, x_] := p[k, x] = ((x - b[k - 1])*p[k - 1, x] - a[k - 2]*p[k - 2, x])/a[k - 1]; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w]

Formula

Recursive polynomial from a tridiagonal matrix version of A123965 ( first number different): p(k, x) = ((x - b(k - 1))*p(k - 1, x) - a(k - 2)*p(k - 2, x))/a(n - 1); a(n)=-1;b(n)=3;
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