A003982 Table read by rows: 1 if x = y, 0 otherwise, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...

1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

0,1

Comments

Also called the delta function.

From Clark Kimberling, Feb 07 2011: (Start)

In rectangular format, the infinite identity matrix and the weight array of A003783(n,k)=min{n,k}; in the accumulation chain

... < A003982 < A003783 < A115262 < A185957 < ... . See A144112 for definitions of weight array and accumulation array. (End)

Links

Antti Karttunen, Table of n, a(n) for n = 0..65537

Index entries for characteristic functions

Formula

n-th 1 is followed by 4*n-1 0's. In the sequence with flattened indices, the 1's are at positions listed in A046092.

G.f.: 1/(1 - x*y). E.g.f.: exp(x*y).

Considered as a linear sequence, expansion of q^(-1/2)*eta(q^8)^2/eta(q^4) in powers of q. If A(x) is the g.f., then B(a) = (q*A(a^2))^2 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2*w - v^3 - 4*v*w^2. Also, given g.f. A(x), then B(q) = q*A(q^2) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u2^2*u6 - u1*u6^3 - u3^3*u2. - Michael Somos, Apr 13 2005

a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 if p>2. - Michael Somos, Jun 06 2005

a(n) = floor(sqrt(2*n+1)) - floor(sqrt(2*n)). - Ridouane Oudra, Oct 09 2020

Example

Table begins
  1;
  0, 0;
  0, 1, 0;
  0, 0, 0, 0;
  0, 0, 1, 0, 0;
  ....
Northwest corner when formatted as a rectangular array:
  1 0 0 0 0 0 0 0
  0 1 0 0 0 0 0 0
  0 0 1 0 0 0 0 0
  0 0 0 1 0 0 0 0
  0 0 0 0 1 0 0 0

Mathematica

f[n_, k_]:=0; f[n_, n_]:=1;
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 10}]] (* array *)
Table[f[n-k+1, k], {n, 10}, {k, n, 1, -1}]//Flatten (*sequence *)
Table[Join[{1}, Table[0, 4n-1]], {n, 10}]//Flatten (* Harvey P. Dale, Dec 21 2016 *)

Prog

(PARI) {a(n) = issquare(2*n + 1)}; /* Michael Somos, Apr 13 2005 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^8 + A)^2 / eta(x^4 + A), n))};
(PARI) A(i, j)=i==j

Crossrefs

Characteristic function of A001844. Antidiagonal sums and main diagonal is A000012.

Cf. also A286100.

Sequence in context: A014065 A014049 A016371 * A126010 A015857 A016343

Adjacent sequences: A003979 A003980 A003981 * A003983 A003984 A003985

Keyword

tabl,nonn,nice,easy

Author

Marc LeBrun

Status

approved