A132918 Identity matrix with interpolated zeros.

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

Offset

1,1

Comments

Identity matrix with interpolated zeros, such that the diagonal of an infinite lower triangular matrix = (1, 0, 1, 0, 1, ...) with the rest zeros.

Given A132918 = matrix M, then M * any sequence as a vector replaces alternate terms with zeros; e.g., M * [1,2,3,...] = [1,0,3,0,5,...]. M * (any infinite lower triangular matrix) replaces alternate rows with zeros; e.g., M * A007318 = (1; 0,0; 1,2,1; 0,0,0,0; ...).

a(n) = 1 if n is a positive hexagonal number, A000384: (1, 6, 15, 28, 45, 66, ...); zero otherwise.

Links

Paolo Xausa, Table of n, a(n) for n = 1..11175 (rows 1..149 of triangle, flattened).

Peter Bala, A continuous interpolation function for A132918.

Index entries for characteristic functions

Formula

a(n) = lim_{x -> n} sqrt(2) * sin(Pi*x)/(u*(sin(Pi*u/4) + cos(Pi*u/4))), where u = sqrt(1 + 8*x). (Put y = x + n. If n is not of the form N*(2*N - 1) then the limit as y -> 0 is easily seen to be 0; if n = N*(2*N - 1) then the limit has the indeterminate form 0/0. Apply L'Hôpital's rule.) - Peter Bala, Sep 13 2025

Example

First few rows of the triangle:
  1;
  0, 0;
  0, 0, 1;
  0, 0, 0, 0;
  0, 0, 0, 0, 1;
  ...

Mathematica

A132918[n_] := Boole[IntegerQ[(Sqrt[8*n + 1] + 1)/4]]; Array[A132918, 120] (* or *)
Normal[SparseArray[Map[# -> 1 &, PolygonalNumber[6, Range[8]]]]] (* or *)
A132918row[n_] := PadLeft[{Boole[OddQ[n]]}, n];
Array[A132918row, 15] (* Paolo Xausa, Oct 31 2025 *)

Prog

(PARI) a(n) = ispolygonal(n, 6); \\ Michel Marcus, Aug 04 2023

Crossrefs

Cf. A000384, A005369, A007318, A132918.

Sequence in context: A015959 A014834 A015659 * A358750 A205809 A380477

Adjacent sequences: A132915 A132916 A132917 * A132919 A132920 A132921

Keyword

nonn,tabl

Author

Gary W. Adamson, Sep 05 2007

Extensions

More terms from Michel Marcus, Mar 06 2022

Status

approved