A389111 Write 0,1,2,3,4,... in a triangular spiral, a(n) is such that n lies geometrically half way between 0 and a(n).

11, 13, 15, 17, 19, 21, 23, 25, 27, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179, 181, 183, 185, 187, 189

Offset

1,1

Links

Table of n, a(n) for n=1..54.

Chinese BBS, Triangular graph of triangular numbers.

Formula

a(n) = 2*n + 9*round(sqrt(2*(n + 1))/3)^2.

G.f. 2/(1-x)^2 + 9*F(x^9)/(1-x), where F(x) = Sum_{n>=0} (2*n+1) * x^(n*(n+1)/2) is the g.f. of A198954. - Paul D. Hanna, Nov 09 2025

Example

The spiral begins:
            15
            / \
          16  14
          /     \
        17   3  13
        /   / \   \
      18   4   2  12
      /   /     \   \
    19   5   0---1  11
    /   /             \
  20   6---7---8---9--10
  /
a(1) = 11, since 1 sits at the midpoint between 0 and 11.
a(2) = 13, since 2 sits at the midpoint between 0 and 13.
a(3) = 15, since 3 sits at the midpoint between 0 and 15.
a(4) = 17, since 4 sits at the midpoint between 0 and 17.

Mathematica

Table[2 n + 9 Round[Sqrt[2 (n + 1)]/3]^2, {n, 60}]

Crossrefs

Cf. A062708.

Sequence in context: A292513 A171491 A277268 * A337254 A382937 A152200

Adjacent sequences: A389108 A389109 A389110 * A389112 A389113 A389114

Keyword

nonn,easy

Author

Zhining Yang, Sep 23 2025

Status

approved