A380763 List triangular numbers up to increasingly large values and back down to 0.

0, 1, 0, 1, 3, 1, 0, 1, 3, 6, 3, 1, 0, 1, 3, 6, 10, 6, 3, 1, 0, 1, 3, 6, 10, 15, 10, 6, 3, 1, 0, 1, 3, 6, 10, 15, 21, 15, 10, 6, 3, 1, 0, 1, 3, 6, 10, 15, 21, 28, 21, 15, 10, 6, 3, 1, 0, 1, 3, 6, 10, 15, 21, 28, 36, 28, 21, 15, 10, 6, 3, 1, 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 36, 28, 21, 15, 10, 6, 3, 1, 0, 1, 3, 6, 10, 15, 21, 28, 36, 45

Offset

0,5

Comments

Analog of the pyramidal sequence A053615 = (0, 1; 0, 1, 2, 1; 0, 1, 2, 3, 2, 1; ...), but with (only) triangular numbers A000217 = (0, 1, 3, 6, 10, ...).

A simple nontrivial example of a sequence in which increasingly large values appear in order, but not all nonnegative integers appear.

RECORDS transform is A000217; RECORD INDICES transform (indices where records occur) is A000290, the squares.

Links

Table of n, a(n) for n=0..99.

Formula

a(n) = A000217(A053615(n)).

a(n) = A000217(m) for n = m^2; for n < m^2, a(n) <= A000217(m-1).

a(n) = 0 iff n = 2*A000217(k) for some k.

Example

The triangular numbers are A000217 = (0, 1, 3, 6, 10, 15, ...)
This sequence lists these numbers up to the first, second, third, ... and back to 0 (excluded in order to not repeat): 0, 1;  0, 1, 3, 1;  0, 1, 3, 6, 3, 1; ....

Prog

(Python) A380763 = lambda n: (n := A053615(n))*(n+1)//2
(Python)
def A380763(upto = 999): # generator of the sequence
    for m in range(0, 2*upto, 2):
        for x in range(m): yield (x := min(x, m-x))*(x+1)//2
list(A380763(upto=5))  # first 4 "peaks"
list(a for a, _ in zip(A380763(), range(99))) # first 99 terms

Crossrefs

Cf. A053615 (pyramidal sequence), A000217 (triangular numbers), A000290 (the squares: indices where records occur).

Sequence in context: A099905 A268441 A264435 * A356656 A085391 A394284

Adjacent sequences: A380760 A380761 A380762 * A380764 A380765 A380766

Keyword

nonn

Author

M. F. Hasler, Feb 01 2025

Status

approved