A364843 Integers are repeated in runs of 1, 2, 3, ... Each new integer (following a run) is given the value of its sequence index value.

1, 2, 2, 4, 4, 4, 7, 7, 7, 7, 11, 11, 11, 11, 11, 16, 16, 16, 16, 16, 16, 22, 22, 22, 22, 22, 22, 22, 29, 29, 29, 29, 29, 29, 29, 29, 37, 37, 37, 37, 37, 37, 37, 37, 37, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56

Offset

1,2

Comments

Omitting repeats yields the triangular numbers plus 1 sequence A000124.

Links

Table of n, a(n) for n=1..66.

Formula

G.f.: x*y*(1 + 2*x^4*y^2 - x*(1 + y) - 2*x^3*y*(1 + y) + x^2*(1 + y + y^2))/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Sep 02 2023

Sum_{k=1..n} k = T(n,k) = A006528(n). - Alois P. Heinz, Sep 15 2023

Example

Illustrated as a triangle begins:
   1;
   2,  2;
   4,  4,  4;
   7,  7,  7,  7;
  11, 11, 11, 11, 11;
  16, 16, 16, 16, 16, 16;
  22, 22, 22, 22, 22, 22, 22;
  ...

Maple

T:= (n, k)-> n*(n-1)/2+1:
seq(seq(T(n, k), k=1..n), n=1..11);  # Alois P. Heinz, Aug 31 2023

Prog

(PARI) a(n) = my(t=(sqrtint(8*n-1)-1)\2); t*(t+1)/2+1 \\ Thomas Scheuerle, Aug 10 2023
(Python)
from math import isqrt
def A364843(n): return ((t:=isqrt((n<<3)-1)-1>>1)*(t+1)>>1)+1 # Chai Wah Wu, Sep 15 2023

Crossrefs

Cf. A000124, A002024, A006528.

Row sums give A006000(n-1).

Sequence in context: A062570 A108514 A317419 * A372678 A120456 A367039

Adjacent sequences: A364840 A364841 A364842 * A364844 A364845 A364846

Keyword

easy,nonn,tabl

Author

Peter Woodward, Aug 10 2023

Status

approved