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

Author

Peter Woodward, Aug 10 2023

Keywords

• easy
• nonn
• tabl

Comments

Omitting repeats yields the triangular numbers plus 1 sequence A000124.

Examples

			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;
  ...
		

Crossrefs

Cf. A000124, A002024, A006528.

Row sums give A006000(n-1).

Programs

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

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

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