A220280 The reluctant sequence for A002260.

1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 1, 1, 2, 1, 2, 3, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3

Offset: 1

Author

Boris Putievskiy, Dec 12 2012

Keywords

• easy
• nonn
• tabl

Comments

The reluctant sequence B for a sequence A is a triangular array in which row k (>= 1) consists of the first k terms of A.

Here A002260 is the reluctant sequence for the sequence 1,2,3,... of positive numbers (A000027).

Examples

			A002260 begins
  1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, ...
so the first few rows of the new triangle are
   1,
   1, 1,
   1, 1, 2,
   1, 1, 2, 1,
   1, 1, 2, 1, 2,
   1, 1, 2, 1, 2, 3,
   ...
                                                                               ~
		

Links

• Boris Putievskiy, Table of n, a(n) for n = 1..10000
• Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Crossrefs

Cf. A002260, A002262, A037126, A059268, A215026.

Programs

Python

t=int((math.sqrt(8*n-7) - 1)/ 2)
n1=n-t*(t+1)/2
t1=int((math.sqrt(8*n1-7) - 1)/ 2)
a=n1-t1*(t1+1)/2

Formula

a(n) = n1 - t1(t1+1)/2, where n1 = n - t(t+1)/2, t1 = floor[(-1+sqrt(8*n1-7))/2], t=floor[(-1+sqrt(8*n-7))/2]. For example, a(6)=2 since t=2, t1=1, n1=3.

Extensions

Edited by N. J. A. Sloane, Jun 07 2024