A002260 Triangle read by rows: T(n,k) = k for n >= 1, k = 1..n.

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, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14

Offset: 1

Author

Angele Hamel (amh(AT)maths.soton.ac.uk)

Keywords

• nonn
• easy
• nice
• tabl
• look

Comments

Old name: integers 1 to k followed by integers 1 to k+1 etc. (a fractal sequence).

Start counting again and again.

This is a "doubly fractal sequence" - see the Franklin T. Adams-Watters link.

The PARI functions t1, t2 can be used to read a square array T(n,k) (n >= 1, k >= 1) by antidiagonals downwards: n -> T(t1(n), t2(n)). - Michael Somos, Aug 23 2002

Reading this sequence as the antidiagonals of a rectangular array, row n is (n,n,n,...); this is the weight array (Cf. A144112) of the array A127779 (rectangular). - Clark Kimberling, Sep 16 2008

The upper trim of an arbitrary fractal sequence s is s, but the lower trim of s, although a fractal sequence, need not be s itself. However, the lower trim of A002260 is A002260. (The upper trim of s is what remains after the first occurrence of each term is deleted; the lower trim of s is what remains after all 0's are deleted from the sequence s-1.) - Clark Kimberling, Nov 02 2009

Eigensequence of the triangle = A001710 starting (1, 3, 12, 60, 360, ...). - Gary W. Adamson, Aug 02 2010

The triangle sums, see A180662 for their definitions, link this triangle of natural numbers with twenty-three different sequences, see the crossrefs. The mirror image of this triangle is A004736. - Johannes W. Meijer, Sep 22 2010

A002260 is the self-fission of the polynomial sequence (q(n,x)), where q(n,x) = x^n + x^(n-1) + ... + x + 1. See A193842 for the definition of fission. - Clark Kimberling, Aug 07 2011

Sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. Sequence A002260 is reluctant sequence of sequence 1,2,3,... (A000027). - Boris Putievskiy, Dec 12 2012

This is the maximal sequence of positive integers, such that once an integer k has occurred, the number of k's always exceeds the number of (k+1)'s for the remainder of the sequence, with the first occurrence of the integers being in order. - Franklin T. Adams-Watters, Oct 23 2013

A002260 are the k antidiagonal numerators of rationals in Cantor's proof of 1-to-1 correspondence between rationals and naturals; the denominators are k-numerator+1. - Adriano Caroli, Mar 24 2015

T(n,k) gives the distance to the largest triangular number < n. - Ctibor O. Zizka, Apr 09 2020

Examples

			First six rows:
  1
  1   2
  1   2   3
  1   2   3   4
  1   2   3   4   5
  1   2   3   4   5   6
		

References

• Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168. (Introduces upper trimming, lower trimming, and signature sequences.)
• M. Myers, Smarandache Crescendo Subsequences, R. H. Wilde, An Anthology in Memoriam, Bristol Banner Books, Bristol, 1998, p. 19.
• F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..11325
• Franklin T. Adams-Watters, Doubly Fractal Sequences
• Matin Amini and Majid Jahangiri, A Novel Proof for Kimberling's Conjecture on Doubly Fractal Sequences, arXiv:1612.09481 [math.NT], 2017.
• Bruno Berselli, Illustration of the initial terms
• Jerry Brown et al., Problem 4619, School Science and Mathematics (USA), Vol. 97(4), 1997, pp. 221-222.
• Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida, and Daisy Ann A. Disu, On Fractal Sequences, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
• Clark Kimberling, Fractal sequences
• Clark Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73 (1995) 103-117.
• Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
• F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
• Aaron Snook, Augmented Integer Linear Recurrences, 2012. - _N. J. A. Sloane_, Dec 19 2012
• Michael Somos, Sequences used for indexing triangular or square arrays
• Eric Weisstein's World of Mathematics, Smarandache Sequences
• Eric Weisstein's World of Mathematics, Unit Fraction

Crossrefs

Cf. A000217, A001710, A002262, A003056, A004736 (ordinal transform), A025581, A056534, A094727, A127779.

Cf. A140756 (alternating signs).

Triangle sums (see the comments): A000217 (Row1, Kn11); A004526 (Row2); A000096 (Kn12); A055998 (Kn13); A055999 (Kn14); A056000 (Kn15); A056115 (Kn16); A056119 (Kn17); A056121 (Kn18); A056126 (Kn19); A051942 (Kn110); A101859 (Kn111); A132754 (Kn112); A132755 (Kn113); A132756 (Kn114); A132757 (Kn115); A132758 (Kn116); A002620 (Kn21); A000290 (Kn3); A001840 (Ca2); A000326 (Ca3); A001972 (Gi2); A000384 (Gi3).

Cf. A108872.

Programs

Haskell

a002260 n k = k
a002260_row n = [1..n]
a002260_tabl = iterate (\row -> map (+ 1) (0 : row)) [1]
-- Reinhard Zumkeller, Aug 04 2014, Jul 03 2012

Maple

at:=0; for n from 1 to 150 do for i from 1 to n do at:=at+1; lprint(at,i); od: od: # N. J. A. Sloane, Nov 01 2006
seq(seq(i,i=1..k),k=1..13); # Peter Luschny, Jul 06 2009

Mathematica

FoldList[{#1, #2} &, 1, Range[2, 13]] // Flatten (* Robert G. Wilson v, May 10 2011 *)
Flatten[Table[Range[n],{n,20}]] (* Harvey P. Dale, Jun 20 2013 *)

Maxima

T(n,k):=sum((i+k)*binomial(i+k-1,i)*binomial(k,n-i-k+1)*(-1)^(n-i-k+1),i,max(0,n+1-2*k),n-k+1); /* Vladimir Kruchinin, Oct 18 2013 */

PARI

t1(n)=n-binomial(floor(1/2+sqrt(2*n)),2) /* this sequence */

PARI

A002260(n)=n-binomial((sqrtint(8*n)+1)\2,2) \\ M. F. Hasler, Mar 10 2014

Python

from math import isqrt, comb
def A002260(n): return n-comb((m:=isqrt(k:=n<<1))+(k>m*(m+1)),2) # Chai Wah Wu, Nov 08 2024

Formula

a(n) = 1 + A002262(n).

n-th term is n - m*(m+1)/2 + 1, where m = floor((sqrt(8*n+1) - 1) / 2).

The above formula is for offset 0; for offset 1, use a(n) = n-m*(m+1)/2 where m = floor((-1+sqrt(8*n-7))/2). - Clark Kimberling, Jun 14 2011

a(k * (k + 1) / 2 + i) = i for k >= 0 and 0 < i <= k + 1. - Reinhard Zumkeller, Aug 14 2001

a(n) = (2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2. - Brian Tenneson, Oct 11 2003

a(n) = n - binomial(floor((1+sqrt(8*n))/2), 2). - Paul Barry, May 25 2004

T(n,k) = A001511(A118413(n,k)); T(n,k) = A003602(A118416(n,k)). - Reinhard Zumkeller, Apr 27 2006

a(A000217(n)) = A000217(n) - A000217(n-1), a(A000217(n-1) + 1) = 1, a(A000217(n) - 1) = A000217(n) - A000217(n-1) - 1. - Alexander R. Povolotsky, May 28 2008

a(A169581(n)) = A038566(n). - Reinhard Zumkeller, Dec 02 2009

T(n,k) = Sum_{i=1..k} i*binomial(k,i)*binomial(n-k,n-i) (regarded as triangle, see the example). - Mircea Merca, Apr 11 2012

T(n,k) = Sum_{i=max(0,n+1-2*k)..n-k+1} (i+k)*binomial(i+k-1,i)*binomial(k,n-i-k+1)*(-1)^(n-i-k+1). - Vladimir Kruchinin, Oct 18 2013

G.f.: x*y / ((1 - x) * (1 - x*y)^2) = Sum_{n,k>0} T(n,k) * x^n * y^k. - Michael Somos, Sep 17 2014

a(n) = n - S(n) where S(n) = sum of distinct terms in {a(1), a(2), ..., a(n-1)}. - David James Sycamore, Mar 10 2025

Extensions

More terms from Reinhard Zumkeller, Apr 27 2006

Incorrect program removed by Franklin T. Adams-Watters, Mar 19 2010

New name from Omar E. Pol, Jul 15 2012