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A002260 Triangle T(n,k) = k for k = 1..n. 398
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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 0s 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

From Paul Curtz, Jul 25 2011: (Start)

Akiyama-Tanigawa algorithm from A000027(n) gives

   1,  2,  3,  4,  5,  6,  7,  8,

  -1, -2, -3, -4, -5, -6, -7, -8,

   1,  2,  3,  4,  5,  6,  7,  8,

  -1, -2, -3, -4, -5, -6, -7, -8.

By antidiagonals:

   1,

  -1,  2,

   1, -2,  3,

  -1,  2, -3,  4,

   1, -2,  3, -4,  5,

  -1,  2, -3,  4, -5,  6.

Row sum = A016116. (End)

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

REFERENCES

Jerry Brown et al., Problem 4619, "School Science and Mathematics", USA, Vol. 97(4), 1997, pp. 221-222.

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

C. Kimberling, Fractal sequences

C. 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]

F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.

Aaron Snook, Augmented Integer Linear Recurrences, 2012 - N. J. A. Sloane, Dec 19 2012

M. 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.

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

EXAMPLE

First six rows:

  1

  1   2

  1   2   3

  1   2   3   4

  1   2   3   4   5

  1   2   3   4   5   6

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 (* Or *)

f[n_] := Block[{m = Floor[(Sqrt[8n - 7] - 1)/2]}, n - m(m + 1)/2]; Array[f, 91]  (* Robert G. Wilson v, May 10 2011 *)

...

m[n_] := Floor[(-1 + Sqrt[8 n - 7])/2]

b[n_] := n - m[n] (m[n] + 1)/2

Table[m[n], {n, 1, 100}]     (* A003056 *)

Table[b[n], {n, 1, 100}]     (* A002260 *)

Table[b[n] - 1, {n, 1, 100}] (* A002262 *)

...

(* Clark Kimberling, Jun 14 2011 *)

z=13; p[0, x_]:=1;

p[n_, x_]:=x*p[n-1, x]+1;

q[n_, x_]:=p[n, x];

p1[n_, k_]:=Coefficient[p[n, x], x^k]; p1[n_, 0]:=p[n, x]/.x->0;

d[n_, x_]:=Sum[p1[n, k]*q[n-1-k, x], {k, 0, n-1}]

h[n_]:=CoefficientList[d[n, x], {x}]

TableForm[Table[Reverse[h[n]], {n, 0, z}]]

Flatten[Table[Reverse[h[n]], {n, -1, z}]] (* A002260 *)

TableForm[Table[h[n], {n, 0, z}]]

Flatten[Table[h[n], {n, -1, z}]] (* A004736 *)

(* Clark Kimberling, Aug 07 2011 *)

Flatten[Table[Range[n], {n, 20}]] (* Harvey P. Dale, Jun 20 2013 *)

PROG

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

(PARI) t2(n)=binomial(floor(3/2+sqrt(2*n)), 2)-n+1 /* A004736 */

(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

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

CROSSREFS

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

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.

Sequence in context: A023121 A136261 A140756 * A243732 A194905 A243730

Adjacent sequences:  A002257 A002258 A002259 * A002261 A002262 A002263

KEYWORD

nonn,easy,nice,tabl,look

AUTHOR

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

EXTENSIONS

More terms from Reinhard Zumkeller, Apr 27 2006

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

New name by Omar E. Pol, Jul 15 2012

STATUS

approved

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Last modified September 19 20:08 EDT 2017. Contains 292247 sequences.