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

%I A002260 #230 Jun 06 2025 08:33:53
%S A002260 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,
%T A002260 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,
%U A002260 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
%N A002260 Triangle read by rows: T(n,k) = k for n >= 1, k = 1..n.
%C A002260 Old name: integers 1 to k followed by integers 1 to k+1 etc. (a fractal sequence).
%C A002260 Start counting again and again.
%C A002260 This is a "doubly fractal sequence" - see the _Franklin T. Adams-Watters_ link.
%C A002260 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
%C A002260 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
%C A002260 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
%C A002260 Eigensequence of the triangle = A001710 starting (1, 3, 12, 60, 360, ...). - _Gary W. Adamson_, Aug 02 2010
%C A002260 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
%C A002260 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
%C A002260 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
%C A002260 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
%C A002260 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
%C A002260 T(n,k) gives the distance to the largest triangular number < n. - _Ctibor O. Zizka_, Apr 09 2020
%D A002260 Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168. (Introduces upper trimming, lower trimming, and signature sequences.)
%D A002260 M. Myers, Smarandache Crescendo Subsequences, R. H. Wilde, An Anthology in Memoriam, Bristol Banner Books, Bristol, 1998, p. 19.
%D A002260 F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.
%H A002260 N. J. A. Sloane, <a href="/A002260/b002260.txt">Table of n, a(n) for n = 1..11325</a>
%H A002260 Franklin T. Adams-Watters, <a href="/A002260/a002260.txt">Doubly Fractal Sequences</a>
%H A002260 Matin Amini and Majid Jahangiri, <a href="https://arxiv.org/abs/1612.09481">A Novel Proof for Kimberling's Conjecture on Doubly Fractal Sequences</a>, arXiv:1612.09481 [math.NT], 2017.
%H A002260 Bruno Berselli, <a href="/A002260/a002260.jpg">Illustration of the initial terms</a>
%H A002260 Jerry Brown et al., <a href="https://doi.org/10.1111/j.1949-8594.1997.tb17373.x">Problem 4619</a>, School Science and Mathematics (USA), Vol. 97(4), 1997, pp. 221-222.
%H A002260 Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida, and Daisy Ann A. Disu, <a href="http://docplayer.net/87034980-Vol-15-no-2-april-2017-dmmmsu-cas-science-monitor.html">On Fractal Sequences</a>, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
%H A002260 Clark Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/fractals.html">Fractal sequences</a>
%H A002260 Clark Kimberling, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa73/aa7321.pdf">Numeration systems and fractal sequences</a>, Acta Arithmetica 73 (1995) 103-117.
%H A002260 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%H A002260 F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/Sequences-book.pdf">Sequences of Numbers Involved in Unsolved Problems</a>.
%H A002260 Aaron Snook, <a href="http://www.cs.cmu.edu/afs/cs/user/mjs/ftp/thesis-program/2012/theses/snook.pdf">Augmented Integer Linear Recurrences</a>, 2012. - _N. J. A. Sloane_, Dec 19 2012
%H A002260 Michael Somos, <a href="/A073189/a073189.txt">Sequences used for indexing triangular or square arrays</a>
%H A002260 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SmarandacheSequences.html">Smarandache Sequences</a>
%H A002260 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnitFraction.html">Unit Fraction</a>
%F A002260 a(n) = 1 + A002262(n).
%F A002260 n-th term is n - m*(m+1)/2 + 1, where m = floor((sqrt(8*n+1) - 1) / 2).
%F A002260 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
%F A002260 a(k * (k + 1) / 2 + i) = i for k >= 0 and 0 < i <= k + 1. - _Reinhard Zumkeller_, Aug 14 2001
%F A002260 a(n) = (2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2. - _Brian Tenneson_, Oct 11 2003
%F A002260 a(n) = n - binomial(floor((1+sqrt(8*n))/2), 2). - _Paul Barry_, May 25 2004
%F A002260 T(n,k) = A001511(A118413(n,k)); T(n,k) = A003602(A118416(n,k)). - _Reinhard Zumkeller_, Apr 27 2006
%F A002260 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
%F A002260 a(A169581(n)) = A038566(n). - _Reinhard Zumkeller_, Dec 02 2009
%F A002260 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
%F A002260 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
%F A002260 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
%F A002260 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
%e A002260 First six rows:
%e A002260   1
%e A002260   1   2
%e A002260   1   2   3
%e A002260   1   2   3   4
%e A002260   1   2   3   4   5
%e A002260   1   2   3   4   5   6
%p A002260 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
%p A002260 seq(seq(i,i=1..k),k=1..13); # _Peter Luschny_, Jul 06 2009
%t A002260 FoldList[{#1, #2} &, 1, Range[2, 13]] // Flatten (* _Robert G. Wilson v_, May 10 2011 *)
%t A002260 Flatten[Table[Range[n],{n,20}]] (* _Harvey P. Dale_, Jun 20 2013 *)
%o A002260 (PARI) t1(n)=n-binomial(floor(1/2+sqrt(2*n)),2) /* this sequence */
%o A002260 (PARI) A002260(n)=n-binomial((sqrtint(8*n)+1)\2,2) \\ _M. F. Hasler_, Mar 10 2014
%o A002260 (Haskell)
%o A002260 a002260 n k = k
%o A002260 a002260_row n = [1..n]
%o A002260 a002260_tabl = iterate (\row -> map (+ 1) (0 : row)) [1]
%o A002260 -- _Reinhard Zumkeller_, Aug 04 2014, Jul 03 2012
%o A002260 (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 */
%o A002260 (Python)
%o A002260 from math import isqrt, comb
%o A002260 def A002260(n): return n-comb((m:=isqrt(k:=n<<1))+(k>m*(m+1)),2) # _Chai Wah Wu_, Nov 08 2024
%Y A002260 Cf. A000217, A001710, A002262, A003056, A004736 (ordinal transform), A025581, A056534, A094727, A127779.
%Y A002260 Cf. A140756 (alternating signs).
%Y A002260 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).
%Y A002260 Cf. A108872.
%K A002260 nonn,easy,nice,tabl,look
%O A002260 1,3
%A A002260 Angele Hamel (amh(AT)maths.soton.ac.uk)
%E A002260 More terms from _Reinhard Zumkeller_, Apr 27 2006
%E A002260 Incorrect program removed by _Franklin T. Adams-Watters_, Mar 19 2010
%E A002260 New name from _Omar E. Pol_, Jul 15 2012