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%I #173 Nov 29 2023 19:11:09
%S 1,2,1,3,2,1,4,3,2,1,5,4,3,2,1,6,5,4,3,2,1,7,6,5,4,3,2,1,8,7,6,5,4,3,
%T 2,1,9,8,7,6,5,4,3,2,1,10,9,8,7,6,5,4,3,2,1,11,10,9,8,7,6,5,4,3,2,1,
%U 12,11,10,9,8,7,6,5,4,3,2,1,13,12,11,10,9,8,7,6,5,4,3,2,1,14,13,12,11,10,9,8,7,6,5,4,3,2,1
%N Triangle read by rows: row n lists the first n positive integers in decreasing order.
%C Old name: Triangle T(n,k) = n-k, n >= 1, 0 <= k < n. Fractal sequence formed by repeatedly appending strings m, m-1, ..., 2, 1.
%C The PARI functions t1 (this sequence), t2 (A002260) can be used to read a square array T(n,k) (n >= 1, k >= 1) by antidiagonals upwards: n -> T(t1(n), t2(n)). - _Michael Somos_, Aug 23 2002, edited by _M. F. Hasler_, Mar 31 2020
%C A004736 is the mirror of the self-fission of the polynomial sequence (q(n,x)) given by q(n,x) = x^n+ x^(n-1) + ... + x + 1. See A193842 for the definition of fission. - _Clark Kimberling_, Aug 07 2011
%C Seen as flattened list: a(A000217(n)) = 1; a(A000124(n)) = n and a(m) <> n for m < A000124(n). - _Reinhard Zumkeller_, Jul 22 2012
%C Sequence B is called a reverse reluctant sequence of sequence A, if B is triangle array read by rows: row number k lists first k elements of the sequence A in reverse order. Sequence A004736 is the reverse reluctant sequence of sequence 1,2,3,... (A000027). - _Boris Putievskiy_, Dec 13 2012
%C The row sums equal A000217(n). The alternating row sums equal A004526(n+1). The antidiagonal sums equal A002620(n+1) respectively A008805(n-1). - _Johannes W. Meijer_, Sep 28 2013
%C From _Peter Bala_, Jul 29 2014: (Start)
%C Riordan array (1/(1-x)^2,x). Call this array M and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
%C /I_k 0\
%C \ 0 M/
%C having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the infinite matrix product M(0)*M(1)*M(2)*... is equal to A078812. (End)
%C T(n, k) gives the number of subsets of [n] := {1, 2, ..., n} with k consecutive numbers (consecutive k-subsets of [n]). - _Wolfdieter Lang_, May 30 2018
%C a(n) gives the distance from (n-1) to the smallest triangular number > (n-1). - _Ctibor O. Zizka_, Apr 09 2020
%C To construct the sequence, start from 1,2,,3,,,4,,,,5,,,,,6... where there are n commas after each "n". Then fill the empty places by the sequence itself. - _Benoit Cloitre_, Aug 17 2021
%C T(n,k) is the number of cycles of length 2*(k+1) in the (n+1)-ladder graph. There are no cycles of odd length. - _Mohammed Yaseen_, Jan 14 2023
%D H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973, pp 159-162.
%H Reinhard Zumkeller, <a href="/A004736/b004736.txt">Rows n = 1..100 of triangle, flattened</a>
%H Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, <a href="https://www.emis.de/journals/JIS/VOL21/Falcao/falcao2.html">Combinatorial Identities Associated with a Multidimensional Polynomial Sequence</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
%H Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Falcao/falcao5.html">Intrinsic Properties of a Non-Symmetric Number Triangle</a>, J. Int. Seq., Vol. 26 (2023), Article 23.4.8.
%H Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida, and Daisy Ann A. Disu, <a href="https://web.archive.org/web/20190407151217/http://www.dmmmsu-sluc.com/wp-content/uploads/2018/03/CAS-Monitor-2016-2017-1.pdf">On Fractal Sequences</a>, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
%H Clark Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/fractals.html">Fractal sequences</a>
%H 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 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%H F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/Sequences-book.pdf">Sequences of Numbers Involved in Unsolved Problems</a>.
%H Michael Somos, <a href="/A073189/a073189.txt">Sequences used for indexing triangular or square arrays</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LadderGraph.html">Ladder Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SmarandacheSequences.html">Smarandache Sequences</a>
%F a(n+1) = 1 + A025581(n).
%F a(n) = (2 - 2*n + round(sqrt(2*n)) + round(sqrt(2*n))^2)/2. - _Brian Tenneson_, Oct 11 2003
%F G.f.: 1 / ((1-x)^2 * (1-x*y)). - _Ralf Stephan_, Jan 23 2005
%F Recursion: e(n,k) = (e(n - 1, k)*e(n, k - 1) + 1)/e(n - 1, k - 1). - _Roger L. Bagula_, Mar 25 2009
%F a(n) = (t*t+3*t+4)/2-n, where t = floor((-1+sqrt(8*n-7))/2). - _Boris Putievskiy_, Dec 13 2012
%F From _Johannes W. Meijer_, Sep 28 2013: (Start)
%F T(n, k) = n - k + 1, n >= 1 and 1 <= k <= n.
%F T(n, k) = A002260(n+k-1, n-k+1). (End)
%F a(n) = A000217(A002024(n)) - n + 1. - _Enrique Pérez Herrero_, Aug 29 2016
%e The triangle T(n, k) starts:
%e n\k 1 2 3 4 5 6 7 8 9 10 11 12 ...
%e 1: 1
%e 2: 2 1
%e 3: 3 2 1
%e 4: 4 3 2 1
%e 5: 5 4 3 2 1
%e 6: 6 5 4 3 2 1
%e 7: 7 6 5 4 3 2 1
%e 8: 8 7 6 5 4 3 2 1
%e 9: 9 8 7 6 5 4 3 2 1
%e 10: 10 9 8 7 6 5 4 3 2 1
%e 11: 11 10 9 8 7 6 5 4 3 2 1
%e 12: 12 11 10 9 8 7 6 5 4 3 2 1
%e ... Reformatted. - _Wolfdieter Lang_, Feb 04 2015
%e T(6, 3) = 4 because the four consecutive 3-subsets of [6] = {1, 2, ..., 6} are {1, 2, 3}, {2, 3, 4}, {3, 4, 5} and {4, 5, 6}. - _Wolfdieter Lang_, May 30 2018
%p A004736 := proc(n,m) n-m+1 ; end:
%p T := (n, k) -> n-k+1: seq(seq(T(n,k), k=1..n), n=1..13); # _Johannes W. Meijer_, Sep 28 2013
%t Flatten[ Table[ Reverse[ Range[n]], {n, 12}]] (* _Robert G. Wilson v_, Apr 27 2004 *)
%t Table[Range[n,1,-1],{n,20}]//Flatten (* _Harvey P. Dale_, May 27 2020 *)
%o (PARI) {a(n) = 1 + binomial(1 + floor(1/2 + sqrt(2*n)), 2) - n}
%o (PARI) {t1(n) = binomial( floor(3/2 + sqrt(2*n)), 2) - n + 1} /* A004736 */
%o (PARI) {t2(n) = n - binomial( floor(1/2 + sqrt(2*n)), 2)} /* A002260 */
%o (PARI) apply( A004736(n)=1-n+(n=sqrtint(8*n)\/2)*(n+1)\2, [1..99]) \\ _M. F. Hasler_, Mar 31 2020
%o (Excel) =if(row()>=column();row()-column()+1;"") [_Mats Granvik_, Jan 19 2009]
%o (Haskell)
%o a004736 n k = n - k + 1
%o a004736_row n = a004736_tabl !! (n-1)
%o a004736_tabl = map reverse a002260_tabl
%o -- _Reinhard Zumkeller_, Aug 04 2014, Jul 22 2012
%o (Python)
%o def agen(rows):
%o for n in range(1, rows+1): yield from range(n, 0, -1)
%o print([an for an in agen(13)]) # _Michael S. Branicky_, Aug 17 2021
%Y Cf. A000217, A002024, A002262, A003056, A025581.
%Y Ordinal transform of A002260. See also A078812.
%Y Cf. A141419 (partial sums per row).
%Y Cf. A134546 (T * A051731, matrix product).
%Y See A001511 for definition of ordinal transform.
%K nonn,easy,tabl,nice
%O 1,2
%A R. Muller
%E New name from _Omar E. Pol_, Jul 15 2012
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