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 A004736 Triangle read by rows: row n lists the first n positive integers in decreasing order. 322
 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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. 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 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 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 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 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 From Peter Bala, Jul 29 2014: (Start) 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 /I_k 0\ \ 0  M/ 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) 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 a(n) gives the distance from (n-1) to the smallest triangular number > (n-1). - Ctibor O. Zizka, Apr 09 2020 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 REFERENCES H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973, pp 159-162. LINKS Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4. 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. Michael Somos, Sequences used for indexing triangular or square arrays Eric Weisstein's World of Mathematics, Smarandache Sequences FORMULA a(n+1) = 1 + A025581(n). a(n) = (2 - 2*n + round(sqrt(2*n)) + round(sqrt(2*n))^2)/2. - Brian Tenneson, Oct 11 2003 G.f.: 1 / ((1-x)^2 * (1-x*y)). - Ralf Stephan, Jan 23 2005 Recursion: e(n,k) = (e(n - 1, k)*e(n, k - 1) + 1)/e(n - 1, k - 1). - Roger L. Bagula, Mar 25 2009 a(n) = (t*t+3*t+4)/2-n, where t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 13 2012 From Johannes W. Meijer, Sep 28 2013: (Start) T(n, k) = n - k + 1, n >= 1 and 1 <= k <= n. T(n, k) = A002260(n+k-1, n-k+1). (End) a(n) = A000217(A002024(n)) - n + 1. - Enrique Pérez Herrero, Aug 29 2016 EXAMPLE The triangle T(n, k) starts: n\k  1   2   3  4  5  6  7  8  9 10 11 12 ... 1:   1 2:   2   1 3:   3   2   1 4:   4   3   2  1 5:   5   4   3  2  1 6:   6   5   4  3  2  1 7:   7   6   5  4  3  2  1 8:   8   7   6  5  4  3  2  1 9:   9   8   7  6  5  4  3  2  1 10: 10   9   8  7  6  5  4  3  2  1 11: 11  10   9  8  7  6  5  4  3  2  1 12: 12  11  10  9  8  7  6  5  4  3  2  1 ... Reformatted. - Wolfdieter Lang, Feb 04 2015 T(6, 3) = 4 because the four consecutive 3-subsets of  = {1, 2, ..., 6} are {1, 2, 3}, {2, 3, 4}, {3, 4, 5} and {4, 5, 6}. - Wolfdieter Lang, May 30 2018 MAPLE A004736 := proc(n, m) n-m+1 ; end: T := (n, k) -> n-k+1: seq(seq(T(n, k), k=1..n), n=1..13); # Johannes W. Meijer, Sep 28 2013 MATHEMATICA Flatten[ Table[ Reverse[ Range[n]], {n, 12}]] (* Robert G. Wilson v, Apr 27 2004 *) Table[Range[n, 1, -1], {n, 20}]//Flatten (* Harvey P. Dale, May 27 2020 *) PROG (PARI) {a(n) = 1 + binomial(1 + floor(1/2 + sqrt(2*n)), 2) - n} (PARI) {t1(n) = binomial( floor(3/2 + sqrt(2*n)), 2) - n + 1} /* A004736 */ (PARI) {t2(n) = n - binomial( floor(1/2 + sqrt(2*n)), 2)} /* A002260 */ (PARI) apply( A004736(n)=1-n+(n=sqrtint(8*n)\/2)*(n+1)\2, [1..99]) \\ M. F. Hasler, Mar 31 2020 (Excel) =if(row()>=column(); row()-column()+1; "") [Mats Granvik, Jan 19 2009] (Haskell) a004736 n k = n - k + 1 a004736_row n = a004736_tabl !! (n-1) a004736_tabl = map reverse a002260_tabl -- Reinhard Zumkeller, Aug 04 2014, Jul 22 2012 (Python) def agen(rows):     for n in range(1, rows+1): yield from range(n, 0, -1) print([an for an in agen(13)]) # Michael S. Branicky, Aug 17 2021 CROSSREFS Cf. A000217, A002024, A002262, A003056, A025581. Ordinal transform of A002260. See also A078812. Cf. A141419 (partial sums per row). Cf. A134546 (T * A051731, matrix product). See A001511 for definition of ordinal transform. Sequence in context: A194877 A102482 A194908 * A200370 A200443 A167288 Adjacent sequences:  A004733 A004734 A004735 * A004737 A004738 A004739 KEYWORD nonn,easy,tabl,nice AUTHOR R. Muller EXTENSIONS New name from Omar E. Pol, Jul 15 2012 STATUS approved

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Last modified October 6 12:25 EDT 2022. Contains 357264 sequences. (Running on oeis4.)