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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A004736 Triangle read by rows: row n lists the first n positive integers in decreasing order.

Original entry on oeis.org

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, 8, 7, 6, 5, 4, 3, 2, 1
Offset: 1

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Author

R. Muller

Keywords

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
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
The first 77 entries are also the signature sequence of log(3)=A002391. Then the two sequences start to differ. - R. J. Mathar, May 27 2024

Examples

			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 [6] = {1, 2, ..., 6} are {1, 2, 3}, {2, 3, 4}, {3, 4, 5} and {4, 5, 6}. - _Wolfdieter Lang_, May 30 2018

References

  • H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973, pp 159-162.

Links

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.
Cf. A128174 (parity).

Programs

  • 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
  • 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 *)
  • 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
  • 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
  • Python
    from math import comb, isqrt
def A004736(n): return comb((m:=isqrt(k:=n<<1))+(k>m*(m+1))+1,2)+1-n # Chai Wah Wu, Nov 08 2024

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

Extensions

New name from Omar E. Pol, Jul 15 2012

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