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A004736 Triangle read by rows: row n lists the first n positive integers in decreasing order. 255
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.

"Smarandache Decrescendo Subsequences".

The PARI functions t1, t2 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

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)

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

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], 2012.

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

M. 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

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 *)

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, Jun 14 2011 *)

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 */

(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

CROSSREFS

Cf. A000217, A002024, A002262, A003056, A025581.

Ordinal transform of A002260. A078812.

Cf. A141419 (partial sums per row).

Cf. A134546 (T * A051731, matrix product).

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 March 26 16:31 EDT 2017. Contains 284137 sequences.