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A002262 Triangle read by rows: T(n,k), n>=0, k>=0, in which row n lists the first n nonnegative integers. 158
0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Old name: Integers 0 to n followed by integers 0 to n+1 etc.

a(n) = n - the largest triangular number <= n. - Amarnath Murthy, Dec 25 2001

The PARI functions t1, t2 can be used to read a square array T(n,k) (n >= 0, k >= 0) by antidiagonals downwards: n -> T(t1(n), t2(n)). - Michael Somos, Aug 23, 2002

Values x of unique solution pair (x,y) to equation T(x+y) + x = n, where T(k)=A000217(k). - Lekraj Beedassy, Aug 21 2004

a(A000217(n)) = 0; a(A000096(n)) = n. - Reinhard Zumkeller, May 20 2009

Concatenation of the set representation of ordinal numbers, where the n_th ordinal number is represented by the set of all ordinals preceding n, 0 being represented by the empty set. - Daniel Forgues, Apr 27 2011

An integer sequence is nonnegative if and only if it is a subsequence of this sequence. - Charles R Greathouse IV, Sep 21 2011

a(A195678(n)) = A000040(n) and a(m) <> A000040(n) for m < A195678(n), an example of the preceding comment. - Reinhard Zumkeller, Sep 23 2011

A 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. A002262 is reluctant sequence of 0,1,2,3,... The nonnegative integers  A001477. - Boris Putievskiy, Dec 12 2012

LINKS

Charles R Greathouse IV, Rows n = 0..100, flattened

Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732, 2012.

M. Somos, Sequences used for indexing triangular or square arrays

FORMULA

a(n) = (n-((trinv(n)*(trinv(n)-1))/2)); trinv := n -> floor((1+sqrt(1+8*n))/2) (cf. A002024); # Gives integral inverses of triangular numbers.

a(n)=n-A000217(A003056(n))=n-A057944(n). - Lekraj Beedassy, Aug 21 2004

a(n) = A140129(A023758(n+2)). - Reinhard Zumkeller, May 14 2008

a(n)=f(n,1) with f(n,m) = if n<m then n else f(n-m,m+1). - Reinhard Zumkeller, May 20 2009

EXAMPLE

From Daniel Forgues, Apr 27 2011: (Start)

  Examples of set-theoretic representation of ordinal numbers:

  0: {}

  1: {0} = {{}}

  2: {0, 1} = {0, {0}} = {{}, {{}}}

  3: {0, 1, 2} = {{}, {0}, {0, 1}} = ... = {{}, {{}}, {{}, {{}}}} (End)

From Omar E. Pol, Jul 15 2012: (Start)

0;

0, 1;

0, 1, 2;

0, 1, 2, 3;

0, 1, 2, 3, 4;

0, 1, 2, 3, 4, 5;

0, 1, 2, 3, 4, 5, 6;

0, 1, 2, 3, 4, 5, 6, 7;

0, 1, 2, 3, 4, 5, 6, 7, 8;

0, 1, 2, 3, 4, 5, 6, 7, 8, 9;

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10;

(End)

MAPLE

seq(seq(i, i=0..n), n=0..12); # Peter Luschny, Sep 22 2011

A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))), 2);

MATHEMATICA

m[n_] := Floor[(-1 + Sqrt[8 n - 7])/2]

b[n_] := n - m[n] (m[n] + 1)/2

Table[m[n], {n, 1, 100}]     (* A003056 *)

Table[b[n], {n, 1, 100}]     (* A002260 *)

Table[b[n] - 1, {n, 1, 100}] (* A002262 *)

(* Clark Kimberling, Jun 14 2011 *)

Flatten[Table[k, {n, 0, 12}, {k, 0, n}]] (* Alonso del Arte, Sep 21 2011 *)

PROG

(PARI) a(n)=n-binomial(round(sqrt(2+2*n)), 2)

(PARI) t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2) /* A002262, this sequence */

(PARI) t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1) /* A025581, cf. comment by Somos for reading arrays by antidiagonals */

(PARI) concat(vector(13, n, vector(n, i, i-1)))  \\ M. F. Hasler, Sep 21 2011

(Haskell)

a002262 n k = a002262_tabl !! n !! k

a002262_row n = a002262_tabl !! n

a002262_tabl = map (enumFromTo 0) [0..]

-- Reinhard Zumkeller, Jul 13 2012, Mar 23 2011, Mar 07 2011

CROSSREFS

A002260(n)=1+a(n).

Cf. A025675, A025682, A025691, A002024, A048645, A004736, A025581. As a sequence, essentially same as A048151.

Cf. A053645, A053186, A056558, A127324.

Sequence in context: A025690 A025668 A048151 * A025675 A025682 A025691

Adjacent sequences:  A002259 A002260 A002261 * A002263 A002264 A002265

KEYWORD

nonn,tabl,easy,nice

AUTHOR

Angele Hamel (amh(AT)maths.soton.ac.uk)

EXTENSIONS

New name by Omar E. Pol, Jul 15 2012

STATUS

approved

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Last modified September 1 09:59 EDT 2014. Contains 246289 sequences.