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A002262
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Triangle read by rows: T(n,k), n>=0, k>=0, in which row n lists the first n nonnegative integers.
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132
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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)
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OFFSET
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0,6
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COMMENTS
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Old name: Integers 0 to n followed by integers 0 to n+1 etc.
a(n) = n - the largest triangular number <= n. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), 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, April 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
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LINKS
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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
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FORMULA
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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]
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EXAMPLE
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[Daniel Forgues, April 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)
Contribution 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)
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MAPLE
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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);
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,tabl,easy,nice,changed
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AUTHOR
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Angele Hamel (amh(AT)maths.soton.ac.uk)
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EXTENSIONS
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New name by Omar E. Pol, Jul 15 2012
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STATUS
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approved
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