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 A002262 Triangle read by rows: T(n,k), 0 <= k <= n, in which row n lists the first n+1 nonnegative integers. 222
 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, 6, 7, 8, 9, 10, 11, 12, 13 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS The point with coordinates (x = A025581(n), y = A002262(n)) sweeps out the first quadrant by upwards antidiagonals. N. J. A. Sloane, Jul 17 2018 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 [math.CO], 2012. Michael Somos, Sequences used for indexing triangular or square arrays FORMULA a(n) = A002260(n) - 1. 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 n - binomial(floor((1/2)+sqrt(2*(1+n))), 2); MATHEMATICA m[n_]:= Floor[(-1 + Sqrt[8n - 7])/2] b[n_]:= n - m[n] (m[n] + 1)/2 Table[m[n], {n, 1, 105}]     (* A003056 *) Table[b[n], {n, 1, 105}]     (* A002260 *) Table[b[n] - 1, {n, 1, 120}] (* A002262 *) (* Clark Kimberling, Jun 14 2011 *) Flatten[Table[k, {n, 0, 14}, {k, 0, n}]] (* Alonso del Arte, Sep 21 2011 *) Flatten[Table[Range[0, n], {n, 0, 15}]] (* Harvey P. Dale, Jan 31 2015 *) 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(15, 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..] a002262_list = concat a002262_tabl -- Reinhard Zumkeller, Aug 05 2015, Jul 13 2012, Mar 07 2011 (Python) for i in range(16):     for j in range(i):         print(j, end=", ") # Mohammad Saleh Dinparvar, May 13 2020 CROSSREFS Cf. A002024, A002260, A004736, A025581, A025675, A025682. Cf. A025691, A048645, A053186, A053645, A056558, A127324. As a sequence, essentially same as A048151. Sequence in context: A025675 A025682 A025691 * A298486 A189768 A262881 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 from Omar E. Pol, Jul 15 2012 Typo in definition fixed by Reinhard Zumkeller, Aug 05 2015 STATUS approved

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Last modified October 29 15:09 EDT 2020. Contains 338066 sequences. (Running on oeis4.)