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A003056
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n appears n+1 times. Also table T(n,k)=n+k read by antidiagonals.
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90
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0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| The PARI functions t1, t2 can be used to read a triangular array T(n,k) (n >= 0, 0 <= k <= n-1) by rows from left to right: n -> T(t1(n), t2(n)). - Michael Somos, Aug 23, 2002
Number of terms in partition of n with greatest number of distinct terms. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 20 2001
Summation table for (x+y) = (0+0),(0+1),(1+0),(0+2),(1+1),(2+0), ...
Also the number of triangular number less than or equal to n, not counting 0 as triangular. - Robert G. Wilson v.
Permutation of A116939: a(n)=A116939(A116941(n)), a(A116942(n))=A116939(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 27 2006
Maximal size of partitions of n into distinct parts, see A000009. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 13 2009]
Also number of digits of A000462(n). [Reinhard Zumkeller, Mar 27 2011]
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
M. Somos, Sequences used for indexing triangular or square arrays
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FORMULA
| a(n) = floor((sqrt(1+8*n)-1)/2) - Antti Karttunen
a(n) = floor(-1/2+sqrt(2*n+b)) with 1/4<=b<9/4 or a(n) = floor((sqrt(8*n+b)-1)/2) with 1<=b<9. - childers_moof(AT)yahoo.com, Nov 11 2001
a(n) = f(n,0) with f(n,k) = if n<=k then k else f(n-k-1,k+1). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 23 2009]
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MAPLE
| A003056 := (n, k) -> n: # Peter Luschny, Oct 29 2011
a := [ 0 ]: for i from 1 to 15 do for j from 1 to i+1 do a := [ op(a), i ]; od: od: a;
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MATHEMATICA
| f[n_] := Floor[(Sqrt[1 + 8n] - 1)/2]; Table[ f[n], {n, 0, 87}] (* from Robert G. Wilson v (rgwv(at)rgwv.com), Oct 21 2005 *)
Table[x, {x, 0, 13}, {y, 0, x}] // Flatten
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PROG
| (PARI) A003056(n)=(sqrtint(8*n+1)-1)\2 } \\ M. F. Hasler, Oct 08 2011
(PARI) t1(n)=floor(-1/2+sqrt(2+2*n)) /* A003056 */
(PARI) t2(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2) /* A002262 */
(Haskell)
a003056 n = a003056_list !! (n-1)
a003056_list = concat $ zipWith ($) (map replicate [1..]) [0..]
-- Reinhard Zumkeller, Oct 17 2010, Mar 18 2011
(MAGMA) [Floor((Sqrt(1+8*n)-1)/2): n in [0..80]]; // Vincenzo Librandi, Oct 23 2011
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CROSSREFS
| a(n) = A002024(n+1)-1. Cf. A004247 (multiplication table), A050600, A050602, A001462, A048645.
Partial sums of A073424.
Cf. A002024, A131507.
Sequence in context: A185283 A083291 A169894 * A117707 A163352 A087834
Adjacent sequences: A003053 A003054 A003055 * A003057 A003058 A003059
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KEYWORD
| nonn,easy,nice,tabl
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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