%I A003056
%S A003056 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,
%T A003056 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,
%U A003056 10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12
%N A003056 n appears n+1 times. Also table T(n,k)=n+k read by antidiagonals.
%C A003056 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
%C A003056 Number of terms in partition of n with greatest number of distinct terms. 
               - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 20 2001
%C A003056 Summation table for (x+y) = (0+0),(0+1),(1+0),(0+2),(1+1),(2+0), ...
%C A003056 Also the number of triangular number less than or equal to n, not counting 
               0 as triangular. - Robert G. Wilson v.
%C A003056 Permutation of A116939: a(n)=A116939(A116941(n)), a(A116942(n))=A116939(n). 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 27 2006
%C A003056 Maximal size of partitions of n into distinct parts, see A000009. [From 
               Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 13 2009]
%H A003056 M. Somos, <a href="a073189.txt">Sequences used for indexing triangular 
               or square arrays</a>
%F A003056 a(n) = floor((sqrt(1+8*n)-1)/2) - Antti Karttunen
%F A003056 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
%F A003056 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]
%p A003056 a := [ 0 ]: for i from 1 to 15 do for j from 1 to i+1 do a := [ op(a),
               i ]; od: od: a;
%t A003056 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)
%o A003056 (PARI) a(n)=floor(-1/2+sqrt(2+2*n))
%o A003056 (PARI) t1(n)=floor(-1/2+sqrt(2+2*n)) /* A003056 */
%o A003056 (PARI) t2(n)=n-binomial(floor(1/2+sqrt(2+2*n)),2) /* A002262 */
%Y A003056 a(n) = A002024(n+1)-1. Cf. A004247 (multiplication table), A050600, A050602, 
               A001462, A048645.
%Y A003056 Partial sums of A073424.
%Y A003056 Cf. A002024.
%Y A003056 Sequence in context: A023966 A088141 A083291 this_sequence A117707 A163352 
               A087834
%Y A003056 Adjacent sequences: A003053 A003054 A003055 this_sequence A003057 A003058 
               A003059
%K A003056 nonn,easy,nice,tabl
%O A003056 0,4
%A A003056 N. J. A. Sloane (njas(AT)research.att.com).