|
%I
%S 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,
%T 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,
%U 10,10,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12
%N n appears n+1 times. Also table T(n,k) = n+k read by antidiagonals.
%C Also triangle read by rows: T(n,k), n>=0, k>=0, in which n appears n+1 times in row n. - _Omar E. Pol_, Jul 15 2012
%C 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 Number of terms in partition of n with greatest number of distinct terms. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 20 2001
%C Summation table for (x+y) = (0+0),(0+1),(1+0),(0+2),(1+1),(2+0), ...
%C Also the number of triangular numbers less than or equal to n, not counting 0 as triangular. - Robert G. Wilson v.
%C Permutation of A116939: a(n) = A116939(A116941(n)), a(A116942(n)) = A116939(n). - _Reinhard Zumkeller_, Feb 27 2006
%C Maximal size of partitions of n into distinct parts, see A000009. [From _Reinhard Zumkeller_, Jun 13 2009]
%C Also number of digits of A000462(n). [_Reinhard Zumkeller_, Mar 27 2011]
%C a(n) = 2*n + 1 - A001614(n+1) = n + 1 - A122797(n+1). [_Reinhard Zumkeller_, Feb 12 2012]
%C Also the maximum number of 1's contained in the list of hook-lengths of a partition of n. E.g. a(4)=2 because hooks of partitions of n=4 comprise of {4,3,2,1}, {4,2,1,1}, {3,2,2,1}, {4,1,2,1}, {4,3,2,1} where the number of 1's in each is 1,2,1,2,1. Hence the maximum is 2. [_T. Amdeberhan_, Jun 03 2012]
%C Fan, Yang, and Yu (2012) prove a conjecture of Amdeberhan on the generating function of a(n). - _Jonathan Sondow_, Dec 17 2012
%H Vincenzo Librandi, <a href="/A003056/b003056.txt">Table of n, a(n) for n = 0..10000</a>
%H Anna R. B. Fan, Harold R. L. Yang, Rebecca T. Yu, <a href="http://arxiv.org/abs/1212.3505">On the Maximum Number of k-Hooks of Partitions of n</a>, arXiv 2012
%H M. Somos, <a href="/A073189/a073189.txt">Sequences used for indexing triangular or square arrays</a>
%F a(n) = floor((sqrt(1+8*n)-1)/2). - _Antti Karttunen_
%F 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 a(n) = f(n,0) with f(n,k) = if n<=k then k else f(n-k-1,k+1). [From _Reinhard Zumkeller_, May 23 2009]
%F a(n) = k if k*(k+1)/2 <= n < (k+1)*(k+2)/2. - _Jonathan Sondow_, Dec 17 2012
%e As triangle, the sequence starts
%e 0,
%e 1, 1,
%e 2, 2, 2,
%e 3, 3, 3, 3,
%e 4, 4, 4, 4, 4,
%e 5, 5, 5, 5, 5, 5,
%e 6, 6, 6, 6, 6, 6, 6,
%e 7, 7, 7, 7, 7, 7, 7, 7,
%e 8, 8, 8, 8, ... etc.
%p A003056 := (n,k) -> n: # Peter Luschny, Oct 29 2011
%p 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 f[n_] := Floor[(Sqrt[1 + 8n] - 1)/2]; Table[ f[n], {n, 0, 87}] (* from _Robert G. Wilson v_, Oct 21 2005 *)
%t Table[x, {x, 0, 13}, {y, 0, x}] // Flatten
%o (PARI) A003056(n)=(sqrtint(8*n+1)-1)\2 } \\ M. F. Hasler, Oct 08 2011
%o (PARI) t1(n)=floor(-1/2+sqrt(2+2*n)) /* A003056 */
%o (PARI) t2(n)=n-binomial(floor(1/2+sqrt(2+2*n)),2) /* A002262 */
%o (Haskell)
%o a003056 n = a003056_list !! (n-1)
%o (Haskell)
%o a003056 = floor . (/ 2) . (subtract 1) .
%o sqrt . (+ 1) . (* 8) . fromIntegral
%o a003056_list = concat $ zipWith ($) (map replicate [1..]) [0..]
%o -- _Reinhard Zumkeller_, Feb 12 2012, Oct 17 2010, Mar 18 2011
%o (MAGMA) [Floor((Sqrt(1+8*n)-1)/2): n in [0..80]]; // Vincenzo Librandi, Oct 23 2011
%Y a(n) = A002024(n+1)-1. Cf. A004247 (multiplication table), A050600, A050602, A001462, A048645.
%Y Partial sums of A073424.
%Y Cf. A002024, A131507.
%K nonn,easy,nice,tabl
%O 0,4
%A _N. J. A. Sloane_.
|
