Search: id:A002024
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%I A002024 M0250 N0089
%S A002024 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,8,8,
%T A002024 8,8,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,
%U A002024 11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13
%N A002024 n appears n times.
%C A002024 The function trinv(n) = floor((1+sqrt(1+8n))/2), n>=0, gives the values 
               1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,..., that is, the same sequence with 
               offset 0. - N. J. A. Sloane, Jun 21 2009
%C A002024 The PARI functions t1, t2 can be used to read a triangular array T(n,
               k) (n >= 1, 1 <= k <= n) by rows from left to right: n -> T(t1(n), 
               t2(n)). - Michael Somos, Aug 23, 2002
%C A002024 The PARI functions t1, t3 can be used to read a triangular array T(n,
               k) (n >= 1, 1 <= k <= n) by rows from right to left: n -> T(t1(n), 
               t3(n)). - Michael Somos, Aug 23, 2002
%C A002024 The PARI functions t1, t4 can be used to read a triangular array T(n,
               k) (n >= 1, 0 <= k <= n-1) by rows from left to right: n -> T(t1(n), 
               t4(n)). - Michael Somos, Aug 23, 2002
%C A002024 Integer inverse function of the triangular numbers A000217.
%C A002024 Array T(k,n) = n+k-1 read by antidiagonals.
%C A002024 Contribution from Clark Kimberling (ck6(AT)evansville.edu), Sep 16 2008: 
               (Start)
%C A002024 As a rectangular array, a northwest corner:
%C A002024 1 2 3 4 5 6
%C A002024 2 3 4 5 6 7
%C A002024 3 4 5 6 7 8
%C A002024 4 5 6 7 8 9
%C A002024 This is the weight array (cf. A144112) of A107985 (formatted as a rectangular 
               array). (End)
%C A002024 Eigensequence of the triangle = A001563 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Dec 29 2008]
%D A002024 E. S. Barbeau et al., Five Hundred Mathematical Challenges, Prob. 441, 
               pp. 41, 194. MAA 1995.
%D A002024 H. W. Gould, Solution to Problem 571, Math. Mag., 38 (1965), 185-187.
%D A002024 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 1990, p. 97.
%D A002024 K. Hardy & K. S. Williams, The Green Book of Mathematical Problems, p. 
               59 Soln. Prob. 14 Dover NY 1985
%D A002024 R. Honsberger, Mathematical Morsels, pp. 133-4 DME no. 3 MAA 1978
%D A002024 J. F. Hurley, Litton's Problematical Recreations, pp. 152;313-4 Prob. 
               22 VNR Co. NY 1971
%D A002024 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, 
               MA, Vol. 1, p. 43.
%D A002024 M. A. Nyblom, Some curious sequences ..., Am. Math. Monthly 109 (#6, 
               200), 559-564.
%D A002024 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002024 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002024 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 129.
%H A002024 Franklin T. Adams-Watters, <a href="b002024.txt">Table of n, a(n) for 
               n = 1..5050</a>
%H A002024 M. Somos, <a href="a073189.txt">Sequences used for indexing triangular 
               or square arrays</a>
%H A002024 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Self-CountingSequence.html">Self-Counting Sequence</a>
%H A002024 <a href="Sindx_Ho.html#Hofstadter">Index entries for Hofstadter-type 
               sequences</a>
%F A002024 a(n) = floor( 1/2 + sqrt(2n) ). Also a(n)=ceil((sqrt(1+8*n)-1)/2).
%F A002024 a((k - 1 ) * k / 2 + i) = k for k > 0 and 0 < i <= k. - Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), Aug 30 2001
%F A002024 a(n) = a(n - a(n-1)) + 1. - Ian M. Levitt (ilevitt(AT)duke.poly.edu), 
               Aug 18 2002
%F A002024 a(n) = round(sqrt(2*n)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), 
               Nov 01 2002
%F A002024 T(n,k)=A003602(A118413(n,k)); = T(n,k)=A001511(A118416(n,k)). - Reinhard 
               Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 27 2006
%F A002024 G.f.: x/(1-x)*Product_{k>0} (1-x^(2*k))/(1-x^(2*k-1)). - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Oct 06 2003
%F A002024 Equals A127899 * A004736 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 
               09 2007
%F A002024 a(n)=sum{i=0..oo, A010054} - Paolo P. Lava (ppl(AT)spl.at), Apr 02 2007
%F A002024 Sum(Sum(T(j,i):i<=j<n+i):1<=i<=n)=A000578(n); Sum(T(n,i):1<=i<=n)=A000290(n). 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 24 2007
%F A002024 a(n)=ceiling( -1/2 + sqrt(2n) ) [From Branko Curgus (curgus(AT)wwu.edu), 
               May 12 2009]
%p A002024 a := [ ]: for i from 1 to 15 do for j from 1 to i do a := [ op(a),i ]; 
               od: od: a;
%p A002024 A002024 := n-> ceil((sqrt(1+8*n)-1)/2);
%t A002024 a[1] = 1; a[n_] := a[n] = a[n - a[n - 1]] + 1 [From Branko Curgus (curgus(AT)wwu.edu), 
               May 12 2009]
%o A002024 (PARI) t1(n)=floor(1/2+sqrt(2*n)) /* A002024 */
%o A002024 (PARI) t2(n)=n-binomial(floor(1/2+sqrt(2*n)),2) /* A002260(n-1) */
%o A002024 (PARI) t3(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1 /* A004736 */
%o A002024 (PARI) t4(n)=n-1-binomial(floor(1/2+sqrt(2*n)),2) /* A002260(n-1)-1 */
%o A002024 (PARI) a(n)=if(n<0,0,floor(1/2+sqrt(2*n)))
%o A002024 (PARI) a(n)=if(n<1,0,(sqrtint(8*n-7)+1)\2)
%Y A002024 a(n+1) = 1+A003056(n), A022846(n)=a(n^2), a(n+1)=A002260(n)+A025581(n).
%Y A002024 Cf. A001462, A002262, A025581, A002260, A004736.
%Y A002024 Cf. A003056, A127899, A004736, A107985, A001563.
%Y A002024 A123578 is an essentially identical sequence.
%Y A002024 Sequence in context: A023965 A087847 A107436 this_sequence A123578 A087845 
               A130146
%Y A002024 Adjacent sequences: A002021 A002022 A002023 this_sequence A002025 A002026 
               A002027
%K A002024 nonn,easy,nice,tabl
%O A002024 1,2
%A A002024 N. J. A. Sloane (njas(AT)research.att.com).

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