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A002024
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n appears n times.
(Formerly M0250 N0089)
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132
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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, 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, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13
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OFFSET
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1,2
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COMMENTS
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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
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
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
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
Integer inverse function of the triangular numbers A000217.
Array T(k,n) = n+k-1 read by antidiagonals.
Contribution from Clark Kimberling, Sep 16 2008: (Start)
As a rectangular array, a northwest corner:
1 2 3 4 5 6
2 3 4 5 6 7
3 4 5 6 7 8
4 5 6 7 8 9
This is the weight array (cf. A144112) of A107985 (formatted as a rectangular array). (End)
Eigensequence of the triangle = A001563 [From Gary W. Adamson, Dec 29 2008]
a(A169581(n)) = A038567(n). [From Reinhard Zumkeller, Dec 02 2009]
If m=(1,2,3,4,..,) then A002024(n)+m=A014132 [From Vincenzo Librandi, Jul 08 2010]
Can apparently also be defined via a(n+1)=b(n) for n>=2 where b(0)=b(1)=1 and b(n) = b(n-b(n-2))+1. Tested to be correct at least up to n<=150000. - José María Grau Ribas, Jun 10 2011
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REFERENCES
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Bae, Jaegug; Choi, Sungjin. A generalization of a subset-sum-distinct sequence. J. Korean Math. Soc. 40 (2003), no. 5, 757--768. MR1996839 (2004d:05198). See b(n).
E. S. Barbeau et al., Five Hundred Mathematical Challenges, Prob. 441, pp. 41, 194. MAA 1995.
H. W. Gould, Solution to Problem 571, Math. Mag., 38 (1965), 185-187.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 97.
K. Hardy & K. S. Williams, The Green Book of Mathematical Problems, p. 59 Soln. Prob. 14 Dover NY 1985
R. Honsberger, Mathematical Morsels, pp. 133-4 DME no. 3 MAA 1978
J. F. Hurley, Litton's Problematical Recreations, pp. 152;313-4 Prob. 22 VNR Co. NY 1971
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 43.
M. A. Nyblom, Some curious sequences ..., Am. Math. Monthly 109 (#6, 200), 559-564.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 129.
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LINKS
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Franklin T. Adams-Watters, Table of n, a(n) for n = 1..5050
Abraham Isgur, Vitaly Kuznetsov, and Stephen Tanny, A combinatorial approach for solving certain nested recursions with non-slow solutions, Arxiv preprint arXiv:1202.0276, 2012
Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732, 2012.
M. Somos, Sequences used for indexing triangular or square arrays
Eric Weisstein's World of Mathematics, Self-Counting Sequence
Index entries for Hofstadter-type sequences
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FORMULA
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a(n) = floor( 1/2 + sqrt(2n) ). Also a(n)=ceil((sqrt(1+8*n)-1)/2).
a((k - 1 ) * k / 2 + i) = k for k > 0 and 0 < i <= k. - Reinhard Zumkeller, Aug 30 2001
a(n) = a(n - a(n-1)) + 1. - Ian M. Levitt (ilevitt(AT)duke.poly.edu), Aug 18 2002
a(n) = round(sqrt(2*n)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002
T(n,k)=A003602(A118413(n,k)); = T(n,k)=A001511(A118416(n,k)). - Reinhard Zumkeller, Apr 27 2006
G.f.: x/(1-x)*Product_{k>0} (1-x^(2*k))/(1-x^(2*k-1)). - Vladeta Jovovic, Oct 06 2003
Equals A127899 * A004736 - Gary W. Adamson, Feb 09 2007
a(n)=sum{i=0..n-1, A010054(i)} - Paolo P. Lava, Apr 02 2007
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, Jun 24 2007
a(n)=ceiling( -1/2 + sqrt(2n) ) [From Branko Curgus, May 12 2009]
We know that a(n)=round(sqrt(2*n))=round(sqrt(2*n-1)); now exist exactly a and b greather than zero, that: 2*n = 2+(a+b)^2 -(a+3*b), we have: a(n)=(a+b-1) in closed formula. [From Fabio Civolani (civox(AT)tiscali.it), Feb 23 2010]
A005318(n+1) = 2*A005318(n)-A205744(n), A205744(n) = A005318(A083920(n)), A083920(n) = n - A002024(n). - N. J. A. Sloane, Feb 11 2012
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MAPLE
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A002024 := n-> ceil((sqrt(1+8*n)-1)/2);
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MATHEMATICA
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a[1] = 1; a[n_] := a[n] = a[n - a[n - 1]] + 1 (* From Branko Curgus, May 12 2009 *)
Table[n, {n, 13}, {n}] // Flatten (* From Robert G. Wilson v, May 11 2010 *)
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PROG
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(PARI) t1(n)=floor(1/2+sqrt(2*n)) /* A002024 */
(PARI) t2(n)=n-binomial(floor(1/2+sqrt(2*n)), 2) /* A002260(n-1) */
(PARI) t3(n)=binomial(floor(3/2+sqrt(2*n)), 2)-n+1 /* A004736 */
(PARI) t4(n)=n-1-binomial(floor(1/2+sqrt(2*n)), 2) /* A002260(n-1)-1 */
(PARI) a(n)=if(n<0, 0, floor(1/2+sqrt(2*n)))
(PARI) a(n)=if(n<1, 0, (sqrtint(8*n-7)+1)\2)
(Haskell)
a002024 = round . sqrt . (* 2) . fromIntegral
a002024_list = concat $ zipWith ($) (map replicate [1..]) [1..]
-- Reinhard Zumkeller, Feb 12 2012, Mar 18 2011
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CROSSREFS
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a(n+1) = 1+A003056(n), A022846(n)=a(n^2), a(n+1)=A002260(n)+A025581(n).
Cf. A001462, A002262, A025581, A002260, A004736.
Cf. A003056, A127899, A004736, A107985, A001563.
A123578 is an essentially identical sequence.
Cf. A014132 [From Vincenzo Librandi, Jul 08 2010]
Cf. A000194, A005145, A131507, A093995.
Sequence in context: A087847 A107436 * A123578 A087845 A130146 A113764
Adjacent sequences: A002021 A002022 A002023 * A002025 A002026 A002027
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KEYWORD
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nonn,easy,nice,tabl,changed
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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