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Search: id:A002024
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| A002024 |
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n appears n times.
(Formerly M0250 N0089)
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+0
73
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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
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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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.
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REFERENCES
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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.
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
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 (reinhard.zumkeller(AT)lhsystems.com), 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 (reinhard.zumkeller(AT)lhsystems.com), Apr 27 2006
G.f.: x/(1-x)*Product_{k>0} (1-x^(2*k))/(1-x^(2*k-1)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 06 2003
Equals A127899 * A004736 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 09 2007
a(n)=sum{i=0..oo, A010054} - Paolo P. Lava (ppl(AT)spl.at), 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 (reinhard.zumkeller(AT)lhsystems.com), Jun 24 2007
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MAPLE
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a := [ ]: for i from 1 to 15 do for j from 1 to i do a := [ op(a), i ]; od: od: a;
A002024 := n-> ceil((sqrt(1+8*n)-1)/2);
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PROGRAM
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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)
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CROSSREFS
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Cf. A001462, A002262, A025581, A002260, A004736.
a(n+1) = 1+A003056(n), A022846(n)=a(n^2), a(n+1)=A002260(n)+A025581(n).
Cf. A003056.
Cf. A127899, A004736.
Adjacent sequences: A002021 A002022 A002023 this_sequence A002025 A002026 A002027
Sequence in context: A023965 A087847 A107436 this_sequence A123578 A087845 A130146
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KEYWORD
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nonn,easy,nice,tabl
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AUTHOR
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njas
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