1,2

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 (ck6(AT)evansville.edu), 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 (qntmpkt(AT)yahoo.com), Dec 29 2008]

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.

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

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)gmail.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)gmail.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.rs), 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)gmail.com), Jun 24 2007

a(n)=ceiling( -1/2 + sqrt(2n) ) [From Branko Curgus (curgus(AT)wwu.edu), May 12 2009]

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);

a[1] = 1; a[n_] := a[n] = a[n - a[n - 1]] + 1 [From Branko Curgus (curgus(AT)wwu.edu), May 12 2009]

(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)

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.

Sequence in context: A023965 A087847 A107436 this_sequence A123578 A087845 A130146

Adjacent sequences: A002021 A002022 A002023 this_sequence A002025 A002026 A002027

nonn,easy,nice,tabl

N. J. A. Sloane (njas(AT)research.att.com).