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]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

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)

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.

Cf. A107985. [From Clark Kimberling (ck6(AT)evansville.edu), Sep 16 2008]

A001563 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 29 2008]

Adjacent sequences: A002021 A002022 A002023 this_sequence A002025 A002026 A002027

Sequence in context: A023965 A087847 A107436 this_sequence A123578 A087845 A130146

nonn,easy,nice,tabl

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