1,3

Start counting again and again.

This is a "doubly fractal sequence" - see the Franklin T. Adams-Watters link.

The PARI functions t1, t2 can be used to read a square array T(n,k) (n >= 1, k >= 1) by antidiagonals downwards: n -> T(t1(n), t2(n)). - Michael Somos, Aug 23, 2002

As a rectangular array, row n is (n,n,n,...); this is the weight array (Cf. A144112) of the array A127779 (rectangular). [From Clark Kimberling (ck6(AT)evansville.edu), Sep 16 2008]

Consider polynomials D(n,z)=sum[((n+2)^2-i^2)*z^(i-1))/i ,i=1,2,..,n+1. Coefficients are 3,8,5/2,15,6=12/2,7/3,24,21/2,16/3,9/4,35,16=32/2,9=27/3,5=20/4,11/5, =A120070/A002260. In A129326=3,5,14,54,. (A120070 leads to Rydberg's formula for spectra of hydrogen). Does this family correspond to a natural phenomena ? Same question with denominators A002260^2=A133819. Remark: in dimension 2, polynomials are P(0,x,y)=3*a1, P(1,x,y)=8*a1+(5/2)*(a2*x+a3*y), P(2,x,y)=15*a1+6*(a2*x+a3*y)+(7/3)*(a4*x^2+a5*x*y+a6*y^2), .. where ai's are parameters. From transformed saddle-points formulae.See reference. [From Paul Curtz (bpcrtz(AT)free.fr), Mar 17 2009]

Contribution from Clark Kimberling (ck6(AT)evansville.edu), Nov 02 2009: (Start)

The upper trim of an arbitrary fractal sequence s is s, but the lower trim

of s, although a fractal sequence, need not be s itself. However, the

lower trim of A002260 is A002260. (The upper trim of s is what remains

after the first occurrence of each term is deleted; the lower trim of s

is what remains after all 0s are deleted from the sequence s-1.) (End)

Jerry Brown et al., Problem 4619, "School Science and Mathematics", USA, Vol. 97(4), 1997, pp. 221-222.

C. Kimberling, "Numeration systems and fractal sequences," Acta Arithmetica 73 (1995) 103-117.

M. Myers, Smarandache Crescendo Subsequences, R.H.Wilde, An Anthology in Memoriam, Bristol Banner Books, Bristol, 1998, p. 19.

F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.

P. Curtz, Stabilite locale des systemes quadratiques. Ann. sc. Ecole Norm. Sup.,1980,293-302. [From Paul Curtz (bpcrtz(AT)free.fr), Mar 17 2009]

Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168. (Introduces upper trimming, lower trimming, and signature sequences.) [From Clark Kimberling (ck6(AT)evansville.edu), Nov 02 2009]

N. J. A. Sloane, Table of n, a(n) for n = 1..11325

Franklin T. Adams-Watters, Doubly Fractal Sequences

C. Kimberling, Fractal sequences

F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.

M. Somos, Sequences used for indexing triangular or square arrays

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

n-th term is n - m(m+1)/2 + 1, where m = [ (sqrt(8n+1) - 1) / 2 ].

a(k * (k + 1) / 2 + i) = i for k >= 0 and 0 < i <= k + 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 14 2001

a(n) = (2 n + round(SQRT(2 n)) - round(SQRT(2 n))^2)/2. E.g. a(47) = 2 - brian tenneson (phoenix(AT)alephnulldimension.net), Oct 11 2003

n+1-binomial(floor((1+sqrt(8n+8))/2), 2). - Paul Barry (pbarry(AT)wit.ie), May 25 2004

T(n,k)=A001511(A118413(n,k)); = T(n,k)=A003602(A118416(n,k)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 27 2006

a(A000217(n)) = A000217(n) - A000217(n-1), a(A000217(n-1) + 1) = 1, a(A000217(n) - 1) = A000217(n) - A000217(n-1) - 1. - Alexander R. Povolotsky (pevnev(AT)juno.com), May 28 2008

at:=0; for n from 1 to 150 do for i from 1 to n do at:=at+1; lprint(at, i); od: od: - N. J. A. Sloane (njas(AT)research.att.com), Nov 01 2006

seq(seq(i, i=1..k), k=1..13); [From Peter Luschny (peter(AT)luschny.de), Jul 06 2009]

(PARI) a(n)=n+1-binomial(floor(1/2+sqrt(2+2*n)), 2)

(PARI) t1(n)=n-binomial(floor(1/2+sqrt(2*n)), 2) /* A002260 */

(PARI) t2(n)=binomial(floor(3/2+sqrt(2*n)), 2)-n+1 /* A004736 */

A002260(n)=1+A002262(n). Cf. A056534.

Cf. A004736 (ordinal transform), A025581, A002262, A003056.

Cf. A000217.

Cf. A127779. [From Clark Kimberling (ck6(AT)evansville.edu), Sep 16 2008]

Adjacent sequences: A002257 A002258 A002259 this_sequence A002261 A002262 A002263

Sequence in context: A023121 A136261 A140756 this_sequence A133994 A066041 A119585

nonn,easy,nice,tabl,new

Angele Hamel (amh(AT)maths.soton.ac.uk)

More terms from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 27 2006