%I A002260
%S A002260 1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,2,3,4,5,6,1,2,3,4,5,6,7,1,2,3,4,5,
%T A002260 6,7,8,1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,10,1,2,3,4,5,6,7,8,9,10,
%U A002260 11,1,2,3,4,5,6,7,8,9,10,11,12,1,2,3,4,5,6,7,8,9,10,11,12,13,1,2,3
%N A002260 Integers 1 to k followed by integers 1 to k+1 etc. (a fractal sequence).
%C A002260 Start counting again and again.
%C A002260 This is a "doubly fractal sequence" - see the Franklin T. Adams-Watters
link.
%C A002260 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
%C A002260 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]
%C A002260 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]
%C A002260 Contribution from Clark Kimberling (ck6(AT)evansville.edu), Nov 02 2009:
(Start)
%C A002260 The upper trim of an arbitrary fractal sequence s is s, but the lower
trim
%C A002260 of s, although a fractal sequence, need not be s itself. However, the
%C A002260 lower trim of A002260 is A002260. (The upper trim of s is what remains
%C A002260 after the first occurrence of each term is deleted; the lower trim of
s
%C A002260 is what remains after all 0s are deleted from the sequence s-1.) (End)
%C A002260 a(A169581(n)) = A038566(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Dec 02 2009]
%D A002260 Jerry Brown et al., Problem 4619, "School Science and Mathematics", USA,
Vol. 97(4), 1997, pp. 221-222.
%D A002260 C. Kimberling, "Numeration systems and fractal sequences," Acta Arithmetica
73 (1995) 103-117.
%D A002260 M. Myers, Smarandache Crescendo Subsequences, R.H.Wilde, An Anthology
in Memoriam, Bristol Banner Books, Bristol, 1998, p. 19.
%D A002260 F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis,
Phoenix, 2006.
%D A002260 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]
%D A002260 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]
%H A002260 N. J. A. Sloane, <a href="b002260.txt">Table of n, a(n) for n = 1..11325</
a>
%H A002260 Franklin T. Adams-Watters, <a href="a002260.txt">Doubly Fractal Sequences</
a>
%H A002260 C. Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/fractals.html">
Fractal sequences</a>
%H A002260 F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/Sequences-book.pdf">
Sequences of Numbers Involved in Unsolved Problems</a>.
%H A002260 M. Somos, <a href="a073189.txt">Sequences used for indexing triangular
or square arrays</a>
%H A002260 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
SmarandacheSequences.html">Link to a section of The World of Mathematics.</
a>
%H A002260 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
UnitFraction.html">Link to a section of The World of Mathematics.</
a>
%F A002260 n-th term is n - m(m+1)/2 + 1, where m = [ (sqrt(8n+1) - 1) / 2 ].
%F A002260 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
%F A002260 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
%F A002260 n+1-binomial(floor((1+sqrt(8n+8))/2), 2). - Paul Barry (pbarry(AT)wit.ie),
May 25 2004
%F A002260 T(n,k)=A001511(A118413(n,k)); = T(n,k)=A003602(A118416(n,k)). - Reinhard
Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 27 2006
%F A002260 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
%p A002260 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
%p A002260 seq(seq(i,i=1..k),k=1..13); [From Peter Luschny (peter(AT)luschny.de),
Jul 06 2009]
%o A002260 (PARI) a(n)=n+1-binomial(floor(1/2+sqrt(2+2*n)),2)
%o A002260 (PARI) t1(n)=n-binomial(floor(1/2+sqrt(2*n)),2) /* A002260 */
%o A002260 (PARI) t2(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1 /* A004736 */
%Y A002260 A002260(n)=1+A002262(n). Cf. A056534.
%Y A002260 Cf. A004736 (ordinal transform), A025581, A002262, A003056.
%Y A002260 Cf. A000217.
%Y A002260 Cf. A127779. [From Clark Kimberling (ck6(AT)evansville.edu), Sep 16 2008]
%Y A002260 Sequence in context: A023121 A136261 A140756 this_sequence A133994 A066041
A119585
%Y A002260 Adjacent sequences: A002257 A002258 A002259 this_sequence A002261 A002262
A002263
%K A002260 nonn,easy,nice,tabl,new
%O A002260 1,3
%A A002260 Angele Hamel (amh(AT)maths.soton.ac.uk)
%E A002260 More terms from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Apr 27 2006
|
