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A002260
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Integers 1 to k followed by integers 1 to k+1 etc. (a fractal sequence).
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120
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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, 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, 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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| 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, Sep 16 2008]
Contribution from Clark Kimberling, 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)
a(A169581(n)) = A038566(n). [From Reinhard Zumkeller, Dec 02 2009]
Eigensequence of the triangle = A001710 starting (1, 3, 12, 60, 360,...). [From Gary W. Adamson, Aug 02 2010]
Contribution from Johannes W. Meijer, Sep 22 2010: (Start)
The triangle or chess sums, see A180662 for their definitions, link this triangle of natural numbers with twenty-three different sequences, see the crossrefs. The knight sums Kn14 - Kn116 have been added. The mirror image of this triangle is A004736. (End)
From Paul Curtz, Jul 25 2011: (Start)
Akiyama-Tanigawa algorithm from A000027(n) gives
1, 2, 3, 4, 5, 6, 7, 8,
-1, -2, -3, -4, -5, -6, -7, -8,
1, 2, 3, 4, 5, 6, 7, 8,
-1, -2, -3, -4, -5, -6, -7, -8.
By antidiagonals :
1,
-1, 2,
1, -2, 3,
-1, 2, -3, 4,
1, -2, 3, -4, 5,
-1, 2, -3, 4, -5, 6.
Row sum=A016116. (End)
A002260 is the self-fission of the polynomial sequence (q(n,x)), where q(n,x)=x^n+x^(n-1)+...+x+1. See A193842 for the definition of fission. [From Clark Kimberling, Aug 7 2011]
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REFERENCES
| 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.
Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168. (Introduces upper trimming, lower trimming, and signature sequences.) [From Clark Kimberling, Nov 02 2009]
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.
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LINKS
| 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.
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FORMULA
| n-th term is n - m*(m+1)/2 + 1, where m = [ (sqrt(8*n+1) - 1) / 2 ].
The above formula is for offset 0; for offset 1, use a(n)=n-m*(m+1)/2 where m=floor[(-1+sqrt(8*n-7))/2]. [Clark Kimberling, Jun 14 2011]
a(k * (k + 1) / 2 + i) = i for k >= 0 and 0 < i <= k + 1. - Reinhard Zumkeller, Aug 14 2001
a(n) = (2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2. - brian tenneson (phoenix(AT)alephnulldimension.net), Oct 11 2003
n+1-binomial(floor((1+sqrt(8*n+8))/2), 2). - Paul Barry, May 25 2004
T(n,k)=A001511(A118413(n,k)); = T(n,k)=A003602(A118416(n,k)). - Reinhard Zumkeller, 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, May 28 2008
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EXAMPLE
| First six rows:
1
1...2
1...2...3
1...2...3...4
1...2...3...4...5
1...2...3...4...5...6
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MAPLE
| 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, Nov 01 2006
seq(seq(i, i=1..k), k=1..13); [From Peter Luschny, Jul 06 2009]
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MATHEMATICA
| FoldList[{#1, #2} &, 1, Range[2, 13]] // Flatten (* Or *)
f[n_] := Block[{m = Floor[(Sqrt[8n - 7] - 1)/2]}, n - m(m + 1)/2]; Array[f, 91] (* Robert G. Wilson v, May 10 2011 *)
...
m[n_] := Floor[(-1 + Sqrt[8 n - 7])/2]
b[n_] := n - m[n] (m[n] + 1)/2
Table[m[n], {n, 1, 100}] (* A003056 *)
Table[b[n], {n, 1, 100}] (* A002260 *)
Table[b[n] - 1, {n, 1, 100}] (* A002262 *)
...
(* Clark Kimberling, Jun 14 2011 *)
z=13; p[0, x_]:=1;
p[n_, x_]:=x*p[n-1, x]+1;
q[n_, x_]:=p[n, x];
p1[n_, k_]:=Coefficient[p[n, x], x^k]; p1[n_, 0]:=p[n, x]/.x->0;
d[n_, x_]:=Sum[p1[n, k]*q[n-1-k, x], {k, 0, n-1}]
h[n_]:=CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]] (* A002260 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]] (* A004736 *)
(* Clark Kimberling, Aug 7 2011 *)
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PROG
| (PARI) t1(n)=n-binomial(floor(1/2+sqrt(2*n)), 2) /* this sequence */
(PARI) t2(n)=binomial(floor(3/2+sqrt(2*n)), 2)-n+1 /* A004736 */
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CROSSREFS
| A002260(n)=1+A002262(n). Cf. A056534.
Cf. A004736 (ordinal transform), A025581, A002262, A003056.
Cf. A000217.
Cf. A127779. [From Clark Kimberling, Sep 16 2008]
A001710 [From Gary W. Adamson, Aug 02 2010]
Contribution from Johannes W. Meijer, Sep 22 2010: (Start)
Triangle sums (see the comments): A000217 (Row1 & Kn11 & Kn4 & Ca1 & Ca4 & Gi1 & Gi4); A004526 (Row2); A000096 (Kn12); A055998 (Kn13); A055999 (Kn14); A056000 (Kn15); A056115 (Kn16); A056119 (Kn17); A056121 (Kn18); A056126 (Kn19); A051942 (Kn110); A101859 (Kn111); A132754 (Kn112); A132755 (Kn113); A132756 (Kn114); A132757 (Kn115); A132758 (Kn116); A002620 (Kn21, Kn22, Kn23, Fi2, Ze2); A000290 (Kn3, Fi1, Ze3); A001840 (Ca2 & Ze4); A000326 (Ca3, Ze1); A001972 (Gi2); A000384 (Gi3).
(End)
Sequence in context: A023121 A136261 A140756 * A194905 A133994 A066041
Adjacent sequences: A002257 A002258 A002259 * A002261 A002262 A002263
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KEYWORD
| nonn,easy,nice,tabl
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AUTHOR
| Angele Hamel (amh(AT)maths.soton.ac.uk)
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EXTENSIONS
| More terms from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 27 2006
Incorrect program removed by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Mar 19 2010
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