Search: id:A004736
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%I A004736
%S A004736 1,2,1,3,2,1,4,3,2,1,5,4,3,2,1,6,5,4,3,2,1,7,6,5,4,3,2,1,8,7,6,5,4,3,2,
%T A004736 1,9,8,7,6,5,4,3,2,1,10,9,8,7,6,5,4,3,2,1,11,10,9,8,7,6,5,4,3,2,1,12,
%U A004736 11,10,9,8,7,6,5,4,3,2,1,13,12,11,10,9,8,7,6,5,4,3,2,1,14,13,12,11,10,
               9
%N A004736 Triangle T(n,k) = n-k, n >= 1, 0<=k<n. Fractal sequence formed by repeatedly 
               appending strings m m-1 . . . 2 1.
%C A004736 "Smarandache Decrescendo Subsequences".
%C A004736 The PARI functions t1, t2 can be used to read a square array T(n,k) (n 
               >= 1, k >= 1) by antidiagonals upwards: n -> T(t1(n), t2(n)). - Michael 
               Somos, Aug 23, 2002
%D A004736 C. Kimberling, "Numeration systems and fractal sequences," Acta Arithmetica 
               73 (1995) 103-117.
%D A004736 F. Smarandache, "Collected Papers", Vol. II, Tempus Publ. Hse., Bucharest, 
               1996; F. Smarandache, "Numerical Sequences", University of Craiova, 
               1975; [ See Arizona State University, Special Collection, Tempe, 
               AZ, USA ].
%D A004736 F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, 
               Phoenix, 2006.
%D A004736 H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973, pp 159-162 
               [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 25 2009]
%H A004736 C. Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/fractals.html">
               Fractal sequences</a>
%H A004736 M. L. Perez et al., eds., <a href="http://www.gallup.unm.edu/~smarandache/
               ">Smarandache Notions Journal</a>
%H A004736 F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/CP2.pdf">
               Collected Papers, Vol. II</a>
%H A004736 F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/Sequences-book.pdf">
               Sequences of Numbers Involved in Unsolved Problems</a>.
%H A004736 M. Somos, <a href="a073189.txt">Sequences used for indexing triangular 
               or square arrays</a>
%H A004736 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SmarandacheSequences.html">Link to a section of The World of Mathematics.</
               a>
%F A004736 a(n) = (2 - 2 n + round(SQRT(2 n)) + round(SQRT(2 n))^2)/2. E.g. a(47) 
               = 9. - Brian Tenneson (phoenix(AT)alephnulldimension.net), Oct 11 
               2003
%F A004736 G.f.: 1 / [(1-x)^2 * (1-xy) ]. - R. Stephan, Jan 23 2005
%F A004736 Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 25 
               2009: (Start)
%F A004736 Recursion:
%F A004736 e(n,k) = (e(n - 1, k)*e(n, k - 1) + 1)/e(n - 1, k - 1); (End)
%e A004736 1; 2,1; 3,2,1; 4,3,2,1; 5,4,3,2,1; ...
%e A004736 Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 25 
               2009: (Start)
%e A004736 {1},
%e A004736 {2, 1},
%e A004736 {3, 2, 1},
%e A004736 {4, 3, 2, 1},
%e A004736 {5, 4, 3, 2, 1},
%e A004736 {6, 5, 4, 3, 2, 1},
%e A004736 {7, 6, 5, 4, 3, 2, 1},
%e A004736 {8, 7, 6, 5, 4, 3, 2, 1},
%e A004736 {9, 8, 7, 6, 5, 4, 3, 2, 1},
%e A004736 {10, 9, 8, 7, 6, 5, 4, 3, 2, 1} (End)
%p A004736 (Excel cell formula) =if(row()>=column();row()-column()+1;"") [From Mats 
               Granvik (mats.granvik(AT)abo.fi), Jan 19 2009]
%t A004736 Flatten[ Table[ Reverse[ Range[n]], {n, 12}]] (from Robert G. Wilson 
               v Apr 27 2004)
%t A004736 Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 25 
               2009: (Start)
%t A004736 Clear[e, n, k];
%t A004736 e[n_, 0] := n;
%t A004736 e[n_, k_] := 0 /; k >= n;
%t A004736 e[n_, k_] := (e[n - 1, k]*e[n, k - 1] + 1)/e[n - 1, k - 1];
%t A004736 Table[Table[e[n, k], {k, 0, n - 1}], {n, 1, 10}];
%t A004736 Flatten[%] (End)
%o A004736 (PARI) a(n)=1+binomial(1+floor(1/2+sqrt(2*n)),2)-n
%o A004736 (PARI) t1(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1 /* A004736 */
%o A004736 (PARI) t2(n)=n-binomial(floor(1/2+sqrt(2*n)),2) /* A002260 */
%Y A004736 a(n+1)=1+A025581(n). Cf. A002262, A025581, A003056.
%Y A004736 Ordinal transform of A002260.
%Y A004736 Sequence in context: A141671 A088643 A102482 this_sequence A167288 A167289 
               A023122
%Y A004736 Adjacent sequences: A004733 A004734 A004735 this_sequence A004737 A004738 
               A004739
%K A004736 nonn,easy,tabl,nice
%O A004736 1,2
%A A004736 R. Muller

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