|
| |
|
|
A004737
|
|
Concatenation of sequences (1,2,..,n-1,n,n-1,..,1) for n >= 1.
|
|
17
| |
|
|
1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,3
|
|
|
COMMENTS
| Also called the Smarandache Crescendo Pyramidal sequence.
The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446. - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), Mar 16 2003
a(A002061(n)) = n; a(A000290(n)) = a(A002522(n)) = 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 10 2006
The ordinal transform of a sequence b_0, b_1, b_2, ... is the sequence a_0, a_1, a_2, ... where a_n is the number of times b_n has occurred in {b_0 ... b_n}.
Contribution from Artur Jasinski (grafix(AT)csl.pl), Mar 07 2010: (Start)
Zeta[2,k/p]+Zeta[2,(p-k)/p]=(Pi/Sin[(Pi*a(n))/p])2, where p=2,3,4, k=1..p-1.
This sequence is even subset of A003983 for odd p=2,4,6,8,....
For odd subset of A003983 see A004739 (End) (End)
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 30 2010: (Start)
Given the triangle rows: (1; 1,2,1; 1,2,3,2,1;...) as polcoeff with offset 0:
q = (1 + 2x + x^2), r = (1 + 2x + 3x^2 + 2x^3 +x^4),...etc; then
(1 + 2x + 3x^2 + ...) = q(x) * q(x^2) * q(x^4) * q(^8) * ...
..................... = r(x) * r(x^3) * r(x^9) * r(x^27) * ...
..................... = s(x) * x(x^4) * s(x^16)* s(x^64) * ...
... (End)
Contribution from L. Edson Jeffery, Jan 13 2012: (Start)
This sequence is related to a certain spectral theory for unit-primitive matrices (see [Jeffery]). Let U_1(t)=1, U_2(2)=2*t, and U_r(t)=2*t*U_(r-1)(t)-U(r-2)(t), r=3,4,..., be Chebyshev polynomials of the second kind. For q>1 an integer, let N=2*q and x_k=cos((2*k-1)*Pi/N), and define the ordered column vectors V_k=[U_k(x_1), U_k(x_2), ..., U_k(x_q)]^T, k=1,2,...,q, where A^T denotes the transpose of matrix A. Let E_N=[V_1, V_2, ..., V_q] be the q X q matrix formed from the ordered components of the vectors V_k (E_N has properties similar to a Vandermonde matrix and is nonsingular), and let M_N=(1/q)*[E_N]^T*E_N. For the trivial case q=1, let M_2=[1]. If N is allowed to increase without bound, then the triangle A004737 emerges in its entirety from the successive antidiagonals containing those entries [M_N]_(i,j) such that i+j=2*v, for each v in {1,2,...,floor((q+1)/2)}, and the antidiagonals are read in the direction from lower left to upper right. For example, for N=18 and q=9 (entries = 0 are not shown, for clarity),
M_18=[
(1 1 1 1 1);
( 2 2 2 2 );
(1 3 3 3 3);
( 2 4 4 4 );
(1 3 5 5 5);
( 2 4 6 6 );
(1 3 5 7 7);
( 2 4 6 8 );
(1 3 5 7 9)],
giving the first five rows of the triangle by reading the nonzero antidiagonals of the matrix in succession. (End)
|
|
|
REFERENCES
| Jerry Brown et al., Problem 4619, School Science and Mathematics, USA, Vol. 97 (4), 1997, pp. 221-222.
F. Smarandache, "Numerical Sequences", University of Craiova, 1975.
F. Smarandache, "Collected Papers", Vol. II, Tempus Publ. Hse., Bucharest, 1996.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.
|
|
|
LINKS
| L. E. Jeffery, Unit-primitive matrices
M. L. Perez et al., eds., Smarandache Notions Journal
F. Smarandache, Collected Papers, Vol. II
F. Smarandache, Collected Papers, Vol. II.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
Eric Weisstein's World of Mathematics, Smarandache Sequences.
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
|
|
|
FORMULA
| a(n) = if n<3 then 1 else (if a(n-1)=1 then 1 + 0^(a(n-2)-1) else a(n-1) - 0^X + (a(n-1)-a(n-2))*(1 - 0^X)), where X = A003059(n-1)-a(n-1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 10 2006
Let b(n) = floor(sqrt(n-1)). Then a(n) = min(n - b(n)^2, (b(n)+1)^2 - n + 1). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 09 2006
Ordinal transform of A004741. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Aug 28 2006
If the sequence is read as a triangular array, beginning [1]; [1,2,1]; [1,2,3,2,1]; ..., then the o.g.f. is (1+qx)/((1-x)(1-qx)(1-q^2x)) = 1 + x(1 + 2q + q^2) + x^2(1 + 2q + 3q^2 + 2q^3 +q^4) + .... The row polynomials for this triangle are (1 + q + ... + q^n)^2 =[n,2]_q + q[n-1,2]_q, where [n,2]_q are Gaussian polynomials (see A008967). - Peter Bala (pbala(AT)toucansurf.com), Sep 23 2007
a(n+1)=a(n)+{floor[sqrt(n)]-floor[sqrt(n-1)]-1}*(-1)^{floor[(n-1)/floor[(1/2)*(1+sqrt(4*n-3))]]-floor[(1/2)*(1+sqrt(4*n-3))]+1}, with a(1)=1 [From Paolo P. Lava (paoloplava(AT)gmail.com), Nov 13 2008]
See Mathematica code. [From Artur Jasinski (grafix(AT)csl.pl), Mar 07 2010]
|
|
|
MAPLE
| P:=proc(i) local a, b, c, n; a:=1; print(a); for n from 1 by 1 to i do a:=a+(floor(sqrt(n))-floor(sqrt(n-1))-1)*(-1)^(floor((n-1)/floor(1/2*(1+sqrt(4*n-3))))-floor(1/2*(1+sqrt(4*n-3)))+1); print(a); od; end: P(100); [From Paolo P. Lava (paoloplava(AT)gmail.com), Nov 13 2008]
|
|
|
MATHEMATICA
| aa = {}; Do[Do[AppendTo[aa, (p/Pi) ArcSin[Sqrt[1/((1/Pi2) (Zeta[2, k/p] + Zeta[2, (p - k)/p]))]]], {k, 1, p - 1}], {p, 2, 50, 2}]; Round[N[aa, 50]] (*Artur Jasinski*) [From Artur Jasinski (grafix(AT)csl.pl), Mar 07 2010]
|
|
|
PROG
| (Haskell)
a004737 n = a004737_list !! (n-1)
a004737_list = concat $ map (\n -> [1..n] ++ [n-1, n-2..1]) [1..]
-- Reinhard Zumkeller, Mar 26 2011
|
|
|
CROSSREFS
| Cf. A004738, A008967.
A003983 [From Artur Jasinski (grafix(AT)csl.pl), Mar 07 2010]
Sequence in context: A122563 A204030 A080345 * A014600 A165475 A098280
Adjacent sequences: A004734 A004735 A004736 * A004738 A004739 A004740
|
|
|
KEYWORD
| nonn,easy,changed
|
|
|
AUTHOR
| R. Muller
|
|
|
EXTENSIONS
| More terms from Patrick De Geest (pdg(AT)worldofnumbers.com), Jun 15 1998.
|
| |
|
|
