A004737 Concatenation of sequences (1,2,...,n-1,n,n-1,...,1) for n >= 1.

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

Offset: 1

Author

R. Muller

Keywords

• nonn
• frac
• easy
• tabf

Comments

The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446. - Jeremy Gardiner, Mar 16 2003

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}.

From Artur Jasinski, Mar 07 2010: (Start)

This sequence is the even subset of A003983 for odd p=2,4,6,8,....

For the odd subset of A003983 see A004739. (End)

From Gary W. Adamson, 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(x^8) * ...

..................... = r(x) * r(x^3) * r(x^9) * r(x^27) * ...

..................... = s(x) * s(x^4) * s(x^16)* s(x^64) * ...

... (End)

From L. Edson Jeffery, Jan 13 2012: (Start)

Let U_1(t)=1, U_2(t)=2*t, and U_r(t)=2*t*U_(r-1)(t)-U(r-2)(t), r>2, 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,...,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 V_k. E_N contains the joint spectra of the Danzer basis (see [Jeffery]) associated with N. Let M_N=(1/q)*[E_N]^T*E_N. For the trivial case q=1, let M_2=[1]. CONJECTURE: E_N and M_N are always integral and symmetric, with M_N having diagonal entries {1,2,...} beginning at entries 1,j (j odd) in the first row and i,1 (i odd) in the first column and with zeros elsewhere. If N is allowed to increase without bound, and assuming the conjecture is true, then 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)}. For example, for N=18 and q=9 (omitting the zeros 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)],

from which the first five rows of the sequence can be read off in succession. (End)

T(n,k) = min(n,k). The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). - Boris Putievskiy, Jan 13 2013

Expanded form of T(2,k) k=0,1,...,2m for ascending m-nomial triangles. - Bob Selcoe, Feb 07 2014

Terms in the first nine rows of the triangle can be duplicated by performing (111...)^2 with <= nine ones. By way of example, (11111)^2 = 123454321. - Gary W. Adamson, Mar 27 2015

Examples

			From _Boris Putievskiy_, Jan 13 2013: (Start)
The start of the sequence as a table:
  1 1 1 1 1 1 ...
  1 2 2 2 2 2 ...
  1 2 3 3 3 3 ...
  1 2 3 4 4 4 ...
  1 2 3 4 5 5 ...
  1 2 3 4 5 6 ...
  ...
The start of the sequence as an irregular triangle array read by rows:
  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;
  ...
Row number k contains 2*k-1 numbers: 1,2,...,k-1,k,k-1,...,1. (End)
The sequence of fractions A196199/A004737 = 0/1, -1/1, 0/2, 1/1, -2/1, -1/2, 0/3, 1/2, 2/1, -3/1, -2/2, -1/3, 0/4, 1/3, 2/2, 3/1, -4/4. -3/2, ... contains every rational number (infinitely often) [Laczkovich]. - _N. J. A. Sloane_, Oct 09 2013
		

References

• Miklós Laczkovich, Conjecture and Proof, TypoTex, Budapest, 1998. See Chapter 10.
• F. Smarandache, "Numerical Sequences", University of Craiova, 1975.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Jerry Brown et al., Problem 4619, School Science and Mathematics, USA, Vol. 97 (4), 1997, pp. 221-222.
• L. E. Jeffery, Danzer matrices (definitions)
• Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
• F. Smarandache, Collected Papers, Vol. II, Tempus Publ. Hse., Bucharest, 1996.
• F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.
• Eric Weisstein's World of Mathematics, Smarandache Sequences.

Crossrefs

Cf. A004738, A008967, A003983, A196199.

Cf. A242357, A000290 (row sums).

Programs

Haskell

import Data.List (inits)
a004737 n = a004737_list !! (n-1)
a004737_list = concatMap f $ tail $ inits [1..]
   where f xs = xs ++ tail (reverse xs)
-- Reinhard Zumkeller, May 11 2014, Mar 26 2011

Mathematica

Table[Min[n - #^2, (# + 1)^2 - n + 1] &@ Floor[Sqrt[n - 1]], {n, 105}] (* or *)
Table[Floor@ # - Abs[n - Floor[#]^2 - Floor@ # - 1] + 1 &@ Sqrt[n - 1], {n, 105}] (* Michael De Vlieger, Oct 21 2016 *)
Table[Join[Range[n],Range[n-1,1,-1]],{n,20}]//Flatten (* Harvey P. Dale, Dec 27 2019 *)

PARI

a(n) = n--;my(m=sqrtint(n));m+1-abs(n-m^2-m) \\ David A. Corneth, Oct 18 2016

Formula

a(A002061(n)) = n; a(A000290(n)) = a(A002522(n)) = 1. - Reinhard Zumkeller, Mar 10 2006

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, Mar 10 2006

Let b(n) = floor(sqrt(n-1)). Then a(n) = min(n - b(n)^2, (b(n)+1)^2 - n + 1). - Franklin T. Adams-Watters, Jun 09 2006

Ordinal transform of A004741. - Franklin T. Adams-Watters, 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]A008967).%20-%20_Peter%20Bala">q + q[n-1,2]_q, where [n,2]_q are Gaussian polynomials (see A008967). - _Peter Bala, Sep 23 2007

a(n) = floor(sqrt(n-1)) - |n - floor(sqrt(n-1))^2 - floor(sqrt(n-1)) - 1| + 1. - Boris Putievskiy, Jan 13 2013

Read as a triangular array, then T(n,k) = n - |n-k-1|; T(n,0) = 1; T(n,n-1) = n. - Juan Pablo Herrera P., Oct 17 2016

Extensions

More terms from Patrick De Geest, Jun 15 1998