OFFSET
1,3
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,....
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
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.
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, 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]_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
EXAMPLE
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
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 *)
PROG
(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
(PARI) a(n) = n--; my(m=sqrtint(n)); m+1-abs(n-m^2-m) \\ David A. Corneth, Oct 18 2016
CROSSREFS
KEYWORD
nonn,frac,easy,tabf
AUTHOR
R. Muller
EXTENSIONS
More terms from Patrick De Geest, Jun 15 1998
STATUS
approved