OFFSET
0,3
COMMENTS
Fill an infinity X infinity matrix with numbers so that 1..n^2 appear in the top left n X n corner for all n; write down the minimal elements in the rows and columns and sort into increasing order; maximize this list in the lexicographic order.
From Donald S. McDonald, Jan 09 2003: (Start)
Numbers of the form n^2 + 1 or n^2 + n + 1.
Locations of right angle turns in Ulam square spiral. (End)
a(n-1) (for n >= 1) is also the number u of unique Fibonacci/Lucas type sequences generated (the total number t of these sequences being a triangular number). Sum(n+1)=t. Then u=Sum((n+1/2) minus 0.5 for odd terms) except for the initial term. E.g., u=13: (n=6)+1 = 7; then 7/2 - 0.5 =3. So u = Sum(1, 1, 1, 2, 2, 3, 3) = 13. - Marco Matosic, Mar 11 2003
Number of (3412,123)-avoiding involutions in S_n.
Schur's Theorem (1905): the maximum number of mutually commuting linearly independent complex matrices of order n is floor((n^2)/4) + 1. Jacobson gave a simpler proof 40 years later, generalizing from algebraically closed fields to arbitrary fields. 54 years after that, Mirzakhani gave an even simpler proof. - Jonathan Vos Post, Apr 03 2007
Let A be the Hessenberg n X n matrix defined by: A[1,j]=j mod 2, A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 1, a(n)=(-1)^(n-1)*coeff(charpoly(A,x),x). - Milan Janjic, Jan 24 2010
Except for the initial two terms, A033638 gives iterates of the nonsquare function: c(n) = f(c(n-1)), where f(n) = A000037(n) = n + floor(1/2 + sqrt(n)) = n-th nonsquare, starting with c(1)=2. - Clark Kimberling, Dec 28 2010
For n >= 1: for all permutations of [0..n-1]: number of distinct values taken by Sum_{k=0..n-1} (k mod 2) * pi(k). - Joerg Arndt, Apr 22 2011
Number of (weakly) unimodal compositions of n with maximal part <= 2, see example. - Joerg Arndt, May 10 2013
Construct an infinite triangular matrix with 1's in the leftmost column and the natural numbers in all other columns but shifted down twice. Square the triangle and the sequence is the leftmost column vector. - Gary W. Adamson, Jan 27 2014
Equals the sum of terms in upward sloping diagonals of an infinite lower triangle with 1's in the leftmost column and the natural numbers in all other columns. - Gary W. Adamson, Jan 29 2014
a(n) is the number of permutations of length n avoiding both 213 and 321 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at A245898. - Manda Riehl, Aug 05 2014
Number of partitions of n with no more than 2 parts > 1. - Wouter Meeussen, Feb 22 2015, revised Apr 24 2023
Number of possible values for the area of a polyomino whose perimeter is 2n + 4. - Luc Rousseau, May 10 2018
a(n) is the number of 231-avoiding even Grassmannian permutations of size n+1. - Juan B. Gil, Mar 10 2023
For n > 0, a(n) is the smallest number that requires n iterations of the map k -> k - floor(sqrt(k)) to reach 0. - Jon E. Schoenfield, Jun 24 2023
a(n) agrees with the lower matching number of the (n + 1) X (n + 1) black bishop graph from n = 1 up to at least n = 14. - Eric W. Weisstein, Dec 23 2024
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Kassie Archer and Aaron Geary, Descents in powers of permutations, arXiv:2406.09369 [math.CO], 2024.
Jonathan Bloom and Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.
H. Cheballah, S. Giraudo, and R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013.
E. S. Egge, Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations, Thm. 6.6, arXiv:math/0307050 [math.CO], 2003.
D. C. Fielder and C. O. Alford, An investigation of sequences derived from Hoggatt Sums and Hoggatt Triangles, Application of Fibonacci Numbers, 3 (1990) 77-88. Proceedings of 'The Third Annual Conference on Fibonacci Numbers and Their Applications,' Pisa, Italy, July 25-29, 1988. (Annotated scanned copy)
Juan B. Gil and Jessica A. Tomasko, Restricted Grassmannian permutations, arXiv:2112.03338 [math.CO], 2021. See Proposition 2.3 p. 4.
Juan B. Gil and Jessica A. Tomasko, Pattern-avoiding even and odd Grassmannian permutations, arXiv:2207.12617 [math.CO], 2022.
Brian Hopkins and Aram Tangboonduangjit, Water Cells in Compositions of 1s and 2s, arXiv:2412.11528 [math.CO], 2024. See p. 3.
Nathan Jacobson, Schur's theorems on commutative matrices, Bull. Amer. Math. Soc. 50 (1944) 431-436.
M. Mirzakhani, A Simple Proof of a Theorem of Schur, The American Mathematical Monthly, Vol. 105, No. 3 (Mar 1998), pp. 260-262.
D. Necas and I. Ohlidal, Consolidated series for efficient calculation of the reflection and transmission in rough multilayers, Optics Express, Vol. 22, 2014, No. 4; DOI:10.1364/OE.22.004499. See Table 1.
I. Schur, Neue Begründung der Theorie der Gruppencharaktere, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin (1905), 406-432.
Harold N. Ward, A Normal Graph Algebra, arXiv:2201.00389 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Black Bishop Graph.
Eric Weisstein's World of Mathematics, Lower Matching Number.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
FORMULA
a(n) = ceiling((n^2+3)/4) = ( (7 + (-1)^n)/2 + n^2 )/4.
a(n) = A001055(prime^n), number of factorizations. - Reinhard Zumkeller, Dec 29 2001
G.f.: (1-x+x^3)/((1-x)^2*(1-x^2)); a(n) = a(n-1) + a(n-2) - a(n-3) + 1. - Jon Perry, Jul 07 2004
a(n) = a(n-2) + n - 1. - Paul Barry, Jul 14 2004
a(0) = 1; a(1) = 1; for n > 1 a(n) = a(n-1) + round(sqrt(a(n-1))). - Jonathan Vos Post, Jan 19 2006
a(n) = floor((n^2)/4) + 1.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 3. - Philippe Deléham, Nov 03 2008
a(0) = a(1) = 1, a(n) = a(n-1) + ceiling(sqrt(a(n-2))) for n > 1. - Jonathan Vos Post, Oct 08 2011
a(n) = floor(b(n)) with b(n) = b(n-1) + n/(1+e^(1/n)) and b(0)= 1. - Richard R. Forberg, Jun 08 2013
a(n) = a(n-1) + floor(n/2). - Michel Lagneau, Jul 11 2014
From Ilya Gutkovskiy, Oct 07 2016: (Start)
E.g.f.: (exp(-x) + (7 + 2*x + 2*x^2)*exp(x))/8.
a(n) = Sum_{k=0..n} A123108(k).
a(n) = (n^2 - n + 4)/2 - a(n-1) for n >= 1. - Kritsada Moomuang, Aug 03 2019
EXAMPLE
First 4 rows can be taken to be 1,2,5,10,17,...; 3,4,6,11,18,...; 7,8,9,12,19,...; 13,14,15,16,20,...
Ulam square spiral = 7 8 9 / 6 1 2 / 5 4 3 /...; changes of direction (right-angle) at 1 2 3 5 7 ...
From Joerg Arndt, May 10 2013: (Start)
The a(7)=13 unimodal compositions of 7 with maximal part <= 2 are
01: [ 1 1 1 1 1 1 1 ]
02: [ 1 1 1 1 1 2 ]
03: [ 1 1 1 1 2 1 ]
04: [ 1 1 1 2 1 1 ]
05: [ 1 1 1 2 2 ]
06: [ 1 1 2 1 1 1 ]
07: [ 1 1 2 2 1 ]
08: [ 1 2 1 1 1 1 ]
09: [ 1 2 2 1 1 ]
10: [ 1 2 2 2 ]
11: [ 2 1 1 1 1 1 ]
12: [ 2 2 1 1 1 ]
13: [ 2 2 2 1 ]
(End)
G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 10*x^6 + 13*x^7 + 17*x^8 + ...
MAPLE
with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r), right=Set(U, card<r), U=Sequence(Z, card>=3)}, unlabeled]: subs(r=1, stack): seq(count(subs(r=2, ZL), size=m), m=6..62); # Zerinvary Lajos, Mar 09 2007
A033638 := proc(n)
1+floor(n^2/4) ;
end proc: # R. J. Mathar, Jul 13 2012
MATHEMATICA
a[n_] := a[n] = 2a[n - 1] - 2a[n - 3] + a[n - 4]; a[0] = a[1] = 1; a[2] = 2; a[3] = 3; Array[a, 54, 0] (* Robert G. Wilson v, Mar 28 2011 *)
LinearRecurrence[{2, 0, -2, 1}, {1, 1, 2, 3}, 60] (* Robert G. Wilson v, Sep 16 2012 *)
PROG
(PARI) {a(n) = n^2\4 + 1} /* Michael Somos, Apr 03 2007 */
(Haskell)
a033638 = (+ 1) . (`div` 4) . (^ 2) -- Reinhard Zumkeller, Apr 06 2012
(Magma) [n^2 div 4 + 1: n in [0.. 50]]; // Vincenzo Librandi, Jul 31 2016
(Python)
def A033638(n): return (n**2>>2)+1 # Chai Wah Wu, Jul 27 2022
CROSSREFS
KEYWORD
easy,nonn,changed
AUTHOR
Tanya Y. Berger-Wolf (tanyabw(AT)uiuc.edu)
STATUS
approved