A033638 Quarter-squares plus 1 (that is, a(n) = A002620(n) + 1).

1, 1, 2, 3, 5, 7, 10, 13, 17, 21, 26, 31, 37, 43, 50, 57, 65, 73, 82, 91, 101, 111, 122, 133, 145, 157, 170, 183, 197, 211, 226, 241, 257, 273, 290, 307, 325, 343, 362, 381, 401, 421, 442, 463, 485, 507, 530, 553, 577, 601, 626, 651, 677, 703, 730, 757, 785, 813, 842

Offset: 0

Author

Tanya Y. Berger-Wolf (tanyabw(AT)uiuc.edu)

Keywords

• easy
• nonn

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. - 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

First differences are A110654. - Jon Perry, Sep 12 2012

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

For n > 0, obtain a positive integer a(n+1) recursively from a(n) by minimizing a(n+1) > a(n) so that each gap between a(k) and a(k+1) for 1 <= k <= n is used at most twice. - Gerold Jäger, Jun 04 2025

From Roger Ford, May 19 2025: (Start)

a(n) = the number of different total arch lengths for the top arches of semi-meanders with n+2 arches.

Example: Each arch length equals 1 + the number of covered arches.

For semi-meanders with 5 top arches there are 3 different values.

/\

//\\ /\ /\

///\\\ //\\ /\ / \

////\\\\ /\ ///\\\ //\\ //\/\\ /\ /\

Total arch lengths: 4+3+2+1 +1= 11 3+2+1 2+1= 9 3+1+1 +1 +1= 7, so a(3) = 3.

For semi-meanders with 6 top arches there are 5 values: 8, 10, 12, 14, 16, so a(4) = 5. (End)

Examples

			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 + ...
		

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.
• Gerold Jäger and Tuomo Lehtilä, The Generalized Double Pouring Problem: Analysis, Bounds and Algorithms, arXiv:2504.03039 [math.CO], 2025. See Definition 25(a), Lemma 26(a) and proof of Theorem 27. p. 9-12.
• 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.
• Julie Zhang, Noah A. Rosenberg, and Julia A. Palacios, The space of multifurcating ranked tree shapes: enumeration, lattice structure, and Markov chains, arXiv:2506.10856 [math.PR], 2025. See p. 8.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Equals A002620 + 1.

Cf. A002878, A004652, A002984, A083479, A080037 (complement, except 2).

A002522 lists the even-indexed terms of this sequence.

Programs

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

Maple

with(combstruct):ZL:=[st,{st=Prod(left,right),left=Set(U,card=r),right=Set(U,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] = 2*a[n - 1] - 2*a[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 *)

PARI

{a(n) = n^2\4 + 1} /* Michael Somos, Apr 03 2007 */

Python

def A033638(n): return (n**2>>2)+1 # Chai Wah Wu, Jul 27 2022

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).

Convolution of A008619 and A179184. (End)

a(n) = (n^2 - n + 4)/2 - a(n-1) for n >= 1. - Kritsada Moomuang, Aug 03 2019