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%I #254 Dec 16 2023 16:38:00
%S 1,1,2,3,5,7,10,13,17,21,26,31,37,43,50,57,65,73,82,91,101,111,122,
%T 133,145,157,170,183,197,211,226,241,257,273,290,307,325,343,362,381,
%U 401,421,442,463,485,507,530,553,577,601,626,651,677,703,730,757,785,813,842
%N Quarter-squares plus 1 (that is, a(n) = A002620(n) + 1).
%C 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.
%C From _Donald S. McDonald_, Jan 09 2003: (Start)
%C Numbers of the form n^2 + 1 or n^2 + n + 1.
%C Locations of right angle turns in Ulam square spiral. (End)
%C 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
%C Number of (3412,123)-avoiding involutions in S_n.
%C 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
%C 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
%C 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
%C 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
%C First differences are A110654. - _Jon Perry_, Sep 12 2012
%C Number of (weakly) unimodal compositions of n with maximal part <= 2, see example. - _Joerg Arndt_, May 10 2013
%C 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
%C 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
%C 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
%C Number of partitions of n with no more than 2 parts > 1. - _Wouter Meeussen_, Feb 22 2015, revised Apr 24 2023
%C Number of possible values for the area of a polyomino whose perimeter is 2n + 4. - _Luc Rousseau_, May 10 2018
%C a(n) is the number of 231-avoiding even Grassmannian permutations of size n+1. - _Juan B. Gil_, Mar 10 2023
%C 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
%H Reinhard Zumkeller, <a href="/A033638/b033638.txt">Table of n, a(n) for n = 0..10000</a>
%H Jonathan Bloom and Nathan McNew, <a href="https://arxiv.org/abs/1908.03953">Counting pattern-avoiding integer partitions</a>, arXiv:1908.03953 [math.CO], 2019.
%H H. Cheballah, S. Giraudo, and R. Maurice, <a href="https://arxiv.org/abs/1306.6605">Combinatorial Hopf algebra structure on packed square matrices</a>, arXiv preprint arXiv:1306.6605 [math.CO], 2013.
%H E. S. Egge, <a href="https://arxiv.org/abs/math/0307050">Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations</a>, Thm. 6.6, arXiv:math/0307050 [math.CO], 2003.
%H D. C. Fielder and C. O. Alford, <a href="/A000108/a000108_19.pdf">An investigation of sequences derived from Hoggatt Sums and Hoggatt Triangles</a>, 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)
%H Juan B. Gil and Jessica A. Tomasko, <a href="https://arxiv.org/abs/2112.03338">Restricted Grassmannian permutations</a>, arXiv:2112.03338 [math.CO], 2021. See Proposition 2.3 p. 4.
%H Juan B. Gil and Jessica A. Tomasko, <a href="https://arxiv.org/abs/2207.12617">Pattern-avoiding even and odd Grassmannian permutations</a>, arXiv:2207.12617 [math.CO], 2022.
%H Nathan Jacobson, <a href="http://dx.doi.org/10.1090/S0002-9904-1944-08169-X">Schur's theorems on commutative matrices</a>, Bull. Amer. Math. Soc. 50 (1944) 431-436.
%H M. Mirzakhani, <a href="http://www.jstor.org/stable/2589084">A Simple Proof of a Theorem of Schur</a>, The American Mathematical Monthly, Vol. 105, No. 3 (Mar 1998), pp. 260-262.
%H D. Necas and I. Ohlidal, <a href="http://dx.doi.org/10.1364/OE.22.004499">Consolidated series for efficient calculation of the reflection and transmission in rough multilayers</a>, Optics Express, Vol. 22, 2014, No. 4; DOI:10.1364/OE.22.004499. See Table 1.
%H I. Schur, <a href="https://archive.org/stream/sitzungsberichte1905deutsch#page/406/mode/2up">Neue Begründung der Theorie der Gruppencharaktere</a>, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin (1905), 406-432.
%H Harold N. Ward, <a href="https://arxiv.org/abs/2201.00389">A Normal Graph Algebra</a>, arXiv:2201.00389 [math.CO], 2022.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F a(n) = ceiling((n^2+3)/4) = ( (7 + (-1)^n)/2 + n^2 )/4.
%F a(n) = A001055(prime^n), number of factorizations. - _Reinhard Zumkeller_, Dec 29 2001
%F 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
%F a(n) = a(n-2) + n - 1. - _Paul Barry_, Jul 14 2004
%F 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
%F a(n) = floor((n^2)/4) + 1.
%F a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 3. - _Philippe Deléham_, Nov 03 2008
%F a(0) = a(1) = 1, a(n) = a(n-1) + ceiling(sqrt(a(n-2))) for n > 1. - _Jonathan Vos Post_, Oct 08 2011
%F 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
%F a(n) = a(n-1) + floor(n/2). - _Michel Lagneau_, Jul 11 2014
%F From _Ilya Gutkovskiy_, Oct 07 2016: (Start)
%F E.g.f.: (exp(-x) + (7 + 2*x + 2*x^2)*exp(x))/8.
%F a(n) = Sum_{k=0..n} A123108(k).
%F Convolution of A008619 and A179184. (End)
%F a(n) = (n^2 - n + 4)/2 - a(n-1) for n >= 1. - _Kritsada Moomuang_, Aug 03 2019
%e 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,...
%e Ulam square spiral = 7 8 9 / 6 1 2 / 5 4 3 /...; changes of direction (right-angle) at 1 2 3 5 7 ...
%e From _Joerg Arndt_, May 10 2013: (Start)
%e The a(7)=13 unimodal compositions of 7 with maximal part <= 2 are
%e 01: [ 1 1 1 1 1 1 1 ]
%e 02: [ 1 1 1 1 1 2 ]
%e 03: [ 1 1 1 1 2 1 ]
%e 04: [ 1 1 1 2 1 1 ]
%e 05: [ 1 1 1 2 2 ]
%e 06: [ 1 1 2 1 1 1 ]
%e 07: [ 1 1 2 2 1 ]
%e 08: [ 1 2 1 1 1 1 ]
%e 09: [ 1 2 2 1 1 ]
%e 10: [ 1 2 2 2 ]
%e 11: [ 2 1 1 1 1 1 ]
%e 12: [ 2 2 1 1 1 ]
%e 13: [ 2 2 2 1 ]
%e (End)
%e 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 + ...
%p 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
%p A033638 := proc(n)
%p 1+floor(n^2/4) ;
%p end proc: # _R. J. Mathar_, Jul 13 2012
%t 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 *)
%t LinearRecurrence[{2, 0, -2, 1}, {1, 1, 2, 3}, 60] (* _Robert G. Wilson v_, Sep 16 2012 *)
%o (PARI) {a(n) = n^2\4 + 1} /* _Michael Somos_, Apr 03 2007 */
%o (Haskell)
%o a033638 = (+ 1) . (`div` 4) . (^ 2) -- _Reinhard Zumkeller_, Apr 06 2012
%o (Magma) [n^2 div 4 + 1: n in [0.. 50]]; // _Vincenzo Librandi_, Jul 31 2016
%o (Python)
%o def A033638(n): return (n**2>>2)+1 # _Chai Wah Wu_, Jul 27 2022
%Y Equals A002620 + 1.
%Y Cf. A002878, A004652, A002984, A083479, A080037 (complement, except 2).
%Y A002522 lists the even-indexed terms of this sequence.
%K easy,nonn
%O 0,3
%A Tanya Y. Berger-Wolf (tanyabw(AT)uiuc.edu)
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