%I A033638
%S A033638 1,1,2,3,5,7,10,13,17,21,26,31,37,43,50,57,65,73,82,91,101,111,122,133,
%T A033638 145,157,170,183,197,211,226,241,257,273,290,307,325,343,362,381,401,
%U A033638 421,442,463,485,507,530,553,577,601,626,651,677,703,730,757,785,813,842
%N A033638 Quarter-squares plus 1 (i.e. A002620 + 1).
%C A033638 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 A033638 a(n) = A001055(prime^n), number of factorizations. - Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), Dec 29, 2001
%C A033638 Locations of right angle turns in Ulam square spiral. - Donald S. McDonald 
               (don.mcdonald(AT)paradise.net.nz), Jan 09 2003
%C A033638 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 (marcomatosic(AT)hotmail.com), 
               Mar 11 2003
%C A033638 Number of (3412,123)-avoiding involutions in S_n.
%C A033638 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 (jvospost3(AT)gmail.com), 
               Apr 03 2007
%D A033638 Nathan Jacobson, Schur's theorems on commutative matrices, Bull. Amer. 
               Math. Soc. 50 (1944) 431-436.
%D A033638 M. Mirzakhani, A Simple Proof of a Theorem of Schur, The American Mathematical 
               Monthly, Vol. 105, No. 3 (Mar 1998), pp. 260-262.
%D A033638 I. Schur, Neue Begrundung der Theorie der Gruppencharaketere, Sitzungberichte 
               der Koniglich Preussischen Akademie der Wissenschaften zu Berlin 
               (1905), 406-432.
%H A033638 E. S. Egge, <a href="http://arXiv.org/abs/math.CO/0307050">Restricted 
               3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials 
               and Enumerations</a>, Thm. 6.6
%F A033638 a(n) = ceil((n^2+3)/4) = ( (7+(-1)^n)/2 + n^2 )/4.
%F A033638 a(0)=1, a(1)=1, a(n)=1+floor(a(n-1)/2). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Nov 06 2002
%F A033638 Numbers of the form n^2+1 or n^2+n+1. - Donald S. McDonald (don.mcdonald(AT)paradise.net.nz), 
               Jan 09 2003
%F A033638 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 (perry(AT)globalnet.co.uk), Jul 07 2004
%F A033638 a(n)=a(n-2)+n-1. - Paul Barry (pbarry(AT)wit.ie), Jul 14 2004
%F A033638 a(0) = 1; a(1) = 1; for n>1 a(n) = a(n-1) + round(sqrt(a(n-1))). - Jonathan 
               Vos Post (jvospost3(AT)gmail.com), Jan 19 2006
%F A033638 a(n) = Floor((n^2)/4)+1.
%F A033638 a(n)=2*a(n-1)-2*a(n-3)+a(n-4); a(0)=1, a(1)=1, a(2)=2, a(3)=3. [From 
               Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 03 2008]
%e A033638 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 A033638 Ulam square spiral = 7 8 9 / 6 1 2 / 5 4 3 /...; changes of direction 
               (right-angle) at 1 2 3 5 7 ...
%p A033638 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 (zerinvarylajos(AT)yahoo.com), 
               Mar 09 2007
%o A033638 (PARI) {a(n)= n^2\4 +1} (Michael Somos)
%Y A033638 Equals A002620 + 1. Cf. A002878, A004652, A002984.
%Y A033638 Sequence in context: A075353 A132278 A025700 this_sequence A136413 A117143 
               A115001
%Y A033638 Adjacent sequences: A033635 A033636 A033637 this_sequence A033639 A033640 
               A033641
%K A033638 easy,nonn
%O A033638 0,3
%A A033638 Tanya Y. Berger-Wolf (tanyabw(AT)uiuc.edu)