0,3

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.

a(n) = A001055(prime^n), number of factorizations. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Dec 29, 2001

Locations of right angle turns in Ulam square spiral. - Donald S. McDonald (don.mcdonald(AT)paradise.net.nz), Jan 09 2003

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 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

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 (jvospost2(AT)yahoo.com), Apr 03 2007

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.

I. Schur, Neue Begrundung der Theorie der Gruppencharaketere, Sitzungberichte der Koniglich Preussischen Akademie der Wissenschaften zu Berlin (1905), 406-432.

E. S. Egge, Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations, Thm. 6.6

a(n) = ceil((n^2+3)/4) = ( (7+(-1)^n)/2 + n^2 )/4.

a(0)=1, a(1)=1, a(n)=1+floor(a(n-1)/2). - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 06 2002

Numbers of the form n^2+1 or n^2+n+1. - Donald S. McDonald (don.mcdonald(AT)paradise.net.nz), Jan 09 2003

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

a(n)=a(n-2)+n-1. - Paul Barry (pbarry(AT)wit.ie), 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 (jvospost2(AT)yahoo.com), Jan 19 2006

a(n) = Floor((n^2)/4)+1.

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 ...

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

(PARI) {a(n)= n^2\4 +1} (Michael Somos)

Equals A002620 + 1. Cf. A002878, A004652, A002984.

Sequence in context: A075353 A132278 A025700 this_sequence A136413 A117143 A115001

Adjacent sequences: A033635 A033636 A033637 this_sequence A033639 A033640 A033641

easy,nonn

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