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A033638
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Quarter-squares plus 1 (i.e. A002620 + 1).
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30
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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
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OFFSET
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0,3
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COMMENTS
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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, 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 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 by 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. [From 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]
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REFERENCES
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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.
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
E. S. Egge, Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations, Thm. 6.6
Index to sequences with linear recurrences with constant coefficients, signature (2,0,-2,1).
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FORMULA
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a(n) = ceil((n^2+3)/4) = ( (7+(-1)^n)/2 + n^2 )/4.
a(0) = a(1) = 1, a(n)=1+floor(a(n-1)/2). - Benoit Cloitre, 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, 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), n>3. [Philippe DELEHAM, Nov 03 2008]
a(0) = a(1) = 1, a(n) = a(n-1) + ceil(sqrt(a(n-2))) for n>1. - Jonathan Vos Post, Oct 8 2011
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EXAMPLE
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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)
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MAPLE
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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
A033638 := proc(n)
1+floor(n^2/4) ;
end proc: # R. J. Mathar, Jul 13 2012
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MATHEMATICA
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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, March 28 2011 *)
LinearRecurrence[{2, 0, -2, 1}, {1, 1, 2, 3}, 60] (* Robert G. Wilson v, Sep 16 2012 *)
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PROG
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(PARI) {a(n)= n^2\4 +1} \\ Michael Somos
(Haskell)
a033638 = (+ 1) . (`div` 4) . (^ 2) -- Reinhard Zumkeller, Apr 06 2012
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CROSSREFS
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Equals A002620 + 1.
Cf. A002878, A004652, A002984.
A083479
Sequence in context: A075353 A132278 A025700 * A194205 A136413 A177337
Adjacent sequences: A033635 A033636 A033637 * A033639 A033640 A033641
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KEYWORD
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easy,nonn
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AUTHOR
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Tanya Y. Berger-Wolf (tanyabw(AT)uiuc.edu)
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STATUS
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approved
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