0,2

An n X n nonnegative matrix A is primitive (see A070322) iff every element of A^k is > 0 for some power k. If A is primitive then the power which should have all positive entries is <= n^2 -2n +2 (Wielandt).

Let Phi_k(x) be the k-th cyclotomic polynomial and form the sequence Phi_k(0), Phi_k(1), Phi_k(2), ... This gives A000027 (k=2), A002061 (k=3), A002522 (k=4), A053699 (k=5), A002061 (k=6), A053716 (k=7), A002523 (k=8), A060883 (k=9), A060884 (k=10), A060885 (k=11), A060886 (k=12), A060887 (k=13), A060888 (k=14), A060889 (k=15), A060890 (k=16), A060891 (k=18), A060892 (k=20), A060893 (k=24), A060894 (k=30), A060895 (k=32), A060896 (k=36).

As the positive solution to x=2n+1/x is x=n+sqrt(a(n)), the continued fraction expansion of sqrt(a(n)) is {n; 2n, 2n, 2n, 2n, ...}. - Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 07 2001

a(n) is one less than the arithmetic mean of its neighbors: a(n) = {a(n-1) + a(n+1)}/2 -1. e.g. 2 = (1+5)/2, 5 = (2+10)/2 -1. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 29 2003. Equivalently, the continued fraction expansion of sqrt(a(n)) is (n;2n,2n,2n,....). - Franz Vrabec (franz.vrabec(AT)planetuniqa.at), Jan 23, 2006.

Number of {12,1*2*,21}-avoiding signed permutations in the hyperoctahedral group.

The number of squares of side 1 which can be drawn without lifting the pencil, starting at one corner of an n X n grid and never visiting an edge twice is n^2-2n+2. - Sebastien Dumortier (sdumortier(AT)ac-limoges.fr), Jun 16 2005

Comment from Cino Hilliard (hillcino368(AT)gmail.com), Feb 21 2006: "Also, except for the first term, numbers that cannot be expressed as a perfect power, i.e. x^2 + 1 != y^n for all x,y,n > 1. Proof. We assume the truth of the following theorem. Proofs can be found in elementary texts on number theory and online. Theorem I: A number N is a sum of two squares if and only if all prime factors of N of the form 4m+3 have even exponents.

"We are now ready to prove x^2 + 1 != y^n for all x,y,n > 1. We assume equality and seek a contradiction for n even and n odd. If n is even = 2k, x^2 + 1 = y^2k = (y^k)^2 and (y^k - x)(y^k + x) = 1. This implies y^k-x = y^k+x = 1 or 2x = 0 contrary to x > 1. So n must be odd for equality to hold.

"Then x^2+1 = y^(2k+1) implies all prime factors of y, including those of the form 4m+3 are raised to an odd exponent contrary to Theorem I. So we have shown x^2+1 = y^n is false for n even or n odd. Therefore x^2 + 1 != y^n as was desired."

Also, numbers m such m^3-m^2 is a perfect square, (n*(1 + n^2))^2. - Zak Seidov (zakseidov(AT)yahoo.com)

1 + 2/2 + 2/5 + 2/10 +...= Pi*coth Pi [Jolley] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 21 2006

For n>=1, a(n-1) is the minimal number of choices from an n-set such that at least one particular element has been chosen at least n times or each of the n elements has been chosen at least once. Some games define "matches" this way; e.g., in the classic Parker Brothers, now Hasbro, board game Risk, a(2)=5 is the number of cards of three available types (suits) required to guarantee at least one match of three different types or of three of the same type (ignoring any jokers or wildcards). - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Nov 18 2007

Sequence allows us to find X values of the equation: X^3 + (X - 1)^2 + X - 2 = Y^2. To prove that X = n^2 + 1: Y^2 = X^3 + (X - 1)^2 + X - 2 = X^3 + X^2 - X - 1 = (X - 1)(X^2 + 2X + 1) = (X - 1)*(X + 1)^2 it means: (X - 1) must be a perfect square, so X = n^2 + 1 and Y = n(n^2 + 2). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Nov 29 2007

For n>0: a(n-1)=A143053(A000290(n))-1, - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 20 2008

A143053(a(n))=A000290(n+1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 20 2008

Also, let Z=a(n), Y=3*a(n)-4, X=[a(n)-4]*sqrt[a(n)-1], then X^2+Y^2=Z^3. Example, Z=17, Y=3*17-4=47, X=(17-4)*4=52, then 52^2+47^2=17^3 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 05 2009]

Except for the first term of [A002522], and [A058331] if X=[A058331], Y=[A087113], A= [A002522], we have, for all other terms, Pell's equation: [A058331]^2 - [A002522]*[A087113]^2=1; (X^2-A*Y^2=1); example: 3^2-2*2^2=1; 9^2-5*4^2=1, 129^2-65*16^2=1, and so on. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 11 2009]

a(n) = A156798(n)/A087475(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 16 2009]

{a(k): 0 <= k < 4} = divisors of 10. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 17 2009]

Number of units of a(n) belongs to a periodic sequence: 1, 2, 5, 0, 7, 6, 7, 0, 5, 2. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009]

S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 120).

L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 176.

Wielandt, H. 1950. Unzerlegbare nicht negativen Matrizen, Math. Z. 52, 642-648.

Index entries for sequences related to linear recurrences with constant coefficients

S. J. Leon, Linear Algebra with Applications: THE PERRON-FROBENIUS THEOREM

T. Mansour and J. West, Avoiding 2-letter signed patterns.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Near-Square Prime

R. Zumkeller, Enumerations of Divisors [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 17 2009]

a(n)=A000290(n)+1. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 07 2004

a(n)=A034262(n)/A001477(n) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 20 2008

Sequences of the form a(n)=n^2+K with offset 0 have o.g.f. (K-2*K*x+K*x^2+x+x^2)/(1-x)^3 and reccurrence a(n)=3a(n-1)-3a(n-2)+a(n-3). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 28 2008

a(n)*a(n-2) = n^4 + 4. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 12 2009]

a(n)=2*n+a(n-1)-3 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 07 2009]

For n=2, a(2)=2*2+1-3=2; n=3, a(3)=2*3+2-3=5; n=4, a(4)=2*4+5-3=10 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 07 2009]

with(combinat, fibonacci):seq(fibonacci(3, i), i=0..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 01 2006

a[n_]:=n^2+1; [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 15 2008]

(PARI) g3(n, p) = for(x=1, n, y=x^2+p; print1(y", ")) - Cino Hilliard (hillcino368(AT)gmail.com), Feb 21 2006

(Other) sage: [lucas_number1(3, n, -1) for n in xrange(0, 51)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 16 2009]

Left edge of A055096.

Cf. A059100, A117950, A087475, A117951, A114949, A117619 (sequences of form n^2 + K).

a(n+1) = A101220(n, n+1, 3).

Cf. A059592, A124808.

Cf. A117950, A132411, A132414, A028872.

Cf. A001477, A034262.

Cf. A087113, A058331 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 11 2009]

A005408, A000124, A016813, A086514, A000125, A058331, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A006261. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 17 2009]

Sequence in context: A059591 A082607 A159547 this_sequence A069987 A119114 A062493

Adjacent sequences: A002519 A002520 A002521 this_sequence A002523 A002524 A002525

nonn,easy,new

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2006