Search: id:A002522
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%I A002522
%S A002522 1,2,5,10,17,26,37,50,65,82,101,122,145,170,197,226,257,290,325,362,401,
%T A002522 442,485,530,577,626,677,730,785,842,901,962,1025,1090,1157,1226,1297,
%U A002522 1370,1445,1522,1601,1682,1765,1850,1937,2026,2117,2210,2305,2402,2501
%N A002522 n^2 + 1.
%C A002522 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).
%C A002522 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).
%C A002522 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
%C A002522 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.
%C A002522 Number of {12,1*2*,21}-avoiding signed permutations in the hyperoctahedral 
               group.
%C A002522 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
%C A002522 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.
%C A002522 "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.
%C A002522 "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."
%C A002522 Also, numbers m such m^3-m^2 is a perfect square, (n*(1 + n^2))^2. - 
               Zak Seidov (zakseidov(AT)yahoo.com)
%C A002522 1 + 2/2 + 2/5 + 2/10 +...= Pi*coth Pi [Jolley] - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Dec 21 2006
%C A002522 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
%C A002522 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
%C A002522 For n>0: a(n-1)=A143053(A000290(n))-1, - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jul 20 2008
%C A002522 A143053(a(n))=A000290(n+1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jul 20 2008
%C A002522 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]
%C A002522 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]
%C A002522 a(n) = A156798(n)/A087475(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Feb 16 2009]
%C A002522 {a(k): 0 <= k < 4} = divisors of 10. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jun 17 2009]
%C A002522 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]
%D A002522 S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, 
               Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see 
               p. 120).
%D A002522 L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 
               176.
%D A002522 Wielandt, H. 1950. Unzerlegbare nicht negativen Matrizen, Math. Z. 52, 
               642-648.
%H A002522 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A002522 S. J. Leon, <a href="http://www.prenhall.com/divisions/esm/app/ph-linear/
               leon/html/perron.html">Linear Algebra with Applications: THE PERRON-FROBENIUS 
               THEOREM</a>
%H A002522 T. Mansour and J. West, <a href="http://arXiv.org/abs/math.CO/0207204">
               Avoiding 2-letter signed patterns</a>.
%H A002522 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               NumberPicking.html">Link to a section of The World of Mathematics.</
               a>
%H A002522 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Near-SquarePrime.html">Near-Square Prime</a>
%H A002522 R. Zumkeller, <a href="a161700.txt">Enumerations of Divisors</a> [From 
               Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 17 2009]
%F A002522 a(n)=A000290(n)+1. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 07 2004
%F A002522 a(n)=A034262(n)/A001477(n) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 20 2008
%F A002522 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
%F A002522 a(n)*a(n-2) = n^4 + 4. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Feb 12 2009]
%F A002522 a(n)=2*n+a(n-1)-3 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 07 2009]
%e A002522 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]
%p A002522 with(combinat, fibonacci):seq(fibonacci(3, i), i=0..50); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Dec 01 2006
%t A002522 a[n_]:=n^2+1; [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 15 
               2008]
%o A002522 (PARI) g3(n,p) = for(x=1,n,y=x^2+p;print1(y",")) - Cino Hilliard (hillcino368(AT)gmail.com), 
               Feb 21 2006
%o A002522 (Other) sage: [lucas_number1(3,n,-1) for n in xrange(0, 51)] # [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 16 2009]
%Y A002522 Left edge of A055096.
%Y A002522 Cf. A059100, A117950, A087475, A117951, A114949, A117619 (sequences of 
               form n^2 + K).
%Y A002522 a(n+1) = A101220(n, n+1, 3).
%Y A002522 Cf. A059592, A124808.
%Y A002522 Cf. A117950, A132411, A132414, A028872.
%Y A002522 Cf. A001477, A034262.
%Y A002522 Cf. A087113, A058331 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Feb 11 2009]
%Y A002522 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]
%Y A002522 Sequence in context: A059591 A082607 A159547 this_sequence A069987 A119114 
               A062493
%Y A002522 Adjacent sequences: A002519 A002520 A002521 this_sequence A002523 A002524 
               A002525
%K A002522 nonn,easy,new
%O A002522 0,2
%A A002522 N. J. A. Sloane (njas(AT)research.att.com).
%E A002522 More terms from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 
               2006

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