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%I #139 Feb 05 2024 02:21:17
%S -1,2,7,14,23,34,47,62,79,98,119,142,167,194,223,254,287,322,359,398,
%T 439,482,527,574,623,674,727,782,839,898,959,1022,1087,1154,1223,1294,
%U 1367,1442,1519,1598,1679,1762,1847,1934,2023,2114,2207,2302,2399,2498
%N a(n) = n^2 - 2.
%C For n >= 2, least m >= 1 such that f(m, n) = 0 where f(m,n) = Sum_{i=0..m} Sum_{k= 0..i} (-1)^k*(floor(i/n^k) - n*floor(i/n^(k+1))). - _Benoit Cloitre_, May 02 2004
%C For n >= 3, the a(n)-th row of Pascal's triangle always contains a triple forming an arithmetic progression. - _Lekraj Beedassy_, Jun 03 2004
%C Let C = 1 + sqrt(2) = 2.414213...; and 1/C = 0.414213... Then a(n) = (n + 1 + 1/C) * (n + 1 - C). Example: a(6) = 34 = (7 + 0.414...) * (7 - 2.414...). - _Gary W. Adamson_, Jul 29 2009
%C The sequence (n-4)^2-2, n = 7, 8, ... enumerates the number of non-isomorphic sequences of length n, with entries from {1, 2, 3} and no two adjacent entries the same, that minimally contain each of the thirteen rankings of three players (111, 121, 112, 211, 122, 212, 221, 123, 132, 213, 231, 312, 321) as embedded order isomorphic subsequences. By "minimally", we mean that the n-th symbol is necessary for complete inclusion of all thirteen words. See the arXiv paper below for proof. If n = 7, these sequences are 1213121, 1213212, 1231213, 1231231, 1231321, 1232123, and 1232132, and for each case, there are 3! = 6 isomorphs. - _Anant Godbole_, Feb 20 2013
%C a(n), n >= 0, with a(0) = -2, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 8 for b = 2*n. In general D = b^2 - 4*a*c > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - _Wolfdieter Lang_, Aug 15 2013
%C With a different offset, this is 2*n^2 - (n + 1)^2, which arises in one explanation of why Bertrand's postulate does not automatically prove Legendre's conjecture: as n gets larger, so does the range of numbers that can have primes that satisfy Bertrand's postulate yet do nothing for Legendre's conjecture. - _Alonso del Arte_, Nov 06 2013
%C x*(x + r*y)^2 + y*(y + r*x)^2 can be written as (x + y)*(x^2 + s*x*y + y^2). For r >= 0, the sequence gives the values of s: in fact, s = (r + 1)^2 - 2. - _Bruno Berselli_, Feb 20 2019
%C For n >= 2, the continued fraction expansion of sqrt(a(n)) is [n-1; {1, n-2, 1, 2n-2}]. For n=2, this collapses to [1; {2}]. - _Magus K. Chu_, Sep 06 2022
%H T. D. Noe, <a href="/A008865/b008865.txt">Table of n, a(n) for n = 1..1000</a>
%H Anant Godbole and Martha Liendo, <a href="http://arxiv.org/abs/1302.4668">Waiting time distribution for the emergence of superpatterns</a>, arxiv 1302.4668 [math.PR], 2013.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Near-SquarePrime.html">Near-Square Prime</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F For n > 1: a(n) = A143053(A000290(n)), A143054(a(n)) = A000290(n). - _Reinhard Zumkeller_, Jul 20 2008
%F G.f.: (x-5*x^2+2*x^3)/(-1+3*x-3*x^2+x^3). - _Klaus Brockhaus_, Oct 17 2008
%F E.g.f.: (x^2 + x -2)*exp(x) + 2. - _G. C. Greubel_, Aug 19 2017
%F a(n+1) = A101986(n) - A101986(n-1) = A160805(n) - A160805(n-1). - _Reinhard Zumkeller_, May 26 2009
%F For n > 1, a(n) = floor(n^5/(n^3 + n + 1)). - _Gary Detlefs_, Feb 10 2010
%F a(n) = a(n-1) + 2*n - 1 for n > 1, a(1) = -1. - _Vincenzo Librandi_, Nov 18 2010
%F Right edge of the triangle in A195437: a(n) = A195437(n-2, n-2). - _Reinhard Zumkeller_, Nov 23 2011
%F a(n)*a(n-1) + 2 = (a(n) - n)^2 = A028552(n-2)^2. - _Bruno Berselli_, Dec 07 2011
%F a(n+1) = A000096(n) + A000096(n-1) for all n in Z. - _Michael Somos_, Nov 11 2015
%F From _Amiram Eldar_, Jul 13 2020: (Start)
%F Sum_{n>=1} 1/a(n) = (1 - sqrt(2)*Pi*cot(sqrt(2)*Pi))/4.
%F Sum_{n>=1} (-1)^n/a(n) = (1 - sqrt(2)*Pi*cosec(sqrt(2)*Pi))/4. (End)
%F Assume offset 0. Then a(n) = 2*LaguerreL(2, 1 - n). - _Peter Luschny_, May 09 2021
%F From _Amiram Eldar_, Feb 05 2024: (Start)
%F Product_{n>=1} (1 - 1/a(n)) = sqrt(2/3)*sin(sqrt(3)*Pi)/sin(sqrt(2)*Pi).
%F Product_{n>=2} (1 + 1/a(n)) = -Pi/(sqrt(2)*sin(sqrt(2)*Pi)). (End)
%e G.f. = -x + 2*x^2 + 7*x^3 + 14*x^4 + 23*x^5 + 34*x^6 + 47*x^7 + 62*x^8 + 79*x^9 + ...
%t Range[50]^2 - 2 (* _Harvey P. Dale_, Mar 14 2011 *)
%o (PARI) {for(n=1, 47, print1(n^2-2, ","))} \\ _Klaus Brockhaus_, Oct 17 2008
%o (Haskell)
%o a008865 = (subtract 2) . (^ 2) :: Integral t => t -> t
%o a008865_list = scanl (+) (-1) [3, 5 ..]
%o -- _Reinhard Zumkeller_, May 06 2013
%o (Magma) [n^2 - 2: n in [1..60]]; // _Vincenzo Librandi_, May 01 2014
%Y Cf. A145067 (Zero followed by partial sums of A008865).
%Y Cf. A000027, A013648.
%Y Cf. A028871 (primes).
%Y Cf. A263766 (partial products).
%Y Cf. A270109. [_Bruno Berselli_, Mar 17 2016]
%K sign,easy
%O 1,2
%A _N. J. A. Sloane_, _R. K. Guy_
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