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%I M0625 N0227 #123 Jul 02 2023 02:21:00
%S 0,0,1,1,2,3,5,6,8,10,13,15,18,21,25,28,32,36,41,45,50,55,61,66,72,78,
%T 85,91,98,105,113,120,128,136,145,153,162,171,181,190,200,210,221,231,
%U 242,253,265,276,288,300,313,325,338,351,365,378,392,406,421,435,450
%N Nearest integer to n^2/8.
%C Restricted partitions.
%C a(0) = a(1) = 0; a(n) are the partitions of floor((3*n+3)/2) with 3 distinct numbers of the set {1, ..., n}; partitions of floor((3*n+3)/2)-C and ceiling((3*n+3)/2)+C have equal numbers. - _Paul Weisenhorn_, Jun 05 2009, corrected by _M. F. Hasler_, Jun 16 2022
%C Odd-indexed terms are the triangular numbers, even-indexed terms are the midpoint (rounded up where necessary) of the surrounding odd-indexed terms. - _Carl R. White_, Aug 12 2010
%C a(n+2) is the number of points one can surround with n stones in Go (including the points under the stones). - _Thomas Dybdahl Ahle_, May 11 2014
%C Corollary of above: a(n) is the number of points one can surround with n+2 stones in Go (excluding the points under the stones). - _Juhani Heino_, Aug 29 2015
%C From _Washington Bomfim_, Jan 13 2021: (Start)
%C For n >= 4, a(n) = A026810(n+2) - A026810(n-4).
%C Let \n,m\ be the number of partitions of n into m non-distinct parts.
%C For n >= 1, \n,4\ = round((n-2)^2/8).
%C For n >= 6, \n,4\ = A026810(n) - A026810(n-6).
%C (End)
%D A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
%D M. Jeger, Einfuehrung in die Kombinatorik, Klett, 1975, Bd.2, pages 110 ff. [_Paul Weisenhorn_, Jun 05 2009]
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A001971/b001971.txt">Table of n, a(n) for n = 0..10000</a>
%H G. Almkvist, <a href="http://projecteuclid.org/euclid.pjm/1102780612">Invariants, mostly old ones</a>, Pacific J. Math. 86 (1980), no. 1, 1-13. MR0586866 (81j:14029)
%H A. Cayley, <a href="/A001971/a001971.pdf">Numerical tables supplementary to second memoir on quantics</a>, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281. [Annotated scanned copy]
%H Shalosh B. Ekhad and Doron Zeilberger, <a href="https://arxiv.org/abs/1901.08172">In How many ways can I carry a total of n coins in my two pockets, and have the same amount in both pockets?</a>, arXiv:1901.08172 [math.CO], 2019.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H D. Vainsencher and A. M. Bruckstein, <a href="https://doi.org/10.1016/j.tcs.2008.06.043">On isoperimetrically optimal polyforms</a>, Theoretical Computer Science 406.1-2, 2008, pp. 146-159.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-2,1).
%F The listed terms through a(20)=50 satisfy a(n+2) = a(n-2) + n. - _John W. Layman_, Dec 16 1999
%F G.f.: x^2 * (1 - x + x^2) / (1 - 2*x + x^2 - x^4 + 2*x^5 - x^6) = x^2 * (1 - x^6) / ((1 - x) * (1 - x^2) * (1 - x^3) * (1 - x^4)). - _Michael Somos_, Feb 07 2004
%F a(n) = floor((n^2+4)/8). - _Paul Weisenhorn_, Jun 05 2009
%F a(2*n+1) = A000217(n), a(2*n) = floor((A000217(n-1)+A000217(n)+1)/2). - _Carl R. White_, Aug 12 2010
%F From _Michael Somos_, Aug 29 2015: (Start)
%F Euler transform of length 6 sequence [ 1, 1, 1, 1, 0, -1].
%F a(n) = a(-n) for all n in Z. (End)
%F a(2n) = A000982(n). - _M. F. Hasler_, Jun 16 2022
%F Sum_{n>=2} 1/a(n) = 2 + Pi^2/12 + tanh(Pi/2)*Pi/2. - _Amiram Eldar_, Jul 02 2023
%p A001971:=-(1-z+z**2)/((z+1)*(z**2+1)*(z-1)**3); # conjectured (correctly) by _Simon Plouffe_ in his 1992 dissertation [Note that this "generating function" is Sum_{n >= 0} a(n+2)*z^n, not a(n)*z^n. - _M. F. Hasler_, Jun 16 2022]
%t LinearRecurrence[{2,-1,0,1,-2,1},{0,0,1,1,2,3},70] (* _Harvey P. Dale_, Jan 30 2014 *)
%o (PARI) {a(n) = round(n^2 / 8)};
%o (PARI) apply( {A001971(n)=n^2\/8}, [0..99]) \\ _M. F. Hasler_, Jun 16 2022
%o (Magma) [Round(n^2/8): n in [0..60]]; // _Vincenzo Librandi_, Jun 23 2011
%o (Haskell)
%o a001971 = floor . (+ 0.5) . (/ 8) . fromIntegral . (^ 2)
%o -- _Reinhard Zumkeller_, May 08 2012
%Y The 4th diagonal of A061857?
%Y Kind of an inverse of A261491 (regarding Go).
%Y Cf. A026810 (partitions with greatest part 4), A001400 (partitions in at most 4 parts), A000217 (a(2n+1): triangular numbers n(n+1)/2), A000982 (a(2n): round(n^2/2)).
%K nonn,easy
%O 0,5
%A _N. J. A. Sloane_
%E Edited Feb 08 2004
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