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%I M1368 N0531 #96 Aug 23 2023 07:10:37
%S 0,2,5,10,16,24,33,44,56,70,85,102,120,140,161,184,208,234,261,290,
%T 320,352,385,420,456,494,533,574,616,660,705,752,800,850,901,954,1008,
%U 1064,1121,1180,1240,1302,1365,1430,1496,1564,1633,1704,1776,1850,1925
%N Triangular numbers plus quarter-squares: n*(n+1)/2 + floor((n+1)^2/4) (i.e., A000217(n) + A002620(n+1)).
%C Number of series-reduced planted trees with n+7 nodes and 3 internal nodes.
%C The trees enumerated with 3 internal nodes are of two types. Those with all internal nodes at different heights are enumerated by the triangular numbers. Those with two internal nodes at the same height are enumerated by the quarter squares. - _Michael Somos_, May 19 2000
%C Number of pairs (x,y) with x in {0,...,n}, y even in {0,...,2n}, and x < y. - _Clark Kimberling_, Jul 02 2012
%D John Riordan, personal communication.
%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 T. D. Noe, <a href="/A001859/b001859.txt">Table of n, a(n) for n = 0..1000</a>
%H D. D. Frey and J. A. Sellers, Jacobsthal Numbers and Alternating Sign Matrices, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/SELLERS/sellers.html">J. Integer Seqs., Vol. 3 (2000), #00.2.3</a>
%H S. V. Gervacio and H. Maehara, <a href="http://dx.doi.org/10.1016/j.disc.2005.12.018">Partial order on a family of k-subsets of a linearly ordered set</a>, Discr. Math., 306 (2006), 413-419.
%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 J. Riordan, <a href="/A000262/a000262_1.pdf">Letter to N. J. A. Sloane, Oct. 1970</a>
%H S. G. Wagner, <a href="http://finanz.math.tu-graz.ac.at/~wagner/identity.pdf">An identity for the cycle indices of rooted tree automorphism groups</a>, Elec. J. Combinat., 13 (2006), #R00.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>.
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F a(n) = A000217(n)+A002620(n+1).
%F a(n) = n + floor( (3n^2+1)/4 ).
%F G.f.: (2*x+x^2)/((1-x)^2*(1-x^2)).
%F a(n) = a(n-1) + a(n-2) - a(n-3) + 3 = A002378(n) - A002620(n) = A006578(n-1) + A004526(n+1) - _Henry Bottomley_, Mar 08 2000
%F a(n) = A006578(-1-n) for all n in Z. - _Michael Somos_, May 10 2006
%F From _Mitch Harris_, Aug 22 2006: (Start)
%F a(n) = (6n^2 + 8n + 1 - (-1)^n)/8;
%F a(n) = Sum_{k=0..n} max(k, n-k). (End)
%F Starting (2, 5, 10, 16, 24, ...), = binomial transform of [2, 3, 2, -1, 2, -4, 8, -16, 32, ...]. - _Gary W. Adamson_, Nov 30 2007
%F a(0)=0, a(1)=2, a(2)=5, a(3)=10, a(n) = 2*a(n-1) + 0*a(n-2) - 2*a(n-3) + a(n-4). - _Harvey P. Dale_, Apr 01 2012
%F a(n) = 3*n*(n+1)/2 - A006578(n). - _Clark Kimberling_, Jul 02 2012
%F a(2*n) = A045944(n), a(2*n - 1) = A049450(n) for all n in Z. - _Michael Somos_, Nov 03 2014
%F 0 = -6 + a(n) - 2*a(n+2) + a(n+4) for all n in Z. - _Michael Somos_, Nov 03 2014
%F 0 = a(n)*(+a(n+1) - a(n+2)) + a(n+1)*(-3 - a(n+1) + a(n+2)) for all n in Z. - _Michael Somos_, Nov 03 2014
%F a(n) = Sum_{k=1..n} floor((n+k+2)/2). - _Wesley Ivan Hurt_, Mar 31 2017
%F Sum_{n>=1} 1/a(n) = 3/4 - Pi/(4*sqrt(3)) + 3*log(3)/4. - _Amiram Eldar_, May 28 2022
%F E.g.f.: (x*(7 + 3*x)*cosh(x) + (1 + 7*x + 3*x^2)*sinh(x))/4. - _Stefano Spezia_, Aug 22 2023
%e For n=1 we find 2 planted trees with 8 nodes, 3 of which are internal (i) and 5 are endpoints (e):
%e .e...e...e...e....e...e....
%e ...i.......i........i...e..
%e .......i..............i...e
%e .......e................i..
%e ........................e..
%e G.f. = 2*x + 5*x^2 + 10*x^3 + 16*x^4 + 24*x^5 + 33*x^6 + 44*x^7 + 56*x^8 + ...
%p A001859:=(-1-z^2-2*z^3+z^4)/(z+1)/(z-1)^3; # conjectured by _Simon Plouffe_ in his 1992 dissertation; gives sequence with an additional leading 1
%p with (combinat):seq(count(Partition((3*n+2)), size=3), n=0..50); # _Zerinvary Lajos_, Mar 28 2008
%t With[{nn=60},Total/@Thread[{Accumulate[Range[0,nn]],Floor[Range[ nn+1]^2/4]}]] (* or *) LinearRecurrence[{2,0,-2,1},{0,2,5,10},60] (* _Harvey P. Dale_, Apr 01 2012 *)
%o (PARI) {a(n) = n + (3*n^2 + 1) \ 4};
%o (Haskell)
%o a001859 n = a000217 n + a002620 (n + 1) -- _Reinhard Zumkeller_, Dec 20 2012
%Y Cf. A000217, A002378, A002620, A004526, A006578, A045944, A049450.
%Y First differences of A045947.
%Y Antidiagonal sums of array A003984.
%Y Cf. A107661, A077043.
%Y Cf. A185212 (odd terms).
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E Entry improved by _Michael Somos_
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