A001859 Triangular numbers plus quarter-squares: n*(n+1)/2 + floor((n+1)^2/4) (i.e., A000217(n) + A002620(n+1)).

0, 2, 5, 10, 16, 24, 33, 44, 56, 70, 85, 102, 120, 140, 161, 184, 208, 234, 261, 290, 320, 352, 385, 420, 456, 494, 533, 574, 616, 660, 705, 752, 800, 850, 901, 954, 1008, 1064, 1121, 1180, 1240, 1302, 1365, 1430, 1496, 1564, 1633, 1704, 1776, 1850, 1925

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Number of series-reduced planted trees with n+7 nodes and 3 internal nodes.

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

Number of pairs (x,y) with x in {0,...,n}, y even in {0,...,2n}, and x < y. - Clark Kimberling, Jul 02 2012

Examples

			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....
...i.......i........i...e..
.......i..............i...e
.......e................i..
........................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 + ...
		

References

• John Riordan, personal communication.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• D. D. Frey and J. A. Sellers, Jacobsthal Numbers and Alternating Sign Matrices, J. Integer Seqs., Vol. 3 (2000), #00.2.3
• S. V. Gervacio and H. Maehara, Partial order on a family of k-subsets of a linearly ordered set, Discr. Math., 306 (2006), 413-419.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• J. Riordan, Letter to N. J. A. Sloane, Oct. 1970
• S. G. Wagner, An identity for the cycle indices of rooted tree automorphism groups, Elec. J. Combinat., 13 (2006), #R00.
• Index entries for sequences related to rooted trees.
• Index entries for sequences related to trees.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A000217, A002378, A002620, A004526, A006578, A045944, A049450.

First differences of A045947.

Antidiagonal sums of array A003984.

Cf. A107661, A077043.

Cf. A185212 (odd terms).

Programs

Haskell

a001859 n = a000217 n + a002620 (n + 1)  -- Reinhard Zumkeller, Dec 20 2012

Maple

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
with (combinat):seq(count(Partition((3*n+2)), size=3), n=0..50); # Zerinvary Lajos, Mar 28 2008

Mathematica

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 *)

PARI

{a(n) = n + (3*n^2 + 1) \ 4};

Formula

a(n) = A000217(n)+A002620(n+1).

a(n) = n + floor( (3n^2+1)/4 ).

G.f.: (2*x+x^2)/((1-x)^2*(1-x^2)).

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

a(n) = A006578(-1-n) for all n in Z. - Michael Somos, May 10 2006

From Mitch Harris, Aug 22 2006: (Start)

a(n) = (6n^2 + 8n + 1 - (-1)^n)/8;

a(n) = Sum_{k=0..n} max(k, n-k). (End)

Starting (2, 5, 10, 16, 24, ...), = binomial transform of [2, 3, 2, -1, 2, -4, 8, -16, 32, ...]. - Gary W. Adamson, Nov 30 2007

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

a(n) = 3*n*(n+1)/2 - A006578(n). - Clark Kimberling, Jul 02 2012

a(2*n) = A045944(n), a(2*n - 1) = A049450(n) for all n in Z. - Michael Somos, Nov 03 2014

0 = -6 + a(n) - 2*a(n+2) + a(n+4) for all n in Z. - Michael Somos, Nov 03 2014

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

a(n) = Sum_{k=1..n} floor((n+k+2)/2). - Wesley Ivan Hurt, Mar 31 2017

Sum_{n>=1} 1/a(n) = 3/4 - Pi/(4*sqrt(3)) + 3*log(3)/4. - Amiram Eldar, May 28 2022

E.g.f.: (x*(7 + 3*x)*cosh(x) + (1 + 7*x + 3*x^2)*sinh(x))/4. - Stefano Spezia, Aug 22 2023

Extensions

Entry improved by Michael Somos