A286778 Sum of the common path length over all 2-tuples of nodes in a complete binary tree of height n.

0, 2, 22, 142, 734, 3390, 14718, 61694, 253438, 1029118, 4151294, 16683006, 66904062, 267993086, 1072791550, 4292935678, 17175543806, 68710301694, 274858508286, 1099470733310, 4397960527870, 17592005689342, 70368366690302, 281474188181502, 1125898262675454, 4503596204818430, 18014391395942398

Offset: 0

Author

F. Skerman, Jul 05 2017

Keywords

• nonn
• easy
• changed

Comments

Let the height of the binary tree be one less than the number of rows; i.e., a complete binary tree of height 2 has one root node, its two descends and four leaf nodes. Any node u has a unique path to the root of the binary tree. Let h(u,v) be the length of the intersection of these paths for nodes u and v. Then a(n) is defined to be the sum of h(u,v) over all ordered 2-tuples of nodes in a binary tree of height n.

Also the sum over all 2-tuples of nodes of the depth of their last common ancestor in the tree. Defined in this way and denoted Q(T) in the Janson link.

Let z(v) be the number of nodes in the subtree rooted at node v (so if u is the root z(u) is the number of nodes in the tree). Then a(n) is also the sum of squares of the z(v) over all non-root nodes v in the tree.

Examples

			A complete binary tree of height two consists of one root node (at depth 0), two children of the root (at depth 1) and four leaf nodes (at depth 2). Notice the common path length of node u with itself, h(u,u), is simply the depth of u.
The only 2-tuples to have common path length two is a leaf with itself (4 such tuples). Each child of the root with itself has common path length one (2 such tuples), as does each leaf with its sibling (4 such tuples) and each leaf with its parent (8 such tuples). All other 2-tuples have only the root as a common ancestor. Hence a(2) = 2*4 + 1*(2 + 4 + 8) + 0 = 22.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Svante Janson, The Wiener index of simply generated random trees, Random Structures Algorithms 22 (2003), no. 4, 337-358. DOI: 10.1002/rsa.10074.
• Index entries for linear recurrences with constant coefficients, signature (9,-28,36,-16).

Crossrefs

Cf. A036799 (total path length of a binary tree of height n).

Programs

Maple

seq( 4*2^(2*n) - (4*n+2)*2^n - 2, n=0..30); # Robert Israel, Jul 05 2017

Mathematica

LinearRecurrence[{9,-28,36,-16},{0,2,22,142},40] (* Harvey P. Dale, Apr 30 2018 *)

PARI

a(n) = sum(d=1, n, 2^d*(2^(n+1-d)-1)^2); \\ Michel Marcus, Jul 05 2017

PARI

concat(0, Vec(2*x*(1 + 2*x) / ((1 - x)*(1 - 2*x)^2*(1 - 4*x)) + O(x^30))) \\ Colin Barker, Jul 05 2017

Sage

[sum(2^d*(2^(n+1-d)-1)^2 for d in range(1,n+1)) for n in range(20)]

Formula

a(n) = Sum_{d=1..n} 2^d*(2^(n+1-d)-1)^2.

From Robert Israel, Jul 05 2017: (Start)

a(n) = 4*2^(2*n) - (4*n+2)*2^n - 2.

G.f.: 2*x*(2*x+1)/((4*x-1)*(x-1)*(2*x-1)^2).

E.g.f.: 4*exp(4*x)-(8*x+2)*exp(2*x)-2*exp(x).

(End)

a(n) = 9*a(n-1) - 28*a(n-2) + 36*a(n-3) - 16*a(n-4) for n>3. - Colin Barker, Jul 05 2017

a(n)-a(n-1) = A050488(n)*2^n . - R. J. Mathar, Aug 26 2025