cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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+$k]) == $r->[$ has authored 2 sequences.

A165910 Wiener indices of Fibonacci trees of order k.

Original entry on oeis.org

1, 4, 18, 62, 210, 666, 2063, 6226, 18484, 54100, 156620, 449268, 1278981, 3617544, 10175590, 28485218, 79406350, 220536910, 610487875, 1684974790, 4638298536, 12737460744, 34902844728, 95449821672, 260554112425, 710056257196
Offset: 1

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Author

K.V.Iyer, K. R. Udaya Kumar Reddy, Sep 30 2009

Keywords

Comments

The Fibonacci trees T(f(k)) of order k is defined as follows: 1. T(f(-1)) and T(f(0)) each consist of a single node. 2. For k >= 1, T(f(k)) is built from copies of T(f(k-1)) and T(f(k-2)) by connecting (by an edge) T(f(k-2)) as the rightmost child of the root of T(f(k-1)).

Examples

			W(T(1)) = 1 because T(1) is a single edge. W(T(2)) = 4 because T(2) is a path on three vertices.

References

  • K. Viswanathan Iyer and K. R. Udaya Kumar Reddy, Wiener index of Binomial trees and Fibonacci trees, Int'l. J. Math. Engin. with Comp., Accepted for publication, Sept. 2009.

Links

Formula

The Wiener index W(T(f(k))) of the Fibonacci tree T(f(k)) satisfies the following recurrence: W(T(f(k))) = W(T(f(k-1))) + W(T(f(k-2))) + F(k+1) D(T(f), (k-2)) + F(k) D(T(f), (k-1)) + F(k+1) F(k), where D(T(f), k) = (1/5) (k F(k+2) + (k+2) F(k)) and F(k) is the k-th Fibonacci number.
D(T(f),k) = A001629(k+1). Conjecture: G.f. x*(1-x+x^2-2*x^3)/( (1-x^2-x) * (1+x)^2 * (x^2-3*x+1)^2 ). [From R. J. Mathar, Apr 19 2010]

A076184 Graph code numbers of simple graphs in numerical order.

Original entry on oeis.org

0, 1, 3, 7, 11, 12, 13, 15, 30, 31, 63, 75, 76, 77, 79, 86, 87, 94, 95, 116, 117, 119, 127, 222, 223, 235, 236, 237, 239, 254, 255, 507, 511, 1023, 1099, 1100, 1101, 1103, 1108, 1109, 1110, 1111, 1118, 1119, 1140, 1141, 1143, 1151, 1182, 1183, 1184, 1185, 1187
Offset: 1

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Author

K. R. Parthasarathy (nuns(AT)vsnl.com), Nov 02 2002

Keywords

Comments

Every simple graph has a symmetric adjacency matrix whose lower triangular part by rows represents a little-endian binary number of which the minimum value over all isomorphic graphs gives the graph code number. Adding isolated vertices will not change the graph code number.
Study of the patterns and gaps in the sequence appears to be quite interesting.
The number of terms that are less than 2^(n*(n-1)/2) is equal to A000088(n). - Vladimir Kulipanov, Oct 13 2015
The numbers of vertices and edges of the graph with code a(n) are A002024(A029837(a(n)+1)) + 1 and A000120(a(n)), respectively. - Pontus von Brömssen, Mar 15 2025
To obtain all unlabeled graphs with n vertices (including graphs with isolated vertices), take the graphs associated with the first A000088(n) terms and add isolated vertices so that each graph gets n vertices. - Pontus von Brömssen, Mar 19 2025

Examples

			a(5)=11 in binary (with 0's prepended to give a triangular number of digits) is 001011 so adjacency matrix [0,1,1,1; 1,0,0,0; 1,0,0,0; 1,0,0,0].
a(6)=12 in binary is 001100 so adjacency matrix [0,0,0,1; 0,0,1,0; 0,1,0,0; 1,0,0,0].

References

  • F. Harary, Problems involving graphical numbers, in Colloq. Math. Soc. Janos Bolyai, 4 (1970) 625-635. Look at his 'mincode numbers'.

Links

Crossrefs

Cf. A000088, A000120, A002024, A029837.

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