K.V.Iyer - cp's OEIS frontend

User: K.V.Iyer

K.V.Iyer has authored 5 sequences.

A191994 (Sum of first n Fibonacci numbers) times (product of first n Fibonacci numbers).

1, 2, 8, 42, 360, 4800, 102960, 3538080, 196035840, 17520703200, 2529842515200, 590412901478400, 222813349683724800, 136001024583142118400, 134285149587387262464000, 214504624277084224347264000, 554361997358383529330695680000

Offset: 1

K.V.Iyer, Venkata Subba Reddy P., Charles R Greathouse IV, Jun 21 2011

Let F(1), F(2), F(3), ... be the Fibonacci numbers 1, 1, 2, .... For k=1, we define the tree T(1) the path on two vertices with one identified as the root r. We assign the edge-weight F(1). T(2) is obtained from T(1) by attaching F(2) vertex to the pendents in T(1) except r. In T(2), r is retained as in T(1) and the new edge-weight is assigned as F(2). For k>1, T(k) is obtained from T(k-1) by attaching F(k) vertices to pendents in T(k-1) except r. In T(k), r is retained as in T(k-1) and all the new edge-weights are assigned F(k). With D(1)=1, for k>1 let D(k)=Sum of all distances d(r,x) taken across all vertices x in T(k). By induction it follows that for k>1, D(k)-D(k-1) is this sequence.

Retaining the notation of D(k) above, it follows, for k>1, that if D(k)=a(1)F(1)+ - - - +a(k)F(k) then D(k+1)=b(1)F(1)+ - - - +b(k)F(k)+b(k+1)F(k+1) where b(k+1) is the number of leaf nodes in T(k+1).

Charles R Greathouse IV, Table of n, a(n) for n = 1..97
Eric Weisstein's World of Mathematics, Fibonacci Factorial Constant

Cf. A000071 (sum of Fibonacci numbers), A003266 (product of Fibonacci numbers).

Cf. A062073 (Fibonacci factorial constant).

s=0;p=1;for(n=1,40,f=fibonacci(n);s+=f;p*=f;print1(s*p", ")) \\ Charles R Greathouse IV, Jun 21 2011

a(n)=prod(k=1,n,fibonacci(k))*(fibonacci(n+2)-1) /* Franklin T. Adams-Watters, Jun 23 2011 */

a(n) ~ C*sqrt(phi^(n^2 + 3*n + 4)/5^(n+1)) where C = A062073 and phi = (1+sqrt(5))/2.

a(n) = (F(n+2)-1) * Product_{k=1..n} F(k). - Franklin T. Adams-Watters, Jun 23 2011

A179023 a(n) = n(F(n+2) - 1) where F(n) is defined by A000045.

Original entry on oeis.org

0, 1, 4, 12, 28, 60, 120, 231, 432, 792, 1430, 2552, 4512, 7917, 13804, 23940, 41328, 71060, 121752, 207955, 354200, 601776, 1020074, 1725552, 2913408, 4910425, 8263060, 13884156, 23297092, 39041772, 65349240, 109261887, 182492352

Offset: 0

K.V.Iyer, Jun 25 2010

Let the 'Fibonacci weighted stars' T(i)'s be defined as: T(1) is an edge with one vertex as a distinguished vertex; the weight on the edge is taken to be F(1); for n>1, T(n) is formed by taking a copy of T(n-1) and attaching an edge to its distinguished vertex; the weight on the new edge is taken to be F(n). The sum of the weighted distances over all pairs of vertices in T(n) is this sequence.

Index entries for linear recurrences with constant coefficients, signature (4, -4, -2, 4, 0, -1).

f[n_] := n(Fibonacci[n + 2] - 1); Array[f, 33, 0] (* Robert G. Wilson v, Jun 27 2010 *)

a(0)=0, a(1)=1 and for n>1, a(n) = a(n-1) + F(n+1) +nF(n) -1.

a(n)= +4*a(n-1) -4*a(n-2) -2*a(n-3) +4*a(n-4) -a(n-6). = -n + A023607(n+1) - A000045(n+2). G.f.: -x*(-1+2*x^3) / ( (x-1)^2*(x^2+x-1)^2 ). - R. J. Mathar, Sep 15 2010

More terms from Robert G. Wilson v, Jun 27 2010

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

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

nonn

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)).

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

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.

K. Viswanathan Iyer and K. R. Udaya Kumar Reddy, Wienerindex of binomial trees and Fibonacci trees, arXiv:0910.4432
Index entries for linear recurrences with constant coefficients, signature (5, -3, -14, 10, 14, -5, -3, 1).

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]

A158681 Wiener indexes of the complete binary trees with n levels, root being at level 0.

Original entry on oeis.org

4, 48, 368, 2304, 12864, 66816, 330496, 1579008, 7353344, 33583104, 151056384, 671219712, 2953068544, 12885491712, 55835820032, 240520790016, 1030797656064, 4398058045440, 18691721789440, 79164887531520, 334251639701504, 1407375101657088, 5910974963908608

Offset: 1

K.V.Iyer, Mar 24 2009

			For n=1, the complete binary tree with level 1 is P_{3} whose Wiener index is 4.

R.Balakrishnan, K.Viswanathan Iyer, K.T.Raghavendra, "Wiener index of two special trees", MATCH Commun. Math. Comput. Chem., 57(2), 2007, 385-392.

Index entries for linear recurrences with constant coefficients, signature (12,-52,96,-64).

LinearRecurrence[{12,-52,96,-64},{4,48,368,2304},40] (* Harvey P. Dale, Nov 05 2015 *)

a(n) = (n+4)2^(n+1) + (n-2)2^(2n+2), n>0.

G.f.: 4*x / ( (4*x-1)^2*(2*x-1)^2 ). [From R. J. Mathar, Sep 15 2010]

A136328 a(n) = Wiener index of the odd graph O_n.

Original entry on oeis.org

0, 3, 75, 1435, 25515, 436821, 7339332, 121782375, 2005392675, 32835436777, 535550923908, 8707954925033, 141270179732500, 2287544190032700, 36988236910737360, 597341791692978975, 9637351741503033075, 155353556752487795625, 2502545930175392062500

Offset: 1

K.V.Iyer, Mar 27 2008

nonn

The odd graph O_n (n>=2) is a graph whose vertices represent the (n-1)-subsets of {1,2,...,2n-1} and two vertices are connected if and only if they correspond to disjoint subsets. It is a distance regular graph. [Emeric Deutsch, Aug 20 2013]

			a(2)=3 is the Wiener index of O_2 which is C_3.
a(3)=75 is the Wiener index of O_3 which is the Petersen graph.

Kailasam Viswanathan Iyer, Some computational and graph theoretical aspects of Wiener index, Ph.D. Dissertation, Dept. of Comp. Sci. & Engg., National Institute of Technology, Trichy, India, 2007.

Andrew Howroyd, Table of n, a(n) for n = 1..200
R. Balakrishnan, N. Sridharan and K. Viswanathan Iyer, The Wiener index of odd graphs, J. Ind. Inst. Sci., vol. 86, no. 5, 2006.
R. Balakrishnan, N. Sridharan and K. Viswanathan Iyer,, The Wiener index of odd graphs, J. Ind. Inst. Sci., vol. 86, no. 5, 2006. [Cached copy]
Eric Weisstein's World of Mathematics, Odd Graph
Eric Weisstein's World of Mathematics, Wiener Index

Cf. A192835, A301566.

A136328d := proc(k) add( (2*j+1)*binomial(k-1,j)^2/(1+j),j=0..(k/2-1) ); %+2*add( (k-1-j)*binomial(k-1,j)^2/(1+j),j=floor(k/2)..(k-2) ); k*% ; end proc:
A136328 := proc(n) binomial(2*n-1,n-1)*A136328d(n)/2 ; end proc: seq(A136328(n),n=1..20) ;
# R. J. Mathar, Sep 15 2010
B := proc (n) options operator, arrow: [seq(n-floor((1/2)*m), m = 1 .. n-1)] end proc: C := proc (n) options operator, arrow: [seq(ceil((1/2)*m), m = 1 .. n-1)] end proc: H := proc (n) options operator, arrow: (1/2)*binomial(2*n-1, n-1)*(sum((product(B(n)[r]/C(n)[r], r = 1 .. j))*t^j, j = 1 .. n-1)) end proc: Wi := proc (n) options operator, arrow: subs(t = 1, diff(H(n), t)) end proc: seq(Wi(n), n = 2 .. 20);
# Emeric Deutsch, Aug 20 2013

Table[Binomial[2 n - 1, n - 1]/2 (Sum[((2 j + 1) n)/(j + 1) Binomial[n - 1, j]^2, {j, 0, Floor[n/2] - 1}] + Sum[(2 (n - 1 - j) n)/(j + 1) Binomial[n - 1, j]^2, {j, Floor[n/2], n - 2}]), {n, 20}] (* Eric W. Weisstein, Sep 08 2017 *)
Table[Sum[k Binomial[n, Ceiling[k/2]] Binomial[n - 1, Floor[k/2]] Binomial[2 n - 1, n - 1]/2, {k, n - 1}], {n, 20}] (* Eric W. Weisstein, Mar 27 2018 *)

a(n) = sum(k=1, n-1, k*binomial(n, ceil(k/2))*binomial(n-1, k\2))*binomial(2*n-1,n-1)/2 \\ Andrew Howroyd, Mar 26 2018

a(n) = binomial(2*n - 1, n - 1)/2*(sum(((2*j + 1)*n)/(j + 1)*binomial(n - 1, j)^2, {j, 0, floor(n/2) - 1}) + sum((2*(n - 1 - j)*n)/(j + 1)*binomial(n - 1, j)^2, {j, floor(n/2), n - 2})). - Eric W. Weisstein, Sep 08 2017
A formula is "hidden" in the 2nd Maple program. B(n) and C(n) are the intersection arrays of O_n, H(n) is the Hosoya-Wiener polynomial of O_n, and Wi(n) is the Wiener index of O_n. - Emeric Deutsch, Aug 20 2013
a(n) = A301566(n)*binomial(2*n-1,n-1)/2. - Eric W. Weisstein, Mar 26 2018

Extended by R. J. Mathar, Sep 15 2010

Terms a(18) and beyond from Andrew Howroyd, Mar 26 2018

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User: K.V.Iyer

A191994 (Sum of first n Fibonacci numbers) times (product of first n Fibonacci numbers).

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A179023 a(n) = n(F(n+2) - 1) where F(n) is defined by A000045.

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A165910 Wiener indices of Fibonacci trees of order k.

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A158681 Wiener indexes of the complete binary trees with n levels, root being at level 0.

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A136328 a(n) = Wiener index of the odd graph O_n.

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