A125051 -id:A125051 - cp's OEIS frontend

A125052 Sum of labels for nodes in generation n of the sub-Fibonacci tree (A125051).

1, 2, 3, 9, 39, 252, 2361, 32077, 631058, 18035534, 751936149, 45973362492, 4144777181393, 554100538432001, 110435083963283354, 32981178674724868365

Offset: 0

Paul D. Hanna, Nov 19 2006

nonn

The sub-Fibonacci tree is a rooted tree in which every node with label k and parent node with label g has g child nodes that are assigned labels beginning with k+1 through k+g; the tree starts at generation n=0 with a root node labeled '1' and a child node labeled '2'. The number of nodes in generation n of the sub-Fibonacci tree is A005270(n+2); the maximum label in generation n is Fibonacci(n+2).

			The initial nodes of the sub-Fibonacci tree for generations 0..5 are:
gen.0: [1];
gen.1: [2];
gen.2: [3];
gen.3: [4,5];
gen.4: (4)->[5,6,7],(5)->[6,7,8];
gen.5: (5)->[6,7,8,9],(6)->[7,8,9,10],(7)->[8,9,10,11],
(6)->[7,8,9,10,11],(7)->[8,9,10,11,12],(8)->[9,10,11,12,13].
The sum of the labels for nodes in generation n+1 >= 2 is equal to:
a(n+1) = sum (parent label)*(label) over all nodes in generation n + sum (parent label)*[label*(label+1)/2] over all nodes in gen. n-1.
For example:
a(2) = 3 = 1*2 + 1*( 1*2/2 );
a(3) = 9 = 2*3 + 1*( 2*3/2 );
a(4) = 39 = 3*(4+5) + 2*( 3*4/2 );
a(5) = 252 = 4*(5+6+7) + 5*(6+7+8) + 3*( 4*5/2 + 5*6/2 );
a(6) = 2361 = 5*(6+7+8+9) + 6*(7+8+9+10) + 7*(8+9+10+11) +
6*(7+8+9+10+11) + 7*(8+9+10+11+12) + 8*(9+10+11+12+13) +
4*( 5*6/2 + 6*7/2 + 7*8/2 ) + 5*( 6*7/2 + 7*8/2 + 8*9/2 ).

Peter C. Fishburn and Fred S. Roberts, Elementary sequences, sub-Fibonacci sequences, Discrete Appl. Math. 44 (1993), no. 1-3, 261-281.

Cf. A125051, A005270.

a(10)-a(15) from Alois P. Heinz, May 03 2015

A005270 Number of sequences s of length n with s[1]=1, s[2]=1, s[j-1]=3.

Original entry on oeis.org

1, 1, 1, 2, 6, 27, 177, 1680, 23009, 455368, 13067353, 546378617, 33472296082, 3021920660821, 404374532614122, 80646410554881100, 24095492607316134304, 10837141045948365696938, 7369252748590790186483284, 7606603491185739308318700818

Offset: 2

N. J. A. Sloane

nonn

The sequences of length n that are counted here are sub-Fibonacci sequences (A005269) with the property that its members, except for the initial two terms, strictly increase. - Emeric Deutsch, Feb 15 2007

			G.f. = x^2 + x^3 + x^4 + 2*x^5 + 6*x^6 + 27*x^7 + 177*x^8 + 1680^x^9 + ...
a(2)=6 because we have (1,1,2,3,4,5), (1,1,2,3,4,6), (1,1,2,3,4,7), (1,1,2,3,5,6), (1,1,2,3,5,7) and (1,1,2,3,5,8).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 2..70
Peter C. Fishburn and Fred S. Roberts, Elementary sequences, sub-Fibonacci sequences, Discrete Appl. Math. 44 (1993), no. 1-3, 261-281.

Cf. A125051, A125052.

Cf. A005269.

g[0]:=1:for k from 0 to 20 do g[k+1]:=expand(sum(subs({x=y, y=z}, g[k]), z=y+1..x+y)) od:seq(subs({x=1, y=1}, g[k]), k=0..20); # Emeric Deutsch, Feb 15 2007

{a(n) = if(n<2, return(0)); my(c, e); forvec(s=vector(n, i, [1, fibonacci(i)]), e=0; for(k=3, n, if( s[k-1]>=s[k] || s[k]>s[k-2]+s[k-1], e=1; break)); if(e, next); c++, 1); c}; /* Michael Somos, Dec 02 2016 */

a(n) equals the number of nodes in generation n-2 of the sub-Fibonacci tree (A125051) for n>=2. - Paul D. Hanna, Nov 19 2006

See the Maple program; g[k](x, y) is the number of sequences s[1], s[2], ..., s[k+2] such that s[1]=x, s[2]=y, s[j-1] =3. - Emeric Deutsch, Feb 15 2007

a(12) from Paul D. Hanna, Nov 19 2006

Edited by Emeric Deutsch, Feb 15 2007

cp's OEIS Frontend

A125052 Sum of labels for nodes in generation n of the sub-Fibonacci tree (A125051).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A005270 Number of sequences s of length n with s[1]=1, s[2]=1, s[j-1]=3.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

PARI

Formula

Extensions