0,3

2-ranks of difference sets constructed from Segre hyperovals.

a(n)=2*Fibonacci(n)-1. - Richard L. Ollerton (r.ollerton(AT)uws.edu.au), Mar 22 2002

Sometimes called Leonardo numbers. - George Pollard (porges+oeis(AT)porg.es), Jan 02 2008

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

R. Evans, H. D. L. Hollmann, C. Krattenthaler, Q. Xiang, Gauss Sums, Jacobi Sums, and p-Ranks of Cyclic Difference Sets, J. Combin. Theory Ser. A 87 (1999), 74-119.

D. Singmaster, Some counterexamples and problems on linear recurrences, Fib. Quart. 8 (1970), 264-267.

Q. Xiang, On Balanced Binary Sequences with Two-Level Autocorrelation Functions, IEEE Trans. Inform. Theory 44 (1998), 3153-3156.

Dijkstra, E. W., 'Smoothsort, an alternative for sorting in situ', Science of Computer Programming, 1(3): 223-233, 1982.

Dijkstra, E. W., 'Fibonacci numbers and Leonardo numbers', circulated privately, July 1981.

T. D. Noe, Table of n, a(n) for n=0..500

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Supplement to "Gauss Sums, Jacobi Sums, and p-Ranks ..."

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1019

E. W. Dijkstra, Smoothsort, an alternative for sorting in situ (EWD796a).

E. W. Dijkstra, Fibonacci numbers and Leonardo numbers (EWD797).

G.f.: (1+x-x^2)/(1-2*x+x^3). a(n) = 2/sqrt(5)*((1+sqrt(5))/2)^(n+1) - 2/sqrt(5)*((1-sqrt(5))/2)^(n+1) - 1.

a(n+1)/a(n) is asymptotic to Phi = (1+sqrt(5))/2. - Jonathan Vos Post (jvospost2(AT)yahoo.com), May 26 2005

a(n) = Sum[A109754(n-k+1,k),{k,0,n+1}] - Sum[A109754(n-k,k),{k,0,n}] = Sum[A101220(n-k+1,0,k),{k,0,n+1}] - Sum[A101220(n-k,0,k),{k,0,n}]. - Ross La Haye (rlahaye(AT)new.rr.com), May 31 2006

a(n)=F(n)+F(n+3)-1 n>=-1 {where F(n) is the n-th Fibonacci number} - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 31 2008

L := 1, 3: for i from 3 to 100 do l := nops([ L ]): L := L, op(l, [ L ])+op(l-1, [ L ])+1: od: [ L ];

A001595:=(1-z+z**2)/(z-1)/(z**2+z-1); [Conjectured by S. Plouffe in his 1992 dissertation.]

with(combinat): seq(fibonacci(n)+fibonacci(n+3)-1, n=-1..32); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 31 2008

Join[ {1, 3}, Table[ a[ 1 ]=1; a[ 2 ]=3; a[ i ]=a[ i-1 ]+a[ i-2 ]+1, {i, 3, 100} ] ]

Cf. A049112, A049114.

Adjacent sequences: A001592 A001593 A001594 this_sequence A001596 A001597 A001598

Sequence in context: A053523 A053522 A053521 this_sequence A092369 A061969 A034084

nonn,easy,nice

njas

Additional comments from Christian Krattenthaler (kratt(AT)ap.univie.ac.at).