0,2

a(n-1)=sum(P(4;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=3. These are the sums over the SW-NE diagonals in P(4;n,k), the (4,1) Pascal triangle A093561. Observation by Paul Barry (pbarry(AT)wit.ie, Apr 29 2004. Proof via recursion relations and comparison of inputs. Also SW-NE diagonal sums in the Pascal (1,3) triangle A095660.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 53.

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 224.

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

A. Brousseau, Seeking the lost gold mine or exploring Fibonacci factorizations, Fib. Quart., 3 (1965), 129-130.

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

Index entries for sequences related to linear recurrences with constant coefficients

Tanya Khovanova, Recursive Sequences

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.

G.f.: (1+3*x)/(1-x-x^2).

Row sums of A131775 starting (1, 4, 5, 9, 14, 23,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 14 2007

a(n)=2*Fibonacci(n-2)+Fibonacci(n), n>=2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 05 2007

a(n)=((1+sqrt5)^n-(1-sqrt5)^n)/(2^n*sqrt5)+ 1.5*((1+sqrt5)^(n-1)-(1-sqrt5)^(n-1))/(2^(n-2)*sqrt5). Offset 1. a(3)=5. [From Al Hakanson (hawkuu(AT)gmail.com), Jan 14 2009]

BB := n->if n=1 then 3; > elif n=2 then 1; > else BB(n-2)+BB(n-1); > fi: > L:=[]: for k from 2 to 34 do L:=[op(L), BB(k)]: od: L; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 19 2007

with(combinat):a:=n->2*fibonacci(n-2)+fibonacci(n): seq(a(n), n=2..34); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 05 2007

A000285:=-(1+3*z)/(-1+z+z**2); [S. Plouffe in his 1992 dissertation.]

a=1; lst={a}; s=6; Do[a=s-(a+1); AppendTo[lst, a]; s+=a, {n, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 27 2009]

Essentially the same as A104449.

a(n) = A101220(3, 0, n+1).

a(n) = A109754(3, n+1).

a(k) = A090888(2, k-1), for k > 0.

Cf. A131775.

Sequence in context: A096818 A038099 A120740 this_sequence A042031 A041493 A042765

Adjacent sequences: A000282 A000283 A000284 this_sequence A000286 A000287 A000288

easy,nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).