OFFSET
0,2
COMMENTS
Shares some properties with Fibonacci sequence.
The sum of any two alternating terms (terms separated by one other term) produces a Fibonacci number (e.g., 2+6=8, 3+10=13, 24+65=89). The product of any two consecutive or alternating Fibonacci terms produces a term from this sequence (e.g., 5*8 = 40, 13*5 = 65, 21*8 = 168).
In Penney's game (see A171861), the number of ways that HTH beats HHH on flip 3,4,5,... - Ed Pegg Jr, Dec 02 2010
The Ca2 sums (see A180662 for the definition of these sums) of triangle A035607 equal the terms of this sequence. - Johannes W. Meijer, Aug 05 2011
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Michael A. Allen and Kenneth Edwards, Identities relating permanents of some classes of (0, 1) Toeplitz matrices to generalized Fibonacci numbers, The Fibonacci Quarterly, Vol. 63, No. 2 (2025), pp. 163-177.
David Applegate, Marc LeBrun and N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq., Vol. 14 (2011), Article 11.9.8.
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol. 25 (2022), Article 22.4.8.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).
FORMULA
a(n) = F(floor(n/2)+1)*F(ceiling(n/2)+2), with F(n) = A000045(n). - Ralf Stephan, Apr 14 2004
G.f.: (1+x)/(1-x-x^3-x^4) = (1+x)/((1+x^2)*(1-x-x^2))
a(n) = A126116(n+4) - F(n+3). - Johannes W. Meijer, Aug 05 2011
a(n) = (1+3*i)/10*(-i)^n + (1-3*i)/10*(i)^n + (2+sqrt(5))/5*((1+sqrt(5))/2)^n + (2-sqrt(5))/5*((1-sqrt(5))/2)^n, where i = sqrt(-1). - Sergei N. Gladkovskii, Jul 16 2013
a(n+1)*a(n+3) = a(n)*a(n+2) + a(n+1)*a(n+2) for all n in Z. - Michael Somos, Jan 19 2014
Sum_{n>=1} 1/a(n) = A290565. - Amiram Eldar, Feb 17 2021
Sum_{n>=0} (-1)^n/a(n)^2 = (13 - 5*sqrt(5))/2 (A226765). - Amiram Eldar, Dec 27 2025
EXAMPLE
G.f.: 1 + 2*x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 15*x^6 + 24*x^7 + ...
MAPLE
with(combinat): A070550 := proc(n): fibonacci(floor(n/2)+1) * fibonacci(ceil(n/2)+2) end: seq(A070550(n), n=0..37); # Johannes W. Meijer, Aug 05 2011
MATHEMATICA
LinearRecurrence[{1, 0, 1, 1}, {1, 2, 2, 3}, 40] (* Jean-François Alcover, Jan 27 2018 *)
nxt[{a_, b_, c_, d_}]:={b, c, d, a+b+d}; NestList[nxt, {1, 2, 2, 3}, 40][[;; , 1]] (* Harvey P. Dale, Jul 16 2024 *)
PROG
(Haskell)
a070550 n = a070550_list !! n
a070550_list = 1 : 2 : 2 : 3 :
zipWith (+) a070550_list
(zipWith (+) (tail a070550_list) (drop 3 a070550_list))
-- Reinhard Zumkeller, Aug 06 2011
(PARI) A070550(n) = fibonacci(n\2+1)*fibonacci((n+5)\2) \\ M. F. Hasler, Aug 06 2011
(PARI) my(x='x+O('x^100)); Vec((1+x)/(1-x-x^3-x^4)) \\ Altug Alkan, Dec 24 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Sreyas Srinivasan (sreyas_srinivasan(AT)hotmail.com), May 02 2002
STATUS
approved