A070550 a(n) = a(n-1) + a(n-3) + a(n-4), starting with a(0..3) = 1, 2, 2, 3.

1, 2, 2, 3, 6, 10, 15, 24, 40, 65, 104, 168, 273, 442, 714, 1155, 1870, 3026, 4895, 7920, 12816, 20737, 33552, 54288, 87841, 142130, 229970, 372099, 602070, 974170, 1576239, 2550408, 4126648, 6677057, 10803704, 17480760, 28284465, 45765226

Offset: 0

Author

Sreyas Srinivasan (sreyas_srinivasan(AT)hotmail.com), May 02 2002

Keywords

• easy
• nonn

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

Examples

			G.f.: 1 + 2*x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 15*x^6 + 24*x^7 + ...
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• 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).

Crossrefs

Bisections: A001654, A059929.

Cf. A049853, A290565.

Programs

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

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 *)

PARI

A070550(n) = fibonacci(n\2+1)*fibonacci((n+5)\2) \\ M. F. Hasler, Aug 06 2011

PARI

x='x+O('x^100); Vec((1+x)/(1-x-x^3-x^4)) \\ Altug Alkan, Dec 24 2015

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