0,2

a(n) = T(n,0) + T(n,1) + ... + T(n,n), where T is the array in A026374.

Number of lattice paths from (0,0) to the line x=n using steps U=(1,1), D=(1,-1) and, at levels ...,-4,-2,0,2,4,..., also H=(2,0). Example: a(2)=5 because we have the following paths from (0,0) to the line x=2: UU, UD, H, DU and DD. - Emeric Deutsch, Jan 25 2004

From Gary W. Adamson, Aug 02 2010: (Start)

Equals eigensequence of a triangle with 1's in even columns starting with k=0 and (1, 2, 2, 2, ...) in odd columns. Example: a(5) = 50 = (1, 2, 1, 2, 1, 1) dot (1, 1, 2, 5, 10, 25) = (1 + 2 + 2 + 10 + 10 + 25) where (1, 2, 1, 2, 1, 1) = row 5 of the generating triangle. (End)

Also related to mixed Ramsey theory (see Chung & Graham reference). - Benoit Cloitre, Oct 22 2016

F. R. K. Chung and R. L. Graham, Edge-colored complete graphs with precisely colored subgraphs, Combinatorica, 3, (3-4) (1983), 315-324.

Index entries for linear recurrences with constant coefficients, signature (0,5).

Also number of integer strings s(0), ...s(n) such that s(0) = 0, where, for 1 <= i <= n, s(i) is even if i is even and |s(i) - s(i-1)| <= 1.

From Emeric Deutsch, Jan 25 2004: (Start)

a(2n) = 5^n, a(2n+1) = 2*5^n.

G.f.: (1+2z)/(1-5z^2). (End)

From Paul Barry, Apr 16 2004: (Start)

Second inverse binomial transform of Fibonacci(3n+3)/2.

a(n) = 5^(n/2)*((1/2 + 1/sqrt(5)) + (1/2 - 1/sqrt(5))*(-1)^n). (End)

From Paul Barry, Jul 14 2004: (Start)

a(n) = a(n-1) + 2*a(n-2) + 5^floor((n-2)/2);

a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n/2), k)*2^(n-2k). (End)

a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011

E.g.f.: 2*sinh(sqrt(5)*x)/sqrt(5) + cosh(sqrt(5)*x). - Ilya Gutkovskiy, Oct 24 2016

From Amiram Eldar, Dec 19 2025: (Start)

Sum_{n>=0} 1/a(n) = 15/8.

Sum_{n>=0} (-1)^n/a(n) = 5/8. (End)

Riffle @@ Transpose@ NestList[5 # &, #, 15] &@ {1, 2} (* or *)

CoefficientList[Series[(1 + 2 x)/(1 - 5 x^2), {x, 0, 31}], x] (* Michael De Vlieger, Oct 23 2016 *)

(PARI) a(n)=(1+n%2)*5^(n\2) \\ Charles R Greathouse IV, Jun 11 2015

Cf. A026374.

Sequence in context: A018262 A336542 A018356 * A162963 A297860 A002094

Adjacent sequences: A026380 A026381 A026382 * A026384 A026385 A026386

nonn,easy

Better name from Ralf Stephan, Jul 17 2013

approved