A026383 a(n) = 5*a(n-2), starting 1,2.

1, 2, 5, 10, 25, 50, 125, 250, 625, 1250, 3125, 6250, 15625, 31250, 78125, 156250, 390625, 781250, 1953125, 3906250, 9765625, 19531250, 48828125, 97656250, 244140625, 488281250, 1220703125, 2441406250, 6103515625, 12207031250

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

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

Links

• 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).

Crossrefs

Cf. A026374.

Programs

Mathematica

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

Formula

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

Extensions

Better name from Ralf Stephan, Jul 17 2013