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A026383 a(n) = 5a(n-2), starting 1,2. 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

REFERENCES

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

LINKS

Table of n, a(n) for n=0..29.

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

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.

a(2n)=5^n, a(2n+1)=2*5^n. G.f.=(1+2z)/(1-5z^2). - Emeric Deutsch, Jan 25 2004

Second inverse binomial transform of Fib(3n+3)/2. a(n)=5^(n/2)((1/2+1/sqrt(5))+(1/2-1/sqrt(5))(-1)^n) - Paul Barry, Apr 16 2004

a(n) = a(n-1)+2a(n-2)+5^floor((n-2)/2); a(n)= sum{k=0..floor(n/2), binomial(floor(n/2), k)2^(n-2k) }. - Paul Barry, Jul 14 2004

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

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 *)

PROG

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

CROSSREFS

Cf. A026374.

Sequence in context: A018262 A336542 A018356 * A162963 A297860 A002094

Adjacent sequences:  A026380 A026381 A026382 * A026384 A026385 A026386

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

EXTENSIONS

Better name from Ralf Stephan, Jul 17 2013

STATUS

approved

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Last modified August 9 22:52 EDT 2020. Contains 336335 sequences. (Running on oeis4.)