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A026383 a(n) = 5*a(n-2), starting 1,2. 10

%I

%S 1,2,5,10,25,50,125,250,625,1250,3125,6250,15625,31250,78125,156250,

%T 390625,781250,1953125,3906250,9765625,19531250,48828125,97656250,

%U 244140625,488281250,1220703125,2441406250,6103515625,12207031250

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

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

%C 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

%C From _Gary W. Adamson_, Aug 02 2010: (Start)

%C 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)

%C Also related to mixed Ramsey theory (see Chung & Graham reference). - _Benoit Cloitre_, Oct 22 2016

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

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,5)

%F 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.

%F From _Emeric Deutsch_, Jan 25 2004: (Start)

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

%F G.f. = (1+2z)/(1-5z^2). (End)

%F From - _Paul Barry_, Apr 16 2004: (Start)

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

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

%F From _Paul Barry_, Jul 14 2004: (Start)

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

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

%F a(n+3) = a(n+2)*a(n+1)/a(n). - _Reinhard Zumkeller_, Mar 04 2011

%F E.g.f.: 2*sinh(sqrt(5)*x)/sqrt(5) + cosh(sqrt(5)*x). - _Ilya Gutkovskiy_, Oct 24 2016

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

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

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

%Y Cf. A026374.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_

%E Better name from _Ralf Stephan_, Jul 17 2013

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Last modified September 24 19:27 EDT 2021. Contains 347651 sequences. (Running on oeis4.)