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 A026383 a(n) = 5*a(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 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 REFERENCES F. R. K. Chung and R. L. Graham, Edge-colored complete graphs with precisely colored subgraphs, Combinatorica, 3, (3-4,) (1983), 315-324. LINKS 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. 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 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 EXTENSIONS Better name from Ralf Stephan, Jul 17 2013 STATUS approved

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Last modified July 26 12:20 EDT 2021. Contains 346294 sequences. (Running on oeis4.)