A026384 a(n) = Sum_{j=0..i, i=0..n} T(i,j), where T is the array in A026374.

1, 3, 8, 18, 43, 93, 218, 468, 1093, 2343, 5468, 11718, 27343, 58593, 136718, 292968, 683593, 1464843, 3417968, 7324218, 17089843, 36621093, 85449218, 183105468, 427246093, 915527343, 2136230468, 4577636718, 10681152343, 22888183593, 53405761718

Offset

0,2

Comments

Partial sums of A026383. Number of lattice paths from (0,0) that do not go to right of the line x=n, using the steps U=(1,1), D=(1,-1) and, at levels ...,-4,-2,0,2,4,..., also H=(2,0). Example: a(2)=8 because we have the empty path, U, D, UU, UD, DD, DU and H. - Emeric Deutsch, Feb 18 2004

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,5,-5).

Formula

G.f.: (1+2*x) / ((1-x)*(1-5*x^2)). - Ralf Stephan, Apr 30 2004

From Colin Barker, Nov 25 2016: (Start)

a(n) = (7*5^(n/2) - 3)/4 for n even.

a(n) = 3*(5^((n+1)/2) - 1)/4 for n odd.

a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) for n>2.

(End)

Maple

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=5*a[n-2]+3 od: seq(a[n], n=1..29); # Zerinvary Lajos, Mar 17 2008

Mathematica

CoefficientList[Series[(1 + 2 x) / ((1 - x) (1 - 5 x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Aug 09 2017 *)
LinearRecurrence[{1, 5, -5}, {1, 3, 8}, 40] (* Harvey P. Dale, May 31 2023 *)

Prog

(PARI) Vec((2*x + 1)/(5*x^3 - 5*x^2 - x + 1) + O(x^40)) \\ Colin Barker, Nov 25 2016
(Magma) I:=[1, 3, 8]; [n le 3 select I[n] else Self(n-1)+5*Self(n-2)-5*Self(n-3): n in [1..35]]; // Vincenzo Librandi, Aug 09 2017

Crossrefs

Cf. A026383.

Sequence in context: A026756 A341583 A216631 * A346397 A322496 A066143

Adjacent sequences: A026381 A026382 A026383 * A026385 A026386 A026387

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved