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

Author

Clark Kimberling

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A026383.

Programs

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

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 *)

PARI

Vec((2*x + 1)/(5*x^3 - 5*x^2 - x + 1) + O(x^40)) \\ Colin Barker, Nov 25 2016

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)