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A193653 Q-residue of the Delannoy triangle A008288, where Q is the triangular array (t(i,j)) given by t(i,j)=1. 2
1, 2, 6, 20, 70, 248, 882, 3140, 11182, 39824, 141834, 505148, 1799110, 6407624, 22821090, 81278516, 289477726, 1030990208, 3671926074, 13077758636, 46577128054, 165886901432, 590814960402, 2104218684068, 7494285973006, 26691295287152, 95062457807466 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

For the definition of Q-residue, see A193649.

This sequence gives the number of closed walks from the two vertices having loops in the digraph defined by its adjacency matrix A=(2,1,1;1,2,1;1,1,0). - David Neil McGrath, Aug 22 2014

LINKS

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

Index entries for linear recurrences with constant coefficients, signature (4,-1,-2).

FORMULA

a(n)=4*a(n-1)-a(n-2)-2*a(n-3); a(n-1)=(1,1) and (2,2) elements of A^(n-1) where A=(2,1,1;1,2,1;1,1,0) and n>1. - David Neil McGrath, Aug 22 2014

G.f.: (1-2*t-t^2)/(1-4*t+t^2+2*t^3). - Robert Israel, Aug 22 2014

a(n) = (34+(17-3*sqrt(17))*((3-sqrt(17))/2)^n+((3+sqrt(17))/2)^n*(17+3*sqrt(17)))/68. - Colin Barker, Sep 02 2016

MATHEMATICA

q[n_, k_] := 1;

r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]

p[n_, 0] := 1; p[n_, n_] := 1;

p[n_, k_] := p[n - 1, k - 1] + p[n - 2, k - 1] +  p[n - 1, k]  (* A008288, Delannoy *)

v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]

Table[v[n], {n, 0, 16}]    (* A193653 *)

TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]

Table[r[k], {k, 0, 8}]  (* 2^k *)

TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]

PROG

(PARI) Vec((1-2*t-t^2)/(1-4*t+t^2+2*t^3) + O(t^40)) \\ Michel Marcus, Aug 23 2014

(PARI) a(n) = round((34+(17-3*sqrt(17))*((3-sqrt(17))/2)^n+((3+sqrt(17))/2)^n*(17+3*sqrt(17)))/68) \\ Colin Barker, Sep 02 2016

CROSSREFS

Cf. A193649, A008288.

Sequence in context: A275046 A229472 A135413 * A147748 A150125 A224514

Adjacent sequences:  A193650 A193651 A193652 * A193654 A193655 A193656

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Aug 02 2011

STATUS

approved

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Last modified May 9 11:58 EDT 2021. Contains 343740 sequences. (Running on oeis4.)