OFFSET
0,3
COMMENTS
For n > 0, also the number of maximum and maximal cliques in the n-Pell graph.
LINKS
E. Munarini, Pell Graphs, Disc. Math., 342 (2019), 2415-2428.
Eric Weisstein's World of Mathematics, Edge Count
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Maximum Clique
Eric Weisstein's World of Mathematics, Pell Graph
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).
FORMULA
a(n) = n*A000129(n+1)/2.
a(n) = 4*a(n-1) - 2*a(n-2) - 4*a(n-3) - a(n-4).
G.f.: x*(1+x)/(-1+2*x+x^2)^2.
From Peter Luschny, Jul 31 2023: (Start)
a(n) = (n/8)*((2 + sqrt(2))*(1 + sqrt(2))^n - (sqrt(2) - 2)*(1 - sqrt(2))^n).
With this formula, the sequence can be continued to the left half of the number line: a(-n) = -(-1)^n*A026937(n-2) for n >= 0.
a(n) = Sum_{k=0..n} k * A008288(n, k). (End)
MAPLE
A364553 := n -> (n/8)*((2 + sqrt(2))*(1 + sqrt(2))^n - (sqrt(2) - 2)*(1 - sqrt(2))^n): seq(simplify(A364553(n)), n=0..29); # Peter Luschny, Jul 30 2023
MATHEMATICA
Table[n Fibonacci[n + 1, 2]/2, {n, 0, 20}]
Table[n (Fibonacci[n, 2] + (-I)^n ChebyshevT[n, I])/2, {n, 0, 20}]
Table[With[{s = Sqrt[2]}, n ((s + 2) (1 + s)^n - (s - 2) (1 - s)^n)/8], {n, 0, 20}] // Expand
LinearRecurrence[{4, -2, -4, -1}, {0, 1, 5, 18}, 20]
CoefficientList[Series[x (1 + x)/(-1 + 2 x + x^2)^2, {x, 0, 20}], x]
PROG
(Python) # Using function 'delannoy_row' from A008288.
def A364553(n:int) -> int:
return sum(k * delannoy_row(n)[k] for k in range(n + 1))
print([A364553(n) for n in range(30)]) # Peter Luschny, Jul 30 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Jul 28 2023
STATUS
approved