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A364553
Number of edges in the n-Pell graph.
4
0, 1, 5, 18, 58, 175, 507, 1428, 3940, 10701, 28705, 76230, 200766, 525083, 1365175, 3531240, 9093512, 23325785, 59625981, 151947066, 386139650, 978834759, 2475645491, 6248406780, 15740857452, 39585199525, 99389810585, 249177006702, 623846750086, 1559888545075
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
FORMULA
a(n) = n*(A000129(n) + A001333(n))/2.
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) = (A093967(n) + A364636(n)) / 2.
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