A364553 Number of edges in the n-Pell graph.

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

Author

Eric W. Weisstein, Jul 28 2023

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000129, A001333, A008288, A026937, A093967, A364636, row sums of A364361.

Programs

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]

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

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)