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

0,3

Comments

For n > 0, also the number of maximum and maximal cliques in the n-Pell graph.

Links

Table of n, a(n) for n=0..29.

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) + 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

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

Sequence in context: A247717 A128553 A190163 * A335551 A235612 A000340

Adjacent sequences: A364550 A364551 A364552 * A364554 A364555 A364556

Keyword

nonn,easy

Author

Eric W. Weisstein, Jul 28 2023

Status

approved