%I #33 Aug 01 2023 08:05:11

%S 0,1,5,18,58,175,507,1428,3940,10701,28705,76230,200766,525083,

%T 1365175,3531240,9093512,23325785,59625981,151947066,386139650,

%U 978834759,2475645491,6248406780,15740857452,39585199525,99389810585,249177006702,623846750086,1559888545075

%N Number of edges in the n-Pell graph.

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

%H E. Munarini, <a href="https://doi.org/10.1016/j.disc.2019.05.008">Pell Graphs</a>, Disc. Math., 342 (2019), 2415-2428.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EdgeCount.html">Edge Count</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximalClique.html">Maximal Clique</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximumClique.html">Maximum Clique</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PellGraph.html">Pell Graph</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-4,-1).

%F a(n) = n*(A000129(n) + A001333(n))/2.

%F a(n) = n*A000129(n+1)/2.

%F a(n) = 4*a(n-1) - 2*a(n-2) - 4*a(n-3) - a(n-4).

%F G.f.: x*(1+x)/(-1+2*x+x^2)^2.

%F From _Peter Luschny_, Jul 31 2023: (Start)

%F a(n) = (n/8)*((2 + sqrt(2))*(1 + sqrt(2))^n - (sqrt(2) - 2)*(1 - sqrt(2))^n).

%F 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.

%F a(n) = (A093967(n) + A364636(n)) / 2.

%F a(n) = Sum_{k=0..n} k * A008288(n, k). (End)

%p 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

%t Table[n Fibonacci[n + 1, 2]/2, {n, 0, 20}]

%t Table[n (Fibonacci[n, 2] + (-I)^n ChebyshevT[n, I])/2, {n, 0, 20}]

%t Table[With[{s = Sqrt[2]}, n ((s + 2) (1 + s)^n - (s - 2) (1 - s)^n)/8], {n, 0, 20}] // Expand

%t LinearRecurrence[{4, -2, -4, -1}, {0, 1, 5, 18}, 20]

%t CoefficientList[Series[x (1 + x)/(-1 + 2 x + x^2)^2, {x, 0, 20}], x]

%o (Python) # Using function 'delannoy_row' from A008288.

%o def A364553(n:int) -> int:

%o return sum(k * delannoy_row(n)[k] for k in range(n + 1))

%o print([A364553(n) for n in range(30)]) # _Peter Luschny_, Jul 30 2023

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

%K nonn,easy

%O 0,3

%A _Eric W. Weisstein_, Jul 28 2023