A096979 Sum of the areas of the first n+1 Pell triangles.

0, 1, 6, 36, 210, 1225, 7140, 41616, 242556, 1413721, 8239770, 48024900, 279909630, 1631432881, 9508687656, 55420693056, 323015470680, 1882672131025, 10973017315470, 63955431761796, 372759573255306, 2172602007770041

Offset: 0

Author

Paul Barry, Jul 17 2004

Keywords

• easy
• nonn

Comments

Convolution of A059841(n) and A001109(n+1).

Partial sums of A084158.

Links

• S. Falcon, On the Sequences of Products of Two k-Fibonacci Numbers, American Review of Mathematics and Statistics, March 2014, Vol. 2, No. 1, pp. 111-120.
• Roger B. Nelson, Multi-Polygonal Numbers, Mathematics Magazine, Vol. 89, No. 3 (June 2016), pp. 159-164.
• Index entries for linear recurrences with constant coefficients, signature (6,0,-6,1).

Crossrefs

Cf. A096977, A064831, A096978.

Programs

Mathematica

Accumulate[LinearRecurrence[{5,5,-1},{0,1,5},30]] (* Harvey P. Dale, Sep 07 2011 *)
LinearRecurrence[{6, 0, -6, 1},{0, 1, 6, 36},22] (* Ray Chandler, Aug 03 2015 *)

Formula

G.f.: x/((1-x)*(1+x)*(1-6*x+x^2)).

a(n) = 6*a(n-1)-6*a(n-3)+a(n-4).

a(n) = (3-2*sqrt(2))^n*(3/32-sqrt(2)/16)+(3+2*sqrt(2))^n*(sqrt(2)/16+3/32)-(-1)^n/16-1/8.

a(n) = Sum_{k=0..n} (sqrt(2)*(sqrt(2)+1)^(2*k)/8-sqrt(2)*(sqrt(2)-1)^(2*k)/8)*(1+(-1)^(n-k))/2.

a(n) = Sum_{k=0..n} A000129(k)*A000129(k+1)/2. [corrected by Jason Yuen, Jan 14 2025]

a(n) = (A001333(n+1)^2 - 1)/8 = ((A000129(n) + A000129(n+1))^2 - 1)/8. - Richard R. Forberg, Aug 25 2013

a(n) = A002620(A000129(n+1)) = A000217(A048739(n-1)), n > 0. - Ivan N. Ianakiev, Jun 21 2014