A096977 -id:A096977 - cp's OEIS frontend

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

Paul Barry, Jul 17 2004

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

Partial sums of A084158.

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).

Cf. A096977, A064831, A096978.

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 *)

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

A096978 Sum of the areas of the first n Jacobsthal rectangles.

Original entry on oeis.org

0, 1, 4, 19, 74, 305, 1208, 4863, 19398, 77709, 310612, 1242907, 4970722, 19884713, 79535216, 318148151, 1272578046, 5090341317, 20361307020, 81445344595, 325781145370, 1303125047521, 5212499258024, 20849998896239, 83399991856694

Offset: 0

Paul Barry, Jul 17 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (4,3,-14,8).

Cf. A096977, A064831.

[8*4^n/27-2*(-2)^n/27-(n+2)/9: n in [0..30]]; // Vincenzo Librandi, May 31 2011

LinearRecurrence[{4,3,-14,8},{0,1,4,19},30] (* or *) Table[(2^(2n+1)-3n - 3+(-2)^n)/27,{n,30}] (* Harvey P. Dale, Aug 08 2011 *)

G.f.: x/((1-x)^2*(1+2*x)*(1-4*x)).
a(n) = 8*4^n/27 - 2*(-2)^n/27 - (n+2)/9;
a(n) = Sum_{k=0..n} (2*4^k/3 + (-2)^k/3)*(n-k).
a(n) = 4*a(n-1) + 3*a(n-2) - 14*a(n-3) + 8*a(n-4).
a(n) = Sum_{k=0..n} A001045(k)*A001045(k+1).
a(n-1) = Sum_{k=0..n} (-1)^(k+1)*A001045(k)*A001045(2*(n-k)). - Paul Barry, Aug 11 2009

cp's OEIS Frontend

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

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