A094706 Convolution of Pell(n) and 2^n.
0, 1, 4, 13, 38, 105, 280, 729, 1866, 4717, 11812, 29365, 72590, 178641, 438064, 1071153, 2613138, 6362965, 15470140, 37565389, 91125206, 220864377, 534951112, 1294960905, 3133261530, 7578261181, 18323338324, 44292046693, 107041649438
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
- S. Falcon, On The Generating Functions of the Powers of the K-Fibonacci Numbers, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.
- Index entries for linear recurrences with constant coefficients, signature (4,-3,-2).
Programs
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Magma
I:=[0, 1, 4]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 24 2012
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Mathematica
LinearRecurrence[{4,-3, -2},{0,1,4},40] (* Vincenzo Librandi, Jun 24 2012 *) -
Sage
[lucas_number1(n+2, 2, -1) - 2^(n+1) for n in (0..30)] # G. C. Greubel, Sep 16 2021
Formula
G.f.: x/((1-2*x-x^2)*(1-2*x)).
a(n) = Sum_{k=0..n} ((1+sqrt(2))^n - (1-sqrt(2))^n)/(2*sqrt(2))*2^(n-k).
a(n) = (1 + 3*sqrt(2)/4)*(1 + sqrt(2))^n + (1 - 3*sqrt(2)/4)*(1-sqrt(2))^n - 2^(n+1).
a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k+1)*2^(n-2k-1);
a(n) = Sum_{k=0..n} binomial(k, n-k+1)*2^k*(1/2)^(n-k+1). - Paul Barry, Oct 07 2004
a(n) = A000129(n+2) - 2^(n+1). - R. J. Mathar, Jan 29 2012
a(n) = 2*a(n-1) + A000129(n), with a(0) = 0, a(1) = 1. - G. C. Greubel, Sep 20 2021