|
%I #50 Mar 09 2024 12:33:36
%S 1,0,1,2,2,4,7,10,17,28,44,72,117,188,305,494,798,1292,2091,3382,5473,
%T 8856,14328,23184,37513,60696,98209,158906,257114,416020,673135,
%U 1089154,1762289,2851444,4613732,7465176,12078909,19544084,31622993,51167078
%N A Fibonacci convolution.
%C Convolution of A000045 and A049347.
%C Diagonal sums of number triangle A116088. - _Paul Barry_, Feb 04 2006
%C Let (b(n)) be the p-INVERT of (1,1,0,0,0,0,0,0,...) using p(S) = 1 - S^2; then b(n) = a(n+1) for n >=0. See A292324. - _Clark Kimberling_, Sep 15 2017
%H G. C. Greubel, <a href="/A094686/b094686.txt">Table of n, a(n) for n = 0..1000</a>
%H Elena Barcucci, Antonio Bernini, Stefano Bilotta, and Renzo Pinzani, <a href="http://arxiv.org/abs/1601.07723">Non-overlapping matrices</a>, arXiv:1601.07723 [cs.DM], 2016 (see 1st column of Table 1 p. 8).
%H Stefano Bilotta, <a href="http://arxiv.org/abs/1605.03785">Variable-length Non-overlapping Codes</a>, arXiv preprint arXiv:1605.03785 [cs.IT], 2016 [See Table 2].
%H Joshua P. Bowman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Bowman/bowman4.html">Compositions with an Odd Number of Parts, and Other Congruences</a>, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 19.
%H David Broadhurst, <a href="http://arxiv.org/abs/1409.7204">Multiple Deligne values: a data mine with empirically tamed denominators</a>, arXiv:1409.7204 [hep-th], 2014 (see p. 10).
%H Leonard Rozendaal, <a href="https://hal.archives-ouvertes.fr/hal-01552281">Pisano word, tesselation, plane-filling fractal</a>, Preprint, 2017.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,2,1).
%F G.f.: 1/((1-x-x^2)*(1+x+x^2)).
%F a(n) = 2*sqrt(3)*Sum_{k=0..n} Fibonacci(k+1)*cos((4*(n-k)+1)*Pi/6)/3.
%F a(n) = a(n-2) + 2*a(n-3) + a(n-4).
%F From _Paul Barry_, Jan 13 2005
%F a(n) = A005252(n) - (-cos((2*n+1)*Pi/3)/2 - sqrt(3)*sin((2*n+1)*Pi/3)/6 + sqrt(3)*cos(Pi*n/3+Pi/6)/6 + sin((2*n+1)*Pi/6)/2).
%F a(n) = Sum_{k=0..floor(n/2)} if(mod(n-k, 2)=0, binomial(n-k, k), 0).
%F a(n) = A093040(n-1) - Fibonacci(n). (End)
%F a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(1+(-1)^(n-k))/2. - _Paul Barry_, Sep 09 2005
%F From _Paul Barry_, Feb 04 2006: (Start)
%F a(n) = Sum_{k=0..floor(n/2)} C(2*k, n-2*k).
%F a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*C(3*k,n-k)/C(3*k,k). (End)
%F 2*a(n) = A000045(n+1) + A049347(n). - _R. J. Mathar_, Feb 13 2020
%F a(n) = (1/2)*(A000045(n+1) + A049347(n)). - _G. C. Greubel_, Feb 09 2023
%t LinearRecurrence[{0,1,2,1}, {1,0,1,2}, 40] (* _Jean-François Alcover_, Sep 21 2017 *)
%o (PARI) Vec(1/((1-x-x^2)*(1+x+x^2)) + O(x^50)) \\ _Michel Marcus_, Sep 27 2014
%o (Magma) [(Fibonacci(n+1) +((n+2) mod 3) -1)/2: n in [0..40]]; // _G. C. Greubel_, Feb 09 2023
%o (SageMath) [(fibonacci(n+1) + (n+2)%3 - 1)/2 for n in range(41)] # _G. C. Greubel_, Feb 09 2023
%Y Cf. A000045, A005252, A049347, A093040, A116088, A292324.
%K easy,nonn
%O 0,4
%A _Paul Barry_, May 19 2004
|
