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%I #48 Dec 31 2023 10:15:31
%S 1,8,56,385,2640,18096,124033,850136,5826920,39938305,273741216,
%T 1876250208,12860010241,88143821480,604146740120,4140883359361,
%U 28382036775408,194533374068496,1333351581704065,9138927697859960
%N a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
%C a(n) such that 9*(T(a(n)-1) + T(a(n+1)-1)) = 7*(T(a(n)+a(n+1)-1)), where T(i) denotes the i-th triangular number.
%C Partial sums of Chebyshev sequence S(n,7) = U(n,7/2) = A004187(n+1). - _Wolfdieter Lang_, Aug 31 2004
%H Michael De Vlieger, <a href="/A092521/b092521.txt">Table of n, a(n) for n = 1..1197</a>
%H Francesca Arici and Jens Kaad, <a href="https://arxiv.org/abs/2012.11186">Gysin sequences and SU(2)-symmetries of C*-algebras</a>, arXiv:2012.11186 [math.OA], 2020.
%H C. Pita, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Pita/pita12.html">On s-Fibonomials</a>, J. Int. Seq. 14 (2011) # 11.3.7.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8,1).
%F G.f.: x/(1 - 8*x + 8*x^2 - x^3) = x/((1 - x)*(1 - 7*x + x^2)).
%F a(n) = 7*a(n-1) - a(n-2) + 1, n>=2, a(0):=0, a(1)=1.
%F a(n) = (S(n, 7)-S(n-1, 7) -1)/5, n>=1, with S(n, 7)=U(n, 7/2)= A004187(n+1).
%F a(n) = A058038(n)/3.
%F a(n) = (1/3)*Sum_{k=0..n} Fibonacci(4*k). - _Gary Detlefs_, Dec 07 2010
%t a[1] = 1; a[2] = 8; a[3] = 56; a[n_] := a[n] = 8 a[n - 1] - 8 a[n - 2] + a[n - 3]; Table[ a[n], {n, 20}] (* _Robert G. Wilson v_, Apr 08 2004 *)
%t Table[(LucasL[4n+2]-3)/15, {n, 1, 20}] (* _Vladimir Reshetnikov_, Oct 28 2015 *)
%t LinearRecurrence[{8,-8,1},{1,8,56},30] (* _Harvey P. Dale_, Dec 27 2015 *)
%o (PARI) Vec(x/((1-x)*(1-7*x+x^2)) + O(x^100)) \\ _Altug Alkan_, Oct 29 2015
%Y Cf. A212336 for more sequences with g.f. of the type 1/(1 - k*x + k*x^2 - x^3).
%K nonn,easy
%O 1,2
%A K. S. Bhanu (bhanu_105(AT)yahoo.com) and M. N. Deshpande, Apr 06 2004
%E Edited and extended by _Robert G. Wilson v_, Apr 08 2004
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