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 A295077 a(n) = 2*n*(n-1) + 2^n - 1. 2

%I

%S 0,1,7,19,39,71,123,211,367,655,1203,2267,4359,8503,16747,33187,66015,

%T 131615,262755,524971,1049335,2097991,4195227,8389619,16778319,

%U 33555631,67110163,134219131,268436967,536872535,1073743563,2147485507

%N a(n) = 2*n*(n-1) + 2^n - 1.

%C We have a(0) = 0, and for n > 0, a(n) is a subsequence of A131098 where the indices are given by the partial sums of A288382.

%C For n > 0, a(n) gives the number of words of length n over the alphabet A = {a,b,c,d} such that: a word containing 'c' does not contain 'b' or 'd'; a word cannot be fully written with 'a'; a word contains letters from {b,d} if and only if it contains exactly a unique couple of letters from {b,d}. Thus a(1) = 1 where the corresponding word is "c" since 'c' is the only letter allowed to be written alone.

%C Primes in the sequence are 7, 19, 71, 211, 367, 2267, 16747, 524971, ... which are of the form 4*k + 3 (A002145).

%C The second difference of this sequence is A140504.

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

%H G. C. Greubel, <a href="/A295077/b295077.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..70 from Franck Maminirina Ramaharo)

%H Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.

%H Franck Ramaharo, <a href="https://arxiv.org/abs/1805.10569">A generating polynomial for the pretzel knot</a>, arXiv:1805.10680 [math.CO], 2018.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,7,-2)

%F G.f.: (x + 2*x^2 - 7*x^3)/((1 - x)^3*(1 - 2*x)).

%F a(0)=0, a(1)=1, a(2)=7, a(3)=19; for n>3, a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4).

%F a(n) = 2*A131924(n-1) - 1 for n>0, a(0)=0.

%F a(n) = a(n-1) + A000079(n-1) + A008586(n-1) for n>0, a(0)=0.

%F a(n) = A126646(n-1) + A046092(n-1) for n>0, a(0)=0.

%F a(n+1) - 2*a(n) + a(n-1) = A140504(n-1) for n>0, a(0)=0.

%F E.g.f.: exp(2*x) - (1 - 2*x^2)*exp(x). - _G. C. Greubel_, Oct 17 2018

%p A295077:=n->2*n*(n-1)+2^n-1; seq(A295077(n), n=0..70);

%t Table[2 n (n - 1) + 2^n - 1, {n, 0, 70}]

%o (PARI) a(n) = 2*n*(n-1) + 2^n - 1; \\ _Michel Marcus_, Nov 14 2017

%o (MAGMA) [2*n*(n-1)+2^n-1 : n in [0..40]]; // _Wesley Ivan Hurt_, Nov 26 2017

%Y Cf. A000079, A002145, A008586, A046092, A126646, A131098, A131924, A140504, A288382.

%K nonn,easy

%O 0,3

%A _Franck Maminirina Ramaharo_, Nov 13 2017

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Last modified February 20 09:15 EST 2019. Contains 320325 sequences. (Running on oeis4.)