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

%I A295077 #55 Sep 08 2022 08:46:20
%S A295077 0,1,7,19,39,71,123,211,367,655,1203,2267,4359,8503,16747,33187,66015,
%T A295077 131615,262755,524971,1049335,2097991,4195227,8389619,16778319,
%U A295077 33555631,67110163,134219131,268436967,536872535,1073743563,2147485507
%N A295077 a(n) = 2*n*(n-1) + 2^n - 1.
%C A295077 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 A295077 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 A295077 Primes in the sequence are 7, 19, 71, 211, 367, 2267, 16747, 524971, ... which are of the form 4*k + 3 (A002145).
%C A295077 The second difference of this sequence is A140504.
%D A295077 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.
%H A295077 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 A295077 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 A295077 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 A295077 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,7,-2)
%F A295077 G.f.: (x + 2*x^2 - 7*x^3)/((1 - x)^3*(1 - 2*x)).
%F A295077 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 A295077 a(n) = 2*A131924(n-1) - 1 for n>0, a(0)=0.
%F A295077 a(n) = a(n-1) + A000079(n-1) + A008586(n-1) for n>0, a(0)=0.
%F A295077 a(n) = A126646(n-1) + A046092(n-1) for n>0, a(0)=0.
%F A295077 a(n+1) - 2*a(n) + a(n-1) = A140504(n-1) for n>0, a(0)=0.
%F A295077 E.g.f.: exp(2*x) - (1 - 2*x^2)*exp(x). - _G. C. Greubel_, Oct 17 2018
%e A295077 a(4) = 39. The corresponding words are aabb, aabd, aadb, aadd, abab, abad, abba, abda, adab, adad, adba, adda, aaac, aaca, aacc, acaa, acac, acca, accc, baab, baad, baba, bada, bbaa, bdaa, caaa, caac, caca, cacc, ccaa, ccac, ccca, cccc, daab, daad, daba, dada, dbaa, ddaa.
%p A295077 A295077:=n->2*n*(n-1)+2^n-1; seq(A295077(n), n=0..70);
%t A295077 Table[2 n (n - 1) + 2^n - 1, {n, 0, 70}]
%o A295077 (PARI) a(n) = 2*n*(n-1) + 2^n - 1; \\ _Michel Marcus_, Nov 14 2017
%o A295077 (Magma) [2*n*(n-1)+2^n-1 : n in [0..40]]; // _Wesley Ivan Hurt_, Nov 26 2017
%Y A295077 Cf. A000079, A002145, A008586, A046092, A126646, A131098, A131924, A140504, A288382.
%K A295077 nonn,easy
%O A295077 0,3
%A A295077 _Franck Maminirina Ramaharo_, Nov 13 2017