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

0, 1, 7, 19, 39, 71, 123, 211, 367, 655, 1203, 2267, 4359, 8503, 16747, 33187, 66015, 131615, 262755, 524971, 1049335, 2097991, 4195227, 8389619, 16778319, 33555631, 67110163, 134219131, 268436967, 536872535, 1073743563, 2147485507

Offset

0,3

Comments

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.

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.

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

The second difference of this sequence is A140504.

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..70 from Franck Maminirina Ramaharo)

Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.

Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.

Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2)

Formula

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

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).

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

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

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

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

E.g.f.: exp(2*x) - (1 - 2*x^2)*exp(x). - G. C. Greubel, Oct 17 2018

Example

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.

Maple

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

Mathematica

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

Prog

(PARI) a(n) = 2*n*(n-1) + 2^n - 1; \\ Michel Marcus, Nov 14 2017
(Magma) [2*n*(n-1)+2^n-1 : n in [0..40]]; // Wesley Ivan Hurt, Nov 26 2017

Crossrefs

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

Sequence in context: A078163 A108766 A303855 * A239359 A120720 A098422

Adjacent sequences: A295074 A295075 A295076 * A295078 A295079 A295080

Keyword

nonn,easy

Author

Franck Maminirina Ramaharo, Nov 13 2017

Status

approved