A293560 -id:A293560 - cp's OEIS frontend

A293506 Decimal expansion of real root of x^5 - x^4 - x^2 - 1.

1, 5, 7, 0, 1, 4, 7, 3, 1, 2, 1, 9, 6, 0, 5, 4, 3, 6, 2, 9, 1, 0, 6, 6, 5, 4, 3, 5, 1, 3, 7, 1, 2, 6, 5, 5, 3, 8, 7, 3, 1, 3, 1, 6, 0, 7, 4, 2, 4, 5, 2, 7, 4, 3, 6, 9, 3, 1, 6, 5, 4, 8, 7, 7, 8, 9, 7, 3, 3, 0, 6, 6, 1, 5, 4, 4, 1, 6, 2, 3, 2, 0, 2, 2, 2, 7, 6

Offset: 1

Iain Fox, Oct 10 2017

This root is also the ninth smallest of the Pisot numbers.

The ratio of successive terms of A122115 converges to this number.

			1.570147312196054362910665...

Iain Fox, Table of n, a(n) for n = 1..20000
Hans Walser, Funktionenfolge [in German]
Eric Weisstein's World of Mathematics, Pisot Number.
Wikipedia, Pisot-Vijayaraghavan number
Index entries for algebraic numbers, degree 5

Cf. A060006, A086106, A228777, A092526, A293508, A293509, A293557, A122115, A293560 (inverse).

First@ RealDigits[Root[#^5 - #^4 - #^2 - 1 &, 1], 10, 87] (* Michael De Vlieger, Oct 23 2017 *)

solve(x=1, 2, x^5 - x^4 - x^2 - 1) \\ Michel Marcus, Oct 11 2017

default(realprecision, 20080); x=solve(x=1, 2, x^5 - x^4 - x^2 - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b293506.txt", n, " ", d));

More terms from Andrey Zabolotskiy, Oct 12 2017

A302118 Number of permutations p of [n] such that |p(i) - p(i-1)| is in {1,3} for all i from 2 to n.

Original entry on oeis.org

1, 1, 2, 2, 8, 12, 32, 40, 88, 118, 244, 338, 642, 912, 1650, 2402, 4182, 6200, 10492, 15786, 26166, 39814, 64994, 99738, 161020, 248670, 398248, 617912, 983890, 1531796, 2428988, 3790980, 5993746, 9371174, 14785512, 23146268, 36465816, 57137316, 89924384

Offset: 0

Alois P. Heinz, Apr 01 2018

			a(3) = 2: 123, 321.
a(4) = 8: 1234, 1432, 2143, 2341, 3214, 3412, 4123, 4321.
a(5) = 12: 12345, 12543, 14325, 14523, 32145, 32541, 34125, 34521, 52143, 52341, 54123, 54321.

Alois P. Heinz, Table of n, a(n) for n = 0..5100
Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1,-1,-3,1,1,3,1,1,0,-2,0,-1)

Cf. A003274, A174700, A293506, A293560, A302119, A307269, A328648.

G.f.: (x^16 -3*x^15 -2*x^14 +3*x^12 +6*x^11 +2*x^10 -6*x^9 -10*x^8 -6*x^7 +6*x^6 +4*x^5 +3*x^4 -x^3 -2*x^2+1) / ((x-1) *(x+1) *(x^5+x^3+x-1) *(x^4+x^2-1)^2).
a(n) = 2 * A302119(n) for n > 1.
Limit_{n->infinity} a(n)/a(n+1) = A293560 = 1/A293506 = 0.63688291680184484849...

A302119 Number of Hamiltonian paths in the graph on n vertices {1,...,n}, with i adjacent to j iff |i-j| in {1,3}.

Original entry on oeis.org

1, 1, 1, 1, 4, 6, 16, 20, 44, 59, 122, 169, 321, 456, 825, 1201, 2091, 3100, 5246, 7893, 13083, 19907, 32497, 49869, 80510, 124335, 199124, 308956, 491945, 765898, 1214494, 1895490, 2996873, 4685587, 7392756, 11573134, 18232908, 28568658, 44962192, 70494629

Offset: 0

Alois P. Heinz, Apr 01 2018

			a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 1: 123.
a(4) = 4: 1234, 1432, 2143, 3214.
a(5) = 6: 12345, 12543, 14325, 14523, 32145, 34125.
a(6) = 16: 123456, 123654, 125436, 125634, 143256, 143652, 145236, 145632, 214365, 214563, 321456, 341256, 365214, 412365, 521436, 541236.

Alois P. Heinz, Table of n, a(n) for n = 0..5102
Eric Weisstein's World of Mathematics, Hamiltonian path
Wikipedia, Hamiltonian path
Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1,-1,-3,1,1,3,1,1,0,-2,0,-1)

Cf. A069241, A293506, A293560, A302118.

G.f.: (x^16 -x^15 +x^13 +x^12 +2*x^11 -x^10 -5*x^9 -6*x^8 -2*x^7 +5*x^6 +3*x^5 +3*x^4 -x^3 -3*x^2+1) / ((x-1) *(x+1) *(x^5+x^3+x-1) *(x^4+x^2-1)^2).
a(n) = ceiling(A302118(n)/2).
limit_{n->infinity} a(n)/a(n+1) = A293560 = 1/A293506 = 0.63688291680184484849...

cp's OEIS Frontend

A293506 Decimal expansion of real root of x^5 - x^4 - x^2 - 1.

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A302118 Number of permutations p of [n] such that |p(i) - p(i-1)| is in {1,3} for all i from 2 to n.

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A302119 Number of Hamiltonian paths in the graph on n vertices {1,...,n}, with i adjacent to j iff |i-j| in {1,3}.

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