A271833 Expansion of (1 + 2*x + 2*x^2 + 2*x^3 - 5*x^4 + 2*x^5 + 2*x^6 + 2*x^7)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7)).

1, 3, 5, 7, 2, 4, 6, 8, 9, 11, 13, 15, 10, 12, 14, 16, 17, 19, 21, 23, 18, 20, 22, 24, 25, 27, 29, 31, 26, 28, 30, 32, 33, 35, 37, 39, 34, 36, 38, 40, 41, 43, 45, 47, 42, 44, 46, 48, 49, 51, 53, 55, 50, 52, 54, 56, 57, 59, 61, 63, 58, 60, 62, 64, 65, 67, 69, 71, 66, 68, 70, 72, 73, 75, 77

Offset: 0

Author

Ilya Gutkovskiy, Apr 15 2016

Keywords

• nonn
• easy

Comments

4 consecutive odds, 4 consecutive evens.

Links

• Ilya Gutkovskiy, Illustration of initial terms.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).
• Index entries for sequences that are permutations of the natural numbers.

Crossrefs

Cf. A000027, A116966, A131793.

Programs

Mathematica

CoefficientList[Series[(1 + 2 x + 2 x^2 + 2 x^3 - 5 x^4 + 2 x^5 + 2 x^6 + 2 x^7)/((1 - x)^2 (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7)), {x, 0, 75}], x]
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 1, -1}, {1, 3, 5, 7, 2, 4, 6, 8, 9}, 75]

PARI

my(x='x+O('x^99)); Vec((1+2*x+2*x^2+2*x^3-5*x^4+2*x^5+2*x^6+2*x^7)/((1-x)^2*(1+x+x^2+x^3+x^4+x^5+x^6+x^7))) \\ Altug Alkan, Apr 15 2016

Formula

G.f.: (1 + 2*x + 2*x^2 + 2*x^3 - 5*x^4 + 2*x^5 + 2*x^6 + 2*x^7)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7)).

a(n) = a(n-1) + a(n-8) - a(n-9).

a(n) = 1 + 2*n + 6*floor(n/8) - 7*floor(n/4). - Vaclav Kotesovec, Apr 15 2016

Sum_{n>=0} (-1)^n/a(n) = Pi/4 + log(2)/2. - Amiram Eldar, Feb 09 2023