A227582 Expansion of (2+3*x+2*x^2+2*x^3+3*x^4+x^5-x^6)/(1-2*x+x^2-x^5+2*x^6-x^7).

2, 7, 14, 23, 35, 50, 67, 86, 107, 131, 158, 187, 218, 251, 287, 326, 367, 410, 455, 503, 554, 607, 662, 719, 779, 842, 907, 974, 1043, 1115, 1190, 1267, 1346, 1427, 1511, 1598, 1687, 1778, 1871, 1967, 2066, 2167, 2270, 2375, 2483, 2594, 2707, 2822, 2939

Offset

1,1

Comments

At A227581, it is conjectured that a(n) = floor[1/(2*H(n) + H(n^2 + n - 1) - g], where H denotes harmonic number and g denotes the Euler-Mascheroni constant.

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1)

Formula

a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7).

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

Mathematica

z = 60; a[1]=2; a[2]=7; a[3]=14; a[4]=23; a[5]=35; a[6]=50; a[7] = 67; a[8]=86; a[n_]:= a[n]= 2*a[n-1] -a[n-2] +a[n-5] -2*a[n-6] + a[n-7]; Table[a[n], {n, 1, z}]  (* A227582 *)
h[n_] := h[n] = HarmonicNumber[n]; t1 = N[Table[2 h[n] - h[n^2 + n - 1] - EulerGamma, {n, 1, z}]]; Floor[1/t1];  (* conjectured A227582 *)
CoefficientList[Series[(1+x)*(2+x+x^2+x^3+2*x^4-x^5)/((1-x)^3*(1+x+x^2+ x^3+x^4)), {x, 0, 50}], x]] (* G. C. Greubel, Aug 04 2018 *)

Prog

(PARI) my(x='x+O('x^50)); Vec((1+x)*(2+x+x^2+x^3+2*x^4-x^5)/((1-x)^3*(1+x + x^2+x^3+x^4))) \\ G. C. Greubel, Aug 04 2018
(Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1+x)*(2+x+x^2+x^3+2*x^4-x^5)/((1-x)^3*(1+x+x^2+x^3+x^4)) )); // G. C. Greubel, Aug 04 2018
(Sage) ((1+x)*(2+x+x^2+x^3+2*x^4-x^5)/((1-x)^3*(1+x+x^2+x^3+x^4)) ).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 06 2019

Crossrefs

Cf. A227581.

Sequence in context: A340664 A261246 A008865 * A249852 A324587 A018392

Adjacent sequences: A227579 A227580 A227581 * A227583 A227584 A227585

Keyword

nonn,easy

Author

Clark Kimberling, Jul 17 2013

Status

approved