A192815 Coefficient of x in the reduction of the polynomial (2*x + 1)^n by x^3 -> x^2 + x + 1.

0, 2, 4, 14, 72, 346, 1612, 7526, 35216, 164786, 770964, 3606974, 16875480, 78953226, 369388508, 1728211222, 8085563168, 37828901730, 176985297700, 828038725486, 3874040046440, 18124981139642, 84799056637292, 396738620092614

Offset: 0

Author

Clark Kimberling, Jul 10 2011

Keywords

• nonn
• easy

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
• Index entries for linear recurrences with constant coefficients, signature (5,-3,7).

Crossrefs

Cf. A192744, A192232, A192815, A192816.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( 2*x*(1-3*x)/(1-5*x+3*x^2-7*x^3) )); // G. C. Greubel, Jan 03 2019

Mathematica

(See A192814.)
LinearRecurrence[{5,-3,7}, {0,2,4}, 30] (* G. C. Greubel, Jan 03 2019 *)

PARI

concat([0], Vec(2*x*(1-3*x)/(1-5*x+3*x^2-7*x^3)+O(x^30))) \\ Charles R Greathouse IV, Jul 11 2011

Sage

(2*x*(1-3*x)/(1-5*x+3*x^2-7*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 03 2019

Formula

a(n) = 5*a(n-1) - 3*a(n-2) + 7*a(n-3).

a(n) = 2*A192816(n).

G.f.: 2*x*(1-3*x)/(1-5*x+3*x^2-7*x^3). - Bruno Berselli, Jul 11 2011