A192814 -id:A192814 - cp's OEIS frontend

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

Clark Kimberling, Jul 10 2011

For discussions of polynomial reduction, see A192232 and A192744.

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).

Cf. A192744, A192232, A192815, A192816.

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

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

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

(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

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

A192816 a(n) = A192815(n)/2.

Original entry on oeis.org

0, 1, 2, 7, 36, 173, 806, 3763, 17608, 82393, 385482, 1803487, 8437740, 39476613, 184694254, 864105611, 4042781584, 18914450865, 88492648850, 414019362743, 1937020023220, 9062490569821, 42399528318646, 198369310046307, 928085399264344

Offset: 0

Clark Kimberling, Jul 10 2011

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).

Cf. A192242, A192744, A192814, A192815.

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

(* See A192814. *)
LinearRecurrence[{5,-3,7}, {0,1,2}, 30] (* G. C. Greubel, Jan 03 2019 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1-3*x)/(1-5*x+3*x^2-7*x^3))) \\ G. C. Greubel, Jan 03 2019

(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

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

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

cp's OEIS Frontend

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

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