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A192750 Define a pair of sequences c_n, d_n by c_0=0, d_0=1 and thereafter c_n = c_{n-1}+d_{n-1}, d_n = c_{n-1}+4*n+2; sequence here is d_n. 5
1, 6, 11, 21, 36, 61, 101, 166, 271, 441, 716, 1161, 1881, 3046, 4931, 7981, 12916, 20901, 33821, 54726, 88551, 143281, 231836, 375121, 606961, 982086, 1589051, 2571141, 4160196, 6731341, 10891541, 17622886, 28514431, 46137321, 74651756 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Old definition was: constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined recursively by p(n,x) = x*p(n-1,x) + 4n+2 for n>0, with p(0,x)=1.

For discussions of polynomial reduction, see A192232 and A192744.

LINKS

Table of n, a(n) for n=0..34.

FORMULA

G.f.: ( 1+4*x-x^2 ) / ( (x-1)*(x^2+x-1) ). The first differences are in A022088. - R. J. Mathar, May 04 2014

a(n) = 5*Fibonacci(n+2)-4. - Gerry Martens, Jul 04 2015

a(n) = A265752(A265750(n)). - Antti Karttunen, Dec 15 2015

MATHEMATICA

q = x^2; s = x + 1; z = 40;

p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 4 n + 2;

Table[Expand[p[n, x]], {n, 0, 7}]

reduce[{p1_, q_, s_, x_}] :=

FixedPoint[(s PolynomialQuotient @@ #1 +

       PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]

t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];

u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]

  (* A192750 *)

u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]

  (* A192751 *)

CROSSREFS

See A192751 for c_n.

Cf. A000045, A192744, A192232, A022088, A265750, A265752.

Sequence in context: A000382 A208670 A208726 * A000383 A205540 A083575

Adjacent sequences:  A192747 A192748 A192749 * A192751 A192752 A192753

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 09 2011

EXTENSIONS

Entry revised by N. J. A. Sloane, Dec 15 2015

STATUS

approved

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Last modified December 13 19:20 EST 2017. Contains 295976 sequences.