cp's OEIS Frontend

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.

%I A192750 #48 Sep 16 2024 12:00:55
%S A192750 1,6,11,21,36,61,101,166,271,441,716,1161,1881,3046,4931,7981,12916,
%T A192750 20901,33821,54726,88551,143281,231836,375121,606961,982086,1589051,
%U A192750 2571141,4160196,6731341,10891541,17622886,28514431,46137321,74651756
%N 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.
%C A192750 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.
%C A192750 For discussions of polynomial reduction, see A192232 and A192744.
%H A192750 Harvey P. Dale, <a href="/A192750/b192750.txt">Table of n, a(n) for n = 0..1000</a>
%H A192750 Feng-Zhen Zhao, <a href="https://math.colgate.edu/~integers/y82/y82.pdf">The log-behavior of some sequences related to the generalized Leonardo numbers</a>, Integers (2024) Vol. 24, Art. No. A82.
%H A192750 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2, 0, -1).
%F A192750 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
%F A192750 a(n) = 5*Fibonacci(n+2)-4. - _Gerry Martens_, Jul 04 2015
%F A192750 a(n) = A265752(A265750(n)). - _Antti Karttunen_, Dec 15 2015
%t A192750 q = x^2; s = x + 1; z = 40;
%t A192750 p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 4 n + 2;
%t A192750 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192750 reduce[{p1_, q_, s_, x_}] :=
%t A192750 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192750        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192750 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192750 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
%t A192750   (* A192750 *)
%t A192750 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t A192750   (* A192751 *)
%t A192750 LinearRecurrence[{2,0,-1},{1,6,11},40] (* _Harvey P. Dale_, Dec 03 2023 *)
%Y A192750 See A192751 for c_n.
%Y A192750 Cf. A000045, A192744, A192232, A022088, A265750, A265752.
%K A192750 nonn
%O A192750 0,2
%A A192750 _Clark Kimberling_, Jul 09 2011
%E A192750 Entry revised by _N. J. A. Sloane_, Dec 15 2015