A192908 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.
1, 1, 3, 7, 17, 43, 111, 289, 755, 1975, 5169, 13531, 35423, 92737, 242787, 635623, 1664081, 4356619, 11405775, 29860705, 78176339, 204668311, 535828593, 1402817467, 3672623807, 9615053953, 25172538051, 65902560199
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (4,-4,1).
Crossrefs
Programs
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GAP
Concatenation([1], List([1..30], n -> 1+2*Fibonacci(2*(n-1)))); # G. C. Greubel, Jan 11 2019
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Magma
[1] cat [1+2*Fibonacci(2*(n-1)): n in [1..30]]; // G. C. Greubel, Jan 11 2019
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Mathematica
u = 1; v = 1; a = 1; b = 1; c = 1; d = 1; e = 0; f = 1; q = x^2; s = u*x + v; z = 26; p[0, x_] := a; p[1, x_] := b*x + c p[n_, x_] := d*(x^2)*p[n - 1, x] + e*x*p[n - 2, x] + f; Table[Expand[p[n, x]], {n, 0, 8}] 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}]; u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192908 *) u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A069403 *) Simplify[FindLinearRecurrence[u0]] (* recurrence for 0-sequence *) Simplify[FindLinearRecurrence[u1]] (* recurrence for 1-sequence *) LinearRecurrence[{4,-4,1}, {1,1,3,7}, 30] (* G. C. Greubel, Jan 11 2019 *) -
PARI
vector(30, n, n--; if(n==0,1,1+2*fibonacci(2*n-2))) \\ G. C. Greubel, Jan 11 2019
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Sage
[1]+[1+2*fibonacci(2*(n-1)) for n in (1..30)] # G. C. Greubel, Jan 11 2019
Formula
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3) for n>3.
G.f.: 1 + x*(1 - x - x^2)/((1 - x)*(1 - 3*x + x^2)). - R. J. Mathar, Jul 13 2011
a(n) = 2*Fibonacci(2*n-2) + 1 for n>0, a(0)=1. - Bruno Berselli, Dec 27 2016
a(n) = -1 + 3*a(n-1) - a(n-2) with a(1) = 1 and a(2) = 3. Cf. A055588 and A097136. - Peter Bala, Nov 12 2017
Comments