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 A192746 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments. 5
 1, 5, 9, 17, 29, 49, 81, 133, 217, 353, 573, 929, 1505, 2437, 3945, 6385, 10333, 16721, 27057, 43781, 70841, 114625, 185469, 300097, 485569, 785669, 1271241, 2056913, 3328157, 5385073, 8713233, 14098309, 22811545, 36909857, 59721405, 96631265 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+3n+1 for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (2,0,-1). FORMULA G.f.: (1+3*x-x^2)/((1-x)*(1-x-x^2)), so the first differences are (essentially) A022087. - R. J. Mathar, May 04 2014 a(n) = 4*Fibonacci(n+2)-3. - Gerry Martens, Jul 04 2015 MATHEMATICA (* First program *) q = x^2; s = x + 1; z = 40; p[0, n_]:= 1; p[n_, x_]:= x*p[n-1, x] +3n +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}] (* A192746 *) u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192747 *) (* Clark Kimberling, Jul 09 2011 *) (* Additional programs *) a[0]=1; a[1]=5; a[n_]:=a[n]=a[n-1]+a[n-2]+3; Table[a[n], {n, 0, 36}] (* Gerry Martens, Jul 04 2015 *) 4*Fibonacci[Range[0, 40]+2]-3 (* G. C. Greubel, Jul 24 2019 *) PROG (PARI) vector(30, n, n--; 4*fibonacci(n+2)-3) \\ G. C. Greubel, Jul 24 2019 (Magma) [4*Fibonacci(n+2)-3: n in [0..30]]; // G. C. Greubel, Jul 24 2019 (Sage) [4*fibonacci(n+2)-3 for n in (0..30)] # G. C. Greubel, Jul 24 2019 (GAP) List([0..30], n-> 4*Fibonacci(n+2)-3); # G. C. Greubel, Jul 24 2019 CROSSREFS Cf. A000045, A192232, A192744. Sequence in context: A200078 A190806 A294774 * A081295 A180565 A233187 Adjacent sequences: A192743 A192744 A192745 * A192747 A192748 A192749 KEYWORD nonn AUTHOR Clark Kimberling, Jul 09 2011 STATUS approved

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Last modified December 8 03:15 EST 2023. Contains 367662 sequences. (Running on oeis4.)