A083083
A diagonal of number array A083075.
Original entry on oeis.org
1, 7, 97, 1831, 43393, 1242375, 41818561, 1620979687, 71191804801, 3496805826823, 190053352702753, 11329044782441127, 735151931535979777, 51600331868857972231, 3896042468112362132353, 314921475287825567805799
Offset: 0
A192383
Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.
Original entry on oeis.org
1, 0, 6, 8, 60, 160, 744, 2496, 10064, 36480, 140512, 522624, 1983168, 7439360, 28091520, 105674752, 398391552, 1500057600, 5652182528, 21288560640, 80200784896, 302101094400, 1138045495296, 4286942363648, 16149041172480, 60833034895360
Offset: 1
The first five polynomials p(n,x) and their reductions are as follows:
p(0, x) = 1 -> 1
p(1, x) = 2*x -> 2*x
p(2, x) = 3 + x + 3*x^2 -> 6 + 4*x
p(3, x) = 12*x + 4*x^2 + 4*x^3 -> 8 + 24*x
p(4, x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 60 + 72*x.
From these, read A192383 = (1, 0, 6, 8, 60, ...) and A192384 = (0, 2, 4, 24, 72, ...).
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R:=PowerSeriesRing(Integers(), 41);
Coefficients(R!( x*(1-2*x-2*x^2)/(1-2*x-8*x^2+4*x^3+4*x^4) )) // G. C. Greubel, Jul 10 2023
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q[x_]:= x+1; d= Sqrt[x+3];
p[n_, x_]:= ((x+d)^n - (x-d)^n)/(2*d); (* suggested by A162517 *)
Table[Expand[p[n, x]], {n,6}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}];
Table[Coefficient[Part[t, n], x, 0], {n,30}] (* A192383 *)
Table[Coefficient[Part[t, n], x, 1], {n,30}] (* A192384 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n,30}] (* A192385 *)
LinearRecurrence[{2,8,-4,-4}, {1,0,6,8}, 40] (* G. C. Greubel, Jul 10 2023 *)
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@CachedFunction
def a(n): # a = A192383
if (n<5): return (0,1,0,6,8)[n]
else: return 2*a(n-1) +8*a(n-2) -4*a(n-3) -4*a(n-4)
[a(n) for n in range(1,41)] # G. C. Greubel, Jul 10 2023
Showing 1-2 of 2 results.
Comments