A228469
a(n) = 6*a(n-2) + a(n-4), where a(0) = 2, a(1) = 8, a(2) = 13, a(3) = 49.
Original entry on oeis.org
2, 8, 13, 49, 80, 302, 493, 1861, 3038, 11468, 18721, 70669, 115364, 435482, 710905, 2683561, 4380794, 16536848, 26995669, 101904649, 166354808, 627964742, 1025124517, 3869693101, 6317101910, 23846123348, 38927735977, 146946433189, 239883517772, 905524722482
Offset: 0
a(3) = 13 because trace(18/13) = 010, and 13 is the least c for which there is a number b such that trace(b/c) = 010. Successive applications of w are indicated by (18,13)->(13,5)->(5,2)->(2,1). Whereas w finds GCD in 3 steps, u takes 4 steps, as indicated by (18,3)->(13,5)->(5,3)->(3,2)->(2,1).
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c1 = CoefficientList[Series[(2 + 8 x + x^2 + x^3)/(1 - 6 x^2 - x^4), {x, 0, 40}], x]; c2 = CoefficientList[Series[(3 + 11 x + 2 x^3)/(1 - 6 x^2 - x^4), {x, 0, 40}], x]; pairs = Transpose[CoefficientList[Series[{-((3 + 11 x + 2 x^3)/(-1 + 6 x^2 + x^4)), -((2 + 8 x + x^2 + x^3)/(-1 + 6 x^2 + x^4))}, {x, 0, 20}], x]]; t[{x_, y_, }] := t[{x, y}]; t[{x, y_}] := Prepend[If[# > y - #, {y - #, 1}, {#, 0}], y] &[Mod[x, y]]; userIn2[{x_, y_}] := Most[NestWhileList[t, {x, y}, (#[[2]] > 0) &]]; Map[Map[#[[3]] &, Rest[userIn2[#]]] &, pairs] (* Peter J. C. Moses, Aug 20 2013 *)
LinearRecurrence[{0, 6, 0, 1}, {2, 8, 13, 49}, 30] (* T. D. Noe, Aug 23 2013 *)
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Vec((x^3+x^2+8*x+2)/(1-6*x^2-x^4)+O(x^99)) \\ Charles R Greathouse IV, Jun 12 2015
A228471
a(n) = 6*a(n-2) + a(n-4), where a(0) = 3, a(1) = 5, a(2) = 19, a(3) = 31.
Original entry on oeis.org
3, 5, 19, 31, 117, 191, 721, 1177, 4443, 7253, 27379, 44695, 168717, 275423, 1039681, 1697233, 6406803, 10458821, 39480499, 64450159, 243289797, 397159775, 1499219281, 2447408809, 9238605483, 15081612629, 56930852179, 92937084583, 350823718557, 572704120127
Offset: 0
a(3) = 19 because trace(30/19) = 101, and 19 is the least c for which there is a number b such that trace(b/c) = 101. Successive applications of w are indicated by (30,19)->(19,11)->(11,3)->(3,1). Whereas w finds GCD in 3 steps, u takes 5 steps, as indicated by (30,19)->(19,11)->(11,8)->(8,3)->(3,2)->(2,1).
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c1 = CoefficientList[Series[(3 + 5 x + x^2 + x^3)/(1 - 6 x^2 - x^4), {x, 0, 40}], x]; c2 = CoefficientList[Series[(5 + 8 x + x^3)/(1 - 6 x^2 - x^4), {x, 0, 40}], x]; pairs = Transpose[CoefficientList[Series[{-((3 + 11 x + 2 x^3)/(-1 + 6 x^2 + x^4)), -((2 + 8 x + x^2 + x^3)/(-1 + 6 x^2 + x^4))}, {x, 0, 20}], x]]; t[{x_, y_, }] := t[{x, y}]; t[{x, y_}] := Prepend[If[# > y - #, {y - #, 1}, {#, 0}], y] &[Mod[x, y]]; userIn2[{x_, y_}] := Most[NestWhileList[t, {x, y}, (#[[2]] > 0) &]]; Map[Map[#[[3]] &, Rest[userIn2[#]]] &, pairs] (* Peter J. C. Moses, Aug 20 2013 *)
LinearRecurrence[{0, 6, 0, 1}, {3, 5, 19, 31}, 30] (* T. D. Noe, Aug 23 2013 *)
A228472
a(n) = 6*a(n-2) + a(n-4), where a(0) = 5, a(1) = 8, a(2) = 30, a(3) = 49.
Original entry on oeis.org
5, 8, 30, 49, 185, 302, 1140, 1861, 7025, 11468, 43290, 70669, 266765, 435482, 1643880, 2683561, 10130045, 16536848, 62424150, 101904649, 384674945, 627964742, 2370473820, 3869693101, 14607517865, 23846123348, 90015581010, 146946433189, 554701003925
Offset: 0
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c1 = CoefficientList[Series[(3 + 5 x + x^2 + x^3)/(1 - 6 x^2 - x^4), {x, 0, 40}], x]; c2 = CoefficientList[Series[(5 + 8 x + x^3)/(1 - 6 x^2 - x^4), {x, 0, 40}], x]; pairs = Transpose[CoefficientList[Series[{-((3 + 11 x + 2 x^3)/(-1 + 6 x^2 + x^4)), -((2 + 8 x + x^2 + x^3)/(-1 + 6 x^2 + x^4))}, {x, 0, 20}], x]]; t[{x_, y_, }] := t[{x, y}]; t[{x, y_}] := Prepend[If[# > y - #, {y - #, 1}, {#, 0}], y] &[Mod[x, y]]; userIn2[{x_, y_}] := Most[NestWhileList[t, {x, y}, (#[[2]] > 0) &]]; Map[Map[#[[3]] &, Rest[userIn2[#]]] &, pairs] (* Peter J. C. Moses, Aug 20 2013 *)
LinearRecurrence[{0, 6, 0, 1}, {5, 8, 30, 49}, 30] (* T. D. Noe, Aug 23 2013 *)
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