Search: kyle maclean smith
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A164544
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a(n) = 2*a(n-1) + 7*a(n-2) for n > 1; a(0) = 1, a(1) = 7.
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+30
5
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1, 7, 21, 91, 329, 1295, 4893, 18851, 71953, 275863, 1055397, 4041835, 15471449, 59235743, 226771629, 868193459, 3323788321, 12724930855, 48716379957, 186507275899, 714029211497, 2733609354287, 10465423189053, 40066111858115
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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a(n) = 2*a(n-1) + 7*a(n-2) for n > 1; a(0) = 1, a(1) = 7.
a(n) = ((2+3*sqrt(2))*(1+2*sqrt(2))^n + (2-3*sqrt(2))*(1-2*sqrt(2))^n)/4.
G.f.: (1+5*x)/(1-2*x-7*x^2).
a(n)/a(n-1) ~ 1 + 2*sqrt(2). - _Kyle MacLean Smith_, Dec 15 2019
E.g.f.: exp(x)*cosh(2*sqrt(2)*x) + 3*exp(x)*sinh(2*sqrt(2)*x)/sqrt(2). - Stefano Spezia, Dec 16 2019
a(n) = (i*sqrt(7))^(n-1)*(i*sqrt(7)*ChebyshevU(n, -i/sqrt(7)) + 5*ChebyshevU(n-1, -i/sqrt(7))).
a(n) = Sum_{j=0..floor(n/2)} binomial(n-k, k)*((7*n -12*k)/(n-k))*7^k*2^(n-2*k-1). (End)
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MATHEMATICA
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LinearRecurrence[{2, 7}, {1, 7}, 40] (* Harvey P. Dale, Jul 15 2012 *)
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PROG
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(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((2+3*r)*(1+2*r)^n+(2-3*r)*(1-2*r)^n)/4: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 19 2009
(Sage) [(i*sqrt(7))^(n-1)*(i*sqrt(7)*chebyshev_U(n, -i/sqrt(7)) + 5*chebyshev_U(n-1, -i/sqrt(7))) for n in (0..40)] # G. C. Greubel, Jul 18 2021
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009
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EXTENSIONS
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STATUS
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approved
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A328604
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G.f.: (1 + 7*x) / (1 - 2*x - 9*x^2).
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+30
3
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1, 9, 27, 135, 513, 2241, 9099, 38367, 158625, 662553, 2752731, 11468439, 47711457, 198638865, 826680843, 3441111471, 14322350529, 59614704297, 248130563355, 1032793465383, 4298762000961, 17892665190369, 74474188389387, 309982363492095, 1290232422488673, 5370306116406201
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = 2*a(n-1) + 9*a(n-2) for n>1. - Colin Barker, Oct 21 2019
a(n)/a(n-1) ~ 1 + sqrt(10).
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PROG
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(PARI) Vec((1 + 7*x) / (1 - 2*x - 9*x^2) + O(x^30)) \\ Colin Barker, Dec 13 2019
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KEYWORD
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nonn,easy
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AUTHOR
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_Kyle MacLean Smith_, Oct 20 2019
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EXTENSIONS
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STATUS
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approved
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A328606
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Expansion of (1 + 9*x) / (1 - 2*x - 11*x^2).
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+30
3
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1, 11, 33, 187, 737, 3531, 15169, 69179, 305217, 1371403, 6100193, 27285819, 121673761, 543491531, 2425394433, 10829195707, 48337730177, 215796613131, 963308258209, 4300379260859, 19197149362017, 85698470593483, 382565584169153, 1707814344866619, 7623850115593921, 34033658024720651
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = 2*a(n-1) + 11*a(n-2) for n>1. - Colin Barker, Oct 21 2019
a(n)/a(n-1) ~ 1 + 2*sqrt(3).
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PROG
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(PARI) Vec((1 + 9*x) / (1 - 2*x - 11*x^2) + O(x^30)) \\ Colin Barker, Dec 13 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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_Kyle MacLean Smith_, Oct 20 2019
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STATUS
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approved
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A328605
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Expansion of (1 + 5*x - 2*x^2 - 15*x^3) / (1 - 12*x^2 + 25*x^4).
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+30
2
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1, 5, 10, 45, 95, 415, 890, 3855, 8305, 35885, 77410, 334245, 721295, 3113815, 6720290, 29009655, 62611105, 270270485, 583326010, 2518004445, 5434634495, 23459291215, 50632463690, 218561383455, 471723701905, 2036254321085, 4394872830610, 18971017266645, 40945381419695
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = 12*a(n-2) - 25*a(n-4) for n>3. - Colin Barker, Oct 21 2019
a(2*n)/a(2*n-1) ~ 2*a(2*n+1)/a(2*n) ~ 1 + sqrt(11).
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PROG
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(PARI) Vec((1 + 5*x - 2*x^2 - 15*x^3) / (1 - 12*x^2 + 25*x^4) + O(x^30)) \\ Colin Barker, Dec 13 2019
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CROSSREFS
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KEYWORD
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nonn,less,easy
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AUTHOR
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_Kyle MacLean Smith_, Oct 20 2019
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STATUS
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approved
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A297189
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Expansion of (x + 3*x^2 - 2*x^3 - 3*x^4)/(1 - 8*x^2 + 9*x^4).
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+30
1
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0, 1, 3, 6, 21, 39, 141, 258, 939, 1713, 6243, 11382, 41493, 75639, 275757, 502674, 1832619, 3340641, 12179139, 22201062, 80939541, 147542727, 537904077, 980532258, 3574776747, 6516373521, 23757077283, 43306197846, 157883627541, 287802221079
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OFFSET
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0,3
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COMMENTS
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Related to a tiling of the plane by heptagons.
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LINKS
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FORMULA
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a(2*n)/a(2*n-1) ~ 2*a(2*n+1)/a(2*n) ~ 1 + sqrt(7). - _Kyle MacLean Smith_, Oct 11 2019
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PROG
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(PARI) concat(0, Vec((x + 3*x^2 - 2*x^3 - 3*x^4)/(1 - 8*x^2 + 9*x^4) + O(x^40))) \\ Colin Barker, Jan 05 2018
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A330390
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G.f.: (1 + 15*x) / (1 - 2*x - 17*x^2).
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+30
1
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1, 17, 51, 391, 1649, 9945, 47923, 264911, 1344513, 7192513, 37241747, 196756215, 1026622129, 5398099913, 28248776019, 148265250559, 776759693441, 4074028646385, 21352972081267, 111964431151079, 586929387683697, 3077254104935737, 16132307800494323
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graph;
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = 2*a(n-1) + 17*a(n-2) for n>1.
a(n)/a(n-1) ~ 1 + 3*sqrt(2).
a(n) = ((1 - 3*sqrt(2))^n*(-16+3*sqrt(2)) + (1+3*sqrt(2))^n*(16 + 3*sqrt(2))) / (6*sqrt(2)). - Colin Barker, Dec 14 2019
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MATHEMATICA
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CoefficientList[Series[(1+15x)/(1-2x-17x^2), {x, 0, 30}], x] (* or *) LinearRecurrence[{2, 17}, {1, 17}, 30] (* Harvey P. Dale, Jul 31 2021 *)
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PROG
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(PARI) Vec((1 + 15*x) / (1 - 2*x - 17*x^2) + O(x^25)) \\ Colin Barker, Jan 25 2020
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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_Kyle MacLean Smith_, Dec 13 2019
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STATUS
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approved
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