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Search: kyle maclean smith
Displaying 1-6 of 6 results found. page 1
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A164544 a(n) = 2*a(n-1) + 7*a(n-2) for n > 1; a(0) = 1, a(1) = 7. +30
5
1, 7, 21, 91, 329, 1295, 4893, 18851, 71953, 275863, 1055397, 4041835, 15471449, 59235743, 226771629, 868193459, 3323788321, 12724930855, 48716379957, 186507275899, 714029211497, 2733609354287, 10465423189053, 40066111858115 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Binomial transform of A164640. Inverse binomial transform of A164545.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..178
Index entries for linear recurrences with constant coefficients, signature (2,7).
FORMULA
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
From G. C. Greubel, Jul 18 2021: (Start)
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)
MATHEMATICA
LinearRecurrence[{2, 7}, {1, 7}, 40] (* Harvey P. Dale, Jul 15 2012 *)
PROG
(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
CROSSREFS
Cf. A164545, A164640.
KEYWORD
nonn,easy
AUTHOR
Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009
EXTENSIONS
Edited and extended beyond a(5) by Klaus Brockhaus, Aug 19 2009
STATUS
approved
A328604 G.f.: (1 + 7*x) / (1 - 2*x - 9*x^2). +30
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,9).
FORMULA
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).
PROG
(PARI) Vec((1 + 7*x) / (1 - 2*x - 9*x^2) + O(x^30)) \\ Colin Barker, Dec 13 2019
KEYWORD
nonn,easy
AUTHOR
_Kyle MacLean Smith_, Oct 20 2019
EXTENSIONS
Edited by N. J. A. Sloane, Dec 05 2019
STATUS
approved
A328606 Expansion of (1 + 9*x) / (1 - 2*x - 11*x^2). +30
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,11).
FORMULA
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).
PROG
(PARI) Vec((1 + 9*x) / (1 - 2*x - 11*x^2) + O(x^30)) \\ Colin Barker, Dec 13 2019
CROSSREFS
Cf. A328604, A328605.
KEYWORD
nonn,easy
AUTHOR
_Kyle MacLean Smith_, Oct 20 2019
STATUS
approved
A328605 Expansion of (1 + 5*x - 2*x^2 - 15*x^3) / (1 - 12*x^2 + 25*x^4). +30
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,12,0,-25).
FORMULA
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).
PROG
(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
CROSSREFS
Cf. A328604, A328606.
KEYWORD
nonn,less,easy
AUTHOR
_Kyle MacLean Smith_, Oct 20 2019
STATUS
approved
A297189 Expansion of (x + 3*x^2 - 2*x^3 - 3*x^4)/(1 - 8*x^2 + 9*x^4). +30
1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Related to a tiling of the plane by heptagons.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. Vol. 70, No. 1, Feb. 1997, 22-31.
Index entries for linear recurrences with constant coefficients, signature (0,8,0,-9).
FORMULA
a(n) = 8*a(n-2) - 9*a(n-4). - Colin Barker, Jan 05 2018
a(2*n)/a(2*n-1) ~ 2*a(2*n+1)/a(2*n) ~ 1 + sqrt(7). - _Kyle MacLean Smith_, Oct 11 2019
PROG
(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
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jan 04 2018, following a suggestion from Roger L. Bagula
STATUS
approved
A330390 G.f.: (1 + 15*x) / (1 - 2*x - 17*x^2). +30
1
1, 17, 51, 391, 1649, 9945, 47923, 264911, 1344513, 7192513, 37241747, 196756215, 1026622129, 5398099913, 28248776019, 148265250559, 776759693441, 4074028646385, 21352972081267, 111964431151079, 586929387683697, 3077254104935737, 16132307800494323 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,17).
FORMULA
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
MATHEMATICA
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 *)
PROG
(PARI) Vec((1 + 15*x) / (1 - 2*x - 17*x^2) + O(x^25)) \\ Colin Barker, Jan 25 2020
CROSSREFS
Cf. A164544, A328606.
KEYWORD
nonn,easy
AUTHOR
_Kyle MacLean Smith_, Dec 13 2019
STATUS
approved
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Last modified May 15 17:39 EDT 2024. Contains 372548 sequences. (Running on oeis4.)