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Search: a127212 -id:a127212
Displaying 1-9 of 9 results found.
page 1
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A127211
a(n) = 4^n*Lucas(n), where Lucas = A000032.
+10
11
2, 4, 48, 256, 1792, 11264, 73728, 475136, 3080192, 19922944, 128974848, 834666496, 5402263552, 34963718144, 226291089408, 1464583847936, 9478992822272, 61349312856064, 397061136580608, 2569833552019456, 16632312393367552, 107646586405781504, 696703343917006848
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1225
Index entries for linear recurrences with constant coefficients, signature (4,16).
FORMULA
a(n) = Trace of matrix [({4,4},{4,0})^n].
a(n) = 4^n * Trace of matrix [({1,1},{1,0})^n].
From Colin Barker, Sep 02 2013: (Start)
a(n) = 4*a(n-1) + 16*a(n-2).
G.f.: 2*x*(2*x-1)/(16*x^2+4*x-1). (End)
From Peter Luschny, Apr 15 2024: (Start)
a(n) = 2^n*((1 - sqrt(5))^n + (1 + sqrt(5))^n).
a(n) = 4^n*(Fibonacci(n+1) + Fibonacci(n-1)). (End)
a(n) = 2^n*A087131(n). - Michel Marcus, Apr 15 2024
MAPLE
a:= n-> 4^n*(<<1|1>, <1|0>>^n. <<2, -1>>)[1, 1]:
seq(a(n), n=0..22); # Alois P. Heinz, Apr 15 2024
MATHEMATICA
Table[4^n Tr[MatrixPower[{{1, 1}, {1, 0}}, n]], {n, 0, 20}]
Table[4^n*LucasL[n], {n, 0, 50}] (* G. C. Greubel, Dec 18 2017 *)
PROG
(PARI) my(x='x + O('x^30)); Vec(-4*x*(8*x+1)/(16*x^2+4*x-1)) \\ G. C. Greubel, Dec 18 2017
(Magma) [4^n*Lucas(n): n in [0..30]]; // G. C. Greubel, Dec 18 2017
CROSSREFS
Cf. A000032, A000204, A087131, A087131, A127210, A127212, A127213, A127214, A127215, A127216.
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Jan 09 2007
EXTENSIONS
a(0)=2 prepended by Alois P. Heinz, Apr 15 2024
STATUS
approved
A127210
a(n) = 3^n*Lucas(n), where Lucas = A000204.
+10
10
3, 27, 108, 567, 2673, 13122, 63423, 308367, 1495908, 7263027, 35252253, 171124002, 830642283, 4032042867, 19571909148, 95004113247, 461159522073, 2238515585442, 10865982454983, 52744587633927, 256027604996628, 1242784103695227, 6032600756055333, 29282859201423042
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1445
Ivica Martinjak, Two Extensions of the Sury's Identity, arXiv:1508.01444 [math.CO], 2015.
Index entries for linear recurrences with constant coefficients, signature (3, 9).
FORMULA
a(n) = Trace of matrix [({3,3},{3,0})^n] = 3^n * Trace of matrix [({1,1},{1,0})^n].
From R. J. Mathar, Oct 27 2008: (Start)
a(n) = 3*a(n-1) + 9*a(n-2).
G.f.: 3*x*(1 + 6*x)/(1 - 3*x - 9*x^2).
a(n) = 3*A099012(n) +18*A099012(n-1). (End)
MATHEMATICA
Table[3^n Tr[MatrixPower[{{1, 1}, {1, 0}}, x]], {x, 1, 20}]
Table[3^n LucasL[n], {n, 25}] (* Vincenzo Librandi, Aug 07 2015 *)
PROG
(PARI) lucas(n) = fibonacci(n-1) + fibonacci(n+1);
vector(30, n, 3^n*lucas(n)) \\ Michel Marcus, Aug 07 2015
(Magma) [3^n*Lucas(n): n in [1..30]]; // Vincenzo Librandi, Aug 07 2015
CROSSREFS
Cf. A000204, A087131, A127211, A127212, A127213, A127214, A127215, A127216.
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Jan 09 2007
EXTENSIONS
More terms from Michel Marcus, Aug 07 2015
STATUS
approved
A127213
a(n) = 6^n*Lucas(n), where Lucas = A000204.
+10
10
6, 108, 864, 9072, 85536, 839808, 8118144, 78941952, 765904896, 7437339648, 72196614144, 700923912192, 6804621582336, 66060990332928, 641332318961664, 6226189565755392, 60445100877152256, 586813429630107648
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,36).
FORMULA
a(n) = Trace of matrix [({6,6},{6,0})^n].
a(n) = 6^n * Trace of matrix [({1,1},{1,0})^n].
From Colin Barker, Sep 02 2013: (Start)
a(n) = 6*a(n-1) + 36*a(n-2).
G.f.: -6*x*(12*x+1)/(36*x^2+6*x-1). (End)
MATHEMATICA
Table[6^n Tr[MatrixPower[{{1, 1}, {1, 0}}, x]], {x, 1, 20}]
Table[6^n*LucasL[n], {n, 1, 50}] (* G. C. Greubel, Dec 18 2017 *)
LinearRecurrence[{6, 36}, {6, 108}, 20] (* Harvey P. Dale, Jan 20 2024 *)
PROG
(PARI) x='x+O('x^30); Vec(-6*x*(12*x+1)/(36*x^2+6*x-1)) \\ G. C. Greubel, Dec 18 2017
(Magma) [6^n*Lucas(n): n in [1..30]]; // G. C. Greubel, Dec 18 2017
CROSSREFS
Cf. A000204, A087131, A127210, A127211, A127212, A127214, A127215, A127216.
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Jan 09 2007
STATUS
approved
A127214
a(n) = 2^n*tribonacci(n) or (2^n)*A001644(n+1).
+10
10
2, 12, 56, 176, 672, 2496, 9088, 33536, 123392, 453632, 1669120, 6139904, 22585344, 83083264, 305627136, 1124270080, 4135714816, 15213527040, 55964073984, 205867974656, 757300461568, 2785785413632, 10247716470784, 37696978288640, 138671105769472
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,4,8).
FORMULA
a(n) = Trace of matrix [({2,2,2},{2,0,0},{0,2,0})^n].
a(n) = 2^n * Trace of matrix [({1,1,1},{1,0,0},{0,1,0})^n].
From Colin Barker, Sep 02 2013: (Start)
a(n) = 2*a(n-1) + 4*a(n-2) + 8*a(n-3).
G.f.: -2*x*(12*x^2+4*x+1)/(8*x^3+4*x^2+2*x-1). (End)
MATHEMATICA
Table[Tr[MatrixPower[2*{{1, 1, 1}, {1, 0, 0}, {0, 1, 0}}, x]], {x, 1, 20}]
LinearRecurrence[{2, 4, 8}, {2, 12, 56}, 50] (* G. C. Greubel, Dec 18 2017 *)
PROG
(PARI) x='x+O('x^30); Vec(-2*x*(12*x^2+4*x+1)/(8*x^3+4*x^2+2*x-1)) \\ G. C. Greubel, Dec 18 2017
(Magma) I:=[2, 12, 56]; [n le 3 select I[n] else 2*Self(n-1) + 4*Self(n-2) + 8*Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 18 2017
CROSSREFS
Cf. A087131, A127210, A127211, A127212, A127213, A127215, A127216.
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Jan 09 2007
EXTENSIONS
More terms from Colin Barker, Sep 02 2013
STATUS
approved
A127216
a(n) = 2^n*tetranacci(n) or (2^n)*A001648(n).
+10
10
2, 12, 56, 240, 832, 3264, 12672, 48896, 187904, 724992, 2795520, 10776576, 41541632, 160153600, 617414656, 2380201984, 9175957504, 35374497792, 136373075968, 525735034880, 2026773676032, 7813464064000, 30121872326656, 116123550875648, 447670682386432
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Indranil Ghosh, Table of n, a(n) for n = 1..1702
Index entries for linear recurrences with constant coefficients, signature (2,4,8,16).
FORMULA
a(n) = Trace of matrix [({2,2,2,2},{2,0,0,0},{0,2,0,0},{0,0,2,0})^n].
a(n) = 2^n * Trace of matrix [({1,1,1,1},{1,0,0,0},{0,1,0,0},{0,0,1,0})^n].
From Colin Barker, Sep 02 2013: (Start)
a(n) = 2*a(n-1) + 4*a(n-2) + 8*a(n-3) + 16*a(n-4).
G.f.: -2*x*(32*x^3+12*x^2+4*x+1) / (16*x^4+8*x^3+4*x^2+2*x-1). (End)
EXAMPLE
a(8) = (2^8) * A001648(8) = 256 * 191 = 48896. - Indranil Ghosh, Feb 09 2017
MATHEMATICA
Table[Tr[MatrixPower[2*{{1, 1, 1, 1}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}}, x]], {x, 1, 20}]
LinearRecurrence[{2, 4, 8, 16}, {2, 12, 56, 240}, 50] (* G. C. Greubel, Dec 19 2017 *)
PROG
(PARI) x='x+O('x^30); Vec(-2*x*(32*x^3+12*x^2+4*x+1)/(16*x^4 +8*x^3 +4*x^2 +2*x -1)) \\ G. C. Greubel, Dec 19 2017
(Magma) I:=[2, 12, 56, 240]; [n le 4 select I[n] else 2*Self(n-1) + 4*Self(n-2) + 8*Self(n-3) + 16*Self(n-4): n in [1..30]]; // G. C. Greubel, Dec 19 2017
CROSSREFS
Cf. A087131, A127210, A127211, A127212, A127213, A127214, A127216, A001648.
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Jan 09 2007
EXTENSIONS
More terms from Colin Barker, Sep 02 2013
STATUS
approved
A127215
a(n) = 3^n*tribonacci(n) or (3^n)*A001644(n+1).
+10
6
3, 27, 189, 891, 5103, 28431, 155277, 859491, 4743603, 26158707, 144374805, 796630059, 4395548511, 24254435799, 133832255589, 738466498755, 4074759563139, 22483948079115, 124063275771981, 684563868232731, 3777327684782127, 20842766314284447
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,9,27).
FORMULA
a(n) = Trace of matrix [({3,3,3},{3,0,0},{0,3,0})^n].
a(n) = 3^n * Trace of matrix [({1,1,1},{1,0,0},{0,1,0})^n].
From Colin Barker, Sep 02 2013: (Start)
a(n) = 3*a(n-1) + 9*a(n-2) + 27*a(n-3).
G.f.: -3*x*(27*x^2+6*x+1)/(27*x^3+9*x^2+3*x-1). (End)
MATHEMATICA
Table[Tr[MatrixPower[3*{{1, 1, 1}, {1, 0, 0}, {0, 1, 0}}, x]], {x, 1, 20}]
LinearRecurrence[{3, 9, 27}, {3, 27, 189}, 50] (* G. C. Greubel, Dec 18 2017 *)
PROG
(PARI) x='x+O('x^30); Vec(-3*x*(27*x^2+6*x+1)/(27*x^3+9*x^2+3*x-1)) \\ G. C. Greubel, Dec 18 2017
(Magma) I:=[3, 27, 189]; [n le 3 select I[n] else 3*Self(n-1) + 9*Self(n-2) + 27*Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 18 2017
CROSSREFS
Cf. A087131, A127210, A127211, A127212, A127213, A127214, A127216.
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Jan 09 2007
EXTENSIONS
More terms from Colin Barker, Sep 02 2013
STATUS
approved
A127220
a(n) = 3^n*tetranacci(n) or (2^n)*A001648(n).
+10
3
3, 27, 189, 1215, 6318, 37179, 216513, 1253151, 7223661, 41806692, 241805655, 1398221271, 8084811933, 46753521975, 270362105694, 1563413859999, 9040715391141, 52279683047127, 302316992442837, 1748203962973380, 10109314209860523, 58458991419115875
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,9,27,81).
FORMULA
a(n) = Trace of matrix [({3,3,3,3},{3,0,0,0},{0,3,0,0},{0,0,3,0})^n].
a(n) = 3^n * Trace of matrix [({1,1,1,1},{1,0,0,0},{0,1,0,0},{0,0,1,0})^n].
From Colin Barker, Sep 02 2013: (Start)
a(n) = 3*a(n-1) + 9*a(n-2) + 27*a(n-3) + 81*a(n-4).
G.f.: -3*x*(108*x^3+27*x^2+6*x+1)/(81*x^4+27*x^3+9*x^2+3*x-1). (End)
MATHEMATICA
Table[Tr[MatrixPower[3*{{1, 1, 1, 1}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}}, x]], {x, 1, 20}]
LinearRecurrence[{3, 9, 27, 81}, {3, 27, 189, 1215}, 50] (* G. C. Greubel, Dec 19 2017 *)
PROG
(PARI) x='x+O('x^30); Vec(-3*x*(108*x^3 +27*x^2 +6*x +1)/(81*x^4 +27*x^3 +9*x^2 +3*x -1)) \\ G. C. Greubel, Dec 19 2017
(Magma) I:=[3, 27, 189, 1215]; [n le 4 select I[n] else 3*Self(n-1) + 9*Self(n-2) + 27*Self(n-3) + 81*Self(n-4): n in [1..30]]; // G. C. Greubel, Dec 19 2017
CROSSREFS
Cf. A087131, A127210, A127211, A127212, A127213, A127214, A127216, A001648, A127221, A127222.
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Jan 09 2007
EXTENSIONS
More terms from Colin Barker, Sep 02 2013
STATUS
approved
A127221
a(n) = 2^n*pentanacci(n) or (2^n)*A023424(n-1).
+10
3
2, 12, 56, 240, 992, 3648, 14464, 57088, 224768, 883712, 3471360, 13651968, 53682176, 211075072, 829915136, 3263102976, 12830244864, 50447253504, 198353354752, 779904614400, 3066503888896, 12057176965120, 47407572189184, 186401664532480, 732912043425792
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,4,8,16,32).
FORMULA
a(n) = Trace of matrix [({2,2,2,2,2},{2,0,0,0,0},{0,2,0,0,0},{0,0,2,0,0},{0,0,0,2,0})^n].
a(n) = 2^n * Trace of matrix [({1,1,1,1,1},{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,0},{0,0,0,1,0})^n].
G.f.: -2*x*(1 +4*x +12*x^2 +32*x^3 +80*x^4)/(-1 +2*x +4*x^2 +8*x^3 +16*x^4 +32*x^5). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009; corrected by R. J. Mathar, Sep 16 2009
a(n) = 2*a(n-1)+4*a(n-2)+8*a(n-3)+16*a(n-4)+32*a(n-5). - Colin Barker, Sep 02 2013
MATHEMATICA
Table[Tr[MatrixPower[2*{{1, 1, 1, 1, 1}, {1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}}, x]], {x, 1, 20}]
LinearRecurrence[{2, 4, 8, 16, 32}, {2, 12, 56, 240, 992}, 50] (* G. C. Greubel, Dec 19 2017 *)
PROG
(PARI) x='x+O('x^30); Vec(-2*x*(1 +4*x +12*x^2 +32*x^3 +80*x^4)/(-1 +2*x +4*x^2 +8*x^3 +16*x^4 +32*x^5)) \\ G. C. Greubel, Dec 19 2017
(Magma) I:=[2, 12, 56, 240, 992]; [n le 5 select I[n] else 2*Self(n-1) + 4*Self(n-2) + 8*Self(n-3) + 16*Self(n-4) + 32*Self(n-5): n in [1..30]]; // G. C. Greubel, Dec 19 2017
CROSSREFS
Cf. A087131, A127210, A127211, A127212, A127213, A127214, A127216, A001648, A127220, A127222.
Cf. A023424, A074048.
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Jan 09 2007
EXTENSIONS
Definition corrected by R. J. Mathar, Sep 17 2009
More terms from Colin Barker, Sep 02 2013
STATUS
approved
A127222
a(n) = 3^n*pentanacci(n) or (3^n)*A023424(n-1).
+10
3
3, 27, 189, 1215, 7533, 41553, 247131, 1463103, 8640837, 50959287, 300264165, 1771292853, 10447598619, 61618989627, 363414767589, 2143339285311, 12641143135581, 74555586323649, 439717218548643, 2593383067853775, 15295369041550269, 90209719910309895
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,9,27,81,243).
FORMULA
a(n) = Trace of matrix [({3,3,3,3,3},{3,0,0,0,0},{0,3,0,0,0},{0,0,3,0,0},{0,0,0,3,0})^n].
a(n) = 3^n * Trace of matrix [({1,1,1,1,1},{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,0},{0,0,0,1,0})^n].
G.f.: -3*x*(1 +6*x +27*x^2 +108*x^3 +405*x^4)/(-1 +3*x +9*x^2 +27*x^3 +81*x^4 +243*x^5). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009
a(n) = 3*a(n-1)+9*a(n-2)+27*a(n-3)+81*a(n-4)+243*a(n-5). - Colin Barker, Sep 02 2013
MATHEMATICA
Table[Tr[MatrixPower[3*{{1, 1, 1, 1, 1}, {1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}}, x]], {x, 1, 20}]
LinearRecurrence[{3, 9, 27, 81, 243}, {3, 27, 189, 1215, 7533}, 50] (* G. C. Greubel, Dec 19 2017 *)
PROG
(PARI) x='x+O('x^30); Vec(-3*x*(1 +6*x +27*x^2 +108*x^3 +405*x^4)/(-1 +3*x +9*x^2 +27*x^3 +81*x^4 +243*x^5)) \\ G. C. Greubel, Dec 19 2017
(Magma) I:=[3, 27, 189, 1215, 7533]; [n le 5 select I[n] else 3*Self(n-1) + 9*Self(n-2) + 27*Self(n-3) + 81*Self(n-4) + 243*Self(n-5): n in [1..30]]; // G. C. Greubel, Dec 19 2017
CROSSREFS
Cf. A087131, A127210, A127211, A127212, A127213, A127214, A127216, A001648, A127220, A127221.
Cf. A023424, A074048.
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Jan 09 2007
EXTENSIONS
G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009
Definition corrected by R. J. Mathar, Sep 17 2009
More terms from Colin Barker, Sep 02 2013
STATUS
approved
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