A270863
Self-composition of the Fibonacci sequence.
Original entry on oeis.org
0, 1, 2, 6, 17, 50, 147, 434, 1282, 3789, 11200, 33109, 97878, 289354, 855413, 2528850, 7476023, 22101326, 65338038, 193158521, 571033600, 1688143881, 4990651642, 14753839486, 43616704857, 128943855250, 381196100507, 1126928202714, 3331532438042, 9848993360069
Offset: 0
a(5) = 3*a(4)+a(3)-3*a(2)-a(1) = 51+6-6-1 = 50.
- Colin Barker, Table of n, a(n) for n = 0..1000
- Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics (2017), Vol. 41, Issue 6, 849-853.
- Oboifeng Dira, Family of composition pairs g(f(x)) generating A270683
- Index entries for linear recurrences with constant coefficients, signature (3,1,-3,-1).
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I:=[0, 1, 2, 6]; [m le 4 select I[m] else 3*Self(m-1)+Self(m-2)-3*Self(m-3)-Self(m-4): m in [1..30]]; // Marius A. Burtea, Aug 03 2019
-
f:= x-> x/(1-x-x^2):
a:= n-> coeff(series(f(f(x)), x, n+1), x, n):
seq(a(n), n=0..30);
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a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,-3,1,3]^(n-1)*[1;2;6;17])[1,1] \\ Charles R Greathouse IV, Mar 24 2016
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concat(0, Vec(x*(1-x-x^2)/(1-3*x-x^2+3*x^3+x^4) + O(x^40))) \\ Colin Barker, Mar 24 2016
A189800
a(n) = 6*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1.
Original entry on oeis.org
0, 1, 6, 44, 312, 2224, 15840, 112832, 803712, 5724928, 40779264, 290475008, 2069084160, 14738305024, 104982503424, 747801460736, 5326668791808, 37942424436736, 270267896954880, 1925146777223168, 13713023838978048, 97679317251653632, 695780094221746176
Offset: 0
Sequences of the form a(n) = c*a(n-1) + d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
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I:=[0,1]; [n le 2 select I[n] else 6*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011
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LinearRecurrence[{6, 8}, {0, 1}, 50]
CoefficientList[Series[-(x/(-1+6 x+8 x^2)),{x,0,50}],x] (* Harvey P. Dale, Jul 26 2011 *)
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a(n)=([0,1; 8,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016
A213895
Fixed points of a sequence h(n) defined by the minimum number of 6's in the relation n*[n,6,6,...,6,n] = [x,...,x] between simple continued fractions.
Original entry on oeis.org
7, 11, 23, 47, 127, 139, 211, 223, 251, 331, 367, 379, 383, 463, 487, 499, 607, 619, 691, 727, 739, 743, 811, 823, 863, 887, 967, 971, 983, 1051, 1063, 1087, 1171, 1291, 1303, 1327, 1367, 1423, 1447, 1451, 1459
Offset: 1
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f[m_, n_] := Block[{c, k = 1}, c[x_, y_] := ContinuedFraction[x FromContinuedFraction[Join[{x}, Table[m, {y}], {x}]]]; While[First@ c[n, k] != Last@ c[n, k], k++]; k]; Select[Range[2, 1000], f[6, #] == # &] (* Michael De Vlieger, Sep 16 2015 *)
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{a(n) = local(t, m=1); if( n<2, 0, while( 1,
t = contfracpnqn( concat([n, vector(m,i,6), n]));
t = contfrac(n*t[1,1]/t[2,1]);
if(t[1]
A218991
Power floor sequence of 3+sqrt(10).
Original entry on oeis.org
6, 36, 221, 1361, 8386, 51676, 318441, 1962321, 12092366, 74516516, 459191461, 2829665281, 17437183146, 107452764156, 662153768081, 4080375372641, 25144406003926, 154946811396196, 954825274381101, 5883898457682801
Offset: 0
a(0) = floor(r) = 6, where r = 3+sqrt(10);
a(1) = floor(6*r) = 36;
a(2) = floor(36*r) = 221.
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[IsZero(n) select Floor(r) else Floor(r*Self(n)) where r is 3+Sqrt(10): n in [0..20]]; // Bruno Berselli, Nov 22 2012
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x = 3 + Sqrt[10]; z = 30; (* z = # terms in sequences *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
t1 = Table[p1[n], {n, 0, z}] (* A218991 *)
t2 = Table[p2[n], {n, 0, z}] (* A005668 *)
t3 = Table[p3[n], {n, 0, z}] (* A015451 *)
t4 = Table[p4[n], {n, 0, z}] (* A218992 *)
A218992
Power ceiling sequence of 3+sqrt(10).
Original entry on oeis.org
7, 44, 272, 1677, 10335, 63688, 392464, 2418473, 14903303, 91838292, 565933056, 3487436629, 21490552831, 132430753616, 816075074528, 5028881200785, 30989362279239, 190965054876220, 1176779691536560, 7251643204095581
Offset: 0
a(0) = ceiling(r) = 7, where r = 3+sqrt(10);
a(1) = ceiling(7*r) = 44;
a(2) = ceiling(44*r) = 272.
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[IsZero(n) select Ceiling(r) else Ceiling(r*Self(n)) where r is 3+Sqrt(10): n in [0..20]]; // Bruno Berselli, Nov 22 2012
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(See A218991.)
LinearRecurrence[{7,-5,-1},{7,44,272},20] (* Harvey P. Dale, Sep 22 2016 *)
A041683
Denominators of continued fraction convergents to sqrt(360).
Original entry on oeis.org
1, 1, 37, 38, 1405, 1443, 53353, 54796, 2026009, 2080805, 76934989, 79015794, 2921503573, 3000519367, 110940200785, 113940720152, 4212806126257, 4326746846409, 159975692596981, 164302439443390, 6074863512559021, 6239165952002411
Offset: 0
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I:=[1,1,37,38]; [n le 4 select I[n] else 38*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 22 2013
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Denominator[Convergents[Sqrt[360], 30]] (* Vincenzo Librandi, Dec 22 2013 *)
A172343
Triangle t(n,k) read by rows: fibonomial ratios c(n)/(c(k)*c(n-k)) where c are partial products of a generalized Fibonacci sequence with multiplier m=6.
Original entry on oeis.org
1, 1, 1, 1, 6, 1, 1, 37, 37, 1, 1, 228, 1406, 228, 1, 1, 1405, 53390, 53390, 1405, 1, 1, 8658, 2027415, 12493260, 2027415, 8658, 1, 1, 53353, 76988379, 2923477635, 2923477635, 76988379, 53353, 1, 1, 328776, 2923530988, 684106251192
Offset: 0
1;
1, 1;
1, 6, 1;
1, 37, 37, 1;
1, 228, 1406, 228, 1;
1, 1405, 53390, 53390, 1405, 1;
1, 8658, 2027415, 12493260, 2027415, 8658, 1;
1, 53353, 76988379, 2923477635, 2923477635, 76988379, 53353, 1;
1, 328776, 2923530988, 684106251192, 4215654749670, 684106251192, 2923530988, 328776, 1;
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Clear[f, c, a, t];
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = a*f[n - 1, a] + f[n - 2, a];
c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
A304255
Triangle read by rows: T(0,0) = 1; T(n,k) = 6*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.
Original entry on oeis.org
1, 6, 36, 1, 216, 12, 1296, 108, 1, 7776, 864, 18, 46656, 6480, 216, 1, 279936, 46656, 2160, 24, 1679616, 326592, 19440, 360, 1, 10077696, 2239488, 163296, 4320, 30, 60466176, 15116544, 1306368, 45360, 540, 1, 362797056, 100776960, 10077696, 435456, 7560, 36
Offset: 0
Triangle begins:
1;
6;
36, 1;
216, 12;
1296, 108, 1;
7776, 864, 18;
46656, 6480, 216, 1;
279936, 46656, 2160, 24;
1679616, 326592, 19440, 360, 1;
10077696, 2239488, 163296, 4320, 30;
60466176, 15116544, 1306368, 45360, 540, 1;
362797056, 100776960, 10077696, 435456, 7560, 36;
2176782336, 665127936, 75582720, 3919104, 90720, 756, 1;
13060694016, 4353564672, 554273280, 33592320, 979776, 12096, 42;
78364164096, 28298170368, 3990767616, 277136640, 9797760, 163296, 1008, 1;
470184984576, 182849716224, 28298170368, 2217093120, 92378880, 1959552, 18144, 48;
- Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 94.
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t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 6 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten
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T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, 6*T(n-1, k) + T(n-2, k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 26 2018
A305534
Index of the smallest prime in the n-Fibonacci sequence, or the Lucas U(n,-1) sequence.
Original entry on oeis.org
3, 2, 2, 3, 2, 3, 2, 5, 29, 3, 2, 5, 2, 3, 23, 3, 2, 7, 2, 3, 29, 19, 2, 3, 83, 3, 53, 19, 2, 5, 2, 5, 5, 5479, 71, 3, 2, 17, 11, 3, 2, 37, 2, 31, 5, 11, 2, 5
Offset: 1
Cf.
A001605,
A096650,
A209493, which are the indices of the primes in the n-Fibonacci sequence for n = 1, 2, 3.
Cf.
A000045,
A000129,
A006190,
A001076,
A052918,
A005668,
A054413,
A041025,
A099371,
A041041,
A049666,
A041061 (the n-Fibonacci sequence for n = 1 to 12).
Cf.
A302990 (for n-step Fibonacci sequence instead of n-Fibonacci sequence).
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b(n,k)=([n,1;1,0]^k)[1,2]
a(n)=for(k=1,2^12,if(ispseudoprime(b(n,k)),return(k)))
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