A190972 -id:A190972 - cp's OEIS frontend

A190958 a(n) = 2a(n-1) - 10a(n-2), with a(0) = 0, a(1) = 1.

0, 1, 2, -6, -32, -4, 312, 664, -1792, -10224, -2528, 97184, 219648, -532544, -3261568, -1197696, 30220288, 72417536, -157367808, -1038910976, -504143872, 9380822016, 23803082752, -46202054656, -330434936832, -198849327104, 2906650714112, 7801794699264

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

For the difference equation a(n) = c*a(n-1) - d*a(n-2), with a(0) = 0, a(1) = 1, the solution is a(n) = d^((n-1)/2) * ChebyshevU(n-1, c/(2*sqrt(d))) and has the alternate form a(n) = ( ((c + sqrt(c^2 - 4*d))/2)^n - ((c - sqrt(c^2 - 4*d))/2)^n )/sqrt(c^2 - 4*d). In the case c^2 = 4*d then the solution is a(n) = n*d^((n-1)/2). The generating function is x/(1 - c*x + d^2) and the exponential generating function takes the form (2/sqrt(c^2 - 4*d))*exp(c*x/2)*sinh(sqrt(c^2 - 4*d)*x/2) for c^2 > 4*d, (2/sqrt(4*d - c^2))*exp(c*x/2)*sin(sqrt(4*d - c^2)*x/2) for 4*d > c^2, and x*exp(sqrt(d)*x) if c^2 = 4*d. - G. C. Greubel, Jun 10 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..190
Index entries for linear recurrences with constant coefficients, signature (2,-10).

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
.A128834,A107920,A106852,A106853,A106854,A145934,A145976,A145978,A146078,A146080
.A000027,A009545,A088137,A088138,A045873,A088139,A089181,A090591,A127357,A190958
.A001906,A000225,A057083,A049072,A190959,A190960,A190961,A190962,A190963,A190964
.A001353,A007070,A003462,A001787,A099456,A190965,A168175,A190966,A190967,A190968
.A004254,A107839,A116415,A002450,A030191,A001047,A099450,A190969,A190970,A190971
.A001109,A154244,A138395,A084326,A003463,A030192,A081179,A006516,A027471,A106392
.A004187,A186446,A190972,A190973,A190974,A003464,A030240,A152268,A099459,A016127
.A001090,A190975,A190976,A099156,A190977,A190978,A023000,A057084,A154245,A154235
.A018913,A190979,A190980,A190981,A190982,A190983,A190984,A023001,A057085,A178869
A004189,A190869,A190985,A190986,A190987,A190988,A190989,A190990,A002452,A057086

I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1)-10*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 17 2011

Mathematica
```
LinearRecurrence[{2,-10}, {0,1}, 50]
```

a(n)=([0,1; -10,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

[lucas_number1(n,2,10) for n in (0..50)] # G. C. Greubel, Jun 10 2022

G.f.: x / ( 1 - 2*x + 10*x^2 ). - R. J. Mathar, Jun 01 2011
E.g.f.: (1/3)*exp(x)*sin(3*x). - Franck Maminirina Ramaharo, Nov 13 2018
a(n) = 10^((n-1)/2) * ChebyshevU(n-1, 1/sqrt(10)). - G. C. Greubel, Jun 10 2022
a(n) = (1/3)*10^(n/2)*sin(n*arctan(3)) = Sum_{k=0..floor(n/2)} (-1)^k*3^(2*k)*binomial(n,2*k+1). - Gerry Martens, Oct 15 2022

A368150 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3x, p(n,x) = up(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - x^2.

Original entry on oeis.org

1, 1, 3, 2, 6, 8, 3, 15, 25, 21, 5, 30, 76, 90, 55, 8, 60, 188, 324, 300, 144, 13, 114, 439, 948, 1251, 954, 377, 21, 213, 961, 2529, 4207, 4527, 2939, 987, 34, 390, 2026, 6246, 12606, 17154, 15646, 8850, 2584, 55, 705, 4136, 14640, 34590, 56970, 65840

Offset: 1

Clark Kimberling, Dec 25 2023

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

			First eight rows:
   1
   1    3
   2    6    8
   3   15   25    21
   5   30   76    90     55
   8   60  188   324    300   144
  13  114  439   948   1251   954   377
  21  213  961  2529   4207  4527  2939   987
Row 4 represents the polynomial p(4,x) = 3 + 15*x + 25*x^2 + 21*x^3, so (T(4,k)) = (3,15,25,21), k=0..3.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Cf. A000045 (column 1); A001906 (p(n,n-1)); A000302 (row sums), (p(n,1)); A122803 (alternating row sums), (p(n,-1)); A190972 (p(n,2)), A116415, (p(n,-2)); A190990, (p(n,3)); A057084, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368151.

p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 3*x, u = p(2,x), and v = 1 - x^2.

p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 6*x + 5*x^2), b = (1/2)*(3*x + 1 - 1/k), c = (1/2)*(3*x + 1 + 1/k).

A368153 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3x - x^2.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 3, 4, -2, 4, 5, 5, 4, -10, 5, 8, 10, -3, 4, -25, 6, 13, 16, 1, -29, 14, -49, 7, 21, 28, -8, -24, -78, 56, -84, 8, 34, 47, -12, -88, -26, -162, 168, -132, 9, 55, 80, -31, -140, -200, 100, -330, 408, -195, 10, 89, 135, -58, -301, -230, -296

Offset: 1

Clark Kimberling, Jan 20 2024

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

			First eight rows:
   1
   1   2
   2   1   3
   3   4  -2    4
   5   5   4  -10    5
   8  10  -3    4  -25    6
  13  16   1  -29   14  -49    7
  21  28  -8  -24  -78   56  -84   8
Row 4 represents the polynomial p(4,x) = 3 + 4*x - 2*x^2 + 4*x^3, so (T(4,k)) = (3,4,-2,4), k=0..3.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), 1-28, Paper No. A14.

Cf. A000045 (column 1); A000027 (p(n,n-1)); A057083 (row sums), (p(n,1)); A182228 (alternating row sums), (p(n,-1)); A190970, (p(n,2)); A030195, (p(n,-2)); A052918, (p(n,-3)); A190972, (p(n,-4)); A057085, (p(n,-5)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152.

p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 - 3x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 2*x, u = p(2,x), and v = 1 - 3*x - x^2.

p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 - 8*x), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

A276288 a(n) = a(n-1) + 3a(n-2) if n is even, otherwise a(n) = 3a(n-1) + a(n-2), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 1, 4, 7, 25, 46, 163, 301, 1066, 1969, 6973, 12880, 45613, 84253, 298372, 551131, 1951765, 3605158, 12767239, 23582713, 83515378, 154263517, 546305929, 1009096480, 3573595369, 6600884809, 23376249796, 43178904223, 152912962465, 282449675134, 1000261987867, 1847611013269, 6543095027674

Offset: 0

Ilya Gutkovskiy, Aug 27 2016

Index entries for linear recurrences with constant coefficients, signature (0,7,0,-3).

Cf. A005824, A010684, A079162, A190972.

LinearRecurrence[{0, 7, 0, -3}, {0, 1, 1, 4}, 34]
RecurrenceTable[{a[0] == 0, a[1] == 1, a[n] == (2 - (-1)^n) a[n - 1] + (2 + (-1)^n) a[n - 2]}, a, {n, 33}]

concat(0, Vec(x*(1+x-3*x^2)/(1-7*x^2+3*x^4) + O(x^99))) \\ Altug Alkan, Aug 27 2016

G.f.: x*(1 + x - 3*x^2)/(1 - 7*x^2 + 3*x^4).
a(n) = 7*a(n-2) - 3*a(n-4).
a(n) = (2 - (-1)^n)*a(n-1) + (2 + (-1)^n)*a(n-2) for n > 1, a(0)=0, a(1)=1.
a(2k) = A190972(k).

A287813 Number of octonary sequences of length n such that no two consecutive terms have distance 2.

Original entry on oeis.org

1, 8, 52, 340, 2224, 14548, 95164, 622504, 4072036, 26636740, 174241072, 1139777284, 7455717772, 48770692552, 319027694548, 2086881784180, 13651089405616, 89296980486772, 584125595190556, 3820988224873576, 24994540788543364, 163498820845182820

Offset: 0

David Nacin, Jun 02 2017

			For n=2 the a(2) = 64 - 12 = 52 sequences contain every combination except these twelve: 02,20,13,31,24,42,35,53,46,64,57,75.

Index entries for linear recurrences with constant coefficients, signature (7, -3).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819.

LinearRecurrence[{7, -3}, {1, 8, 52}, 40]

def a(n):
 if n in [0, 1, 2]:
  return [1, 8, 52][n]
 return 7*a(n-1)-3*a(n-2)

For n>2, a(n) = 7*a(n-1) - 3*a(n-2), a(0)=1, a(1)=8, a(2)=52.
G.f.: (1 + x - x^2)/(1 - 7 x + 3 x^2).
a(n) = A190972(n) + A190972(n+1) - A190972(n-1). - R. J. Mathar, Oct 20 2019

cp's OEIS Frontend

A190958 a(n) = 2*a(n-1) - 10*a(n-2), with a(0) = 0, a(1) = 1.

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A368150 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - x^2.

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A368153 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3*x - x^2.

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A276288 a(n) = a(n-1) + 3*a(n-2) if n is even, otherwise a(n) = 3*a(n-1) + a(n-2), a(0)=0, a(1)=1.

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A287813 Number of octonary sequences of length n such that no two consecutive terms have distance 2.

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Formula

A190958 a(n) = 2a(n-1) - 10a(n-2), with a(0) = 0, a(1) = 1.

A368150 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3x, p(n,x) = up(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - x^2.

A368153 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3x - x^2.

A276288 a(n) = a(n-1) + 3a(n-2) if n is even, otherwise a(n) = 3a(n-1) + a(n-2), a(0)=0, a(1)=1.