A291660 -id:A291660 - cp's OEIS frontend

A292521 a(n) = a(n-2) - 2a(n-3) + a(n-4) for n>3, with a(0)=2, a(1)=0, a(2)=1, a(3)=-1, a sequence related to Pellian numbers.

2, 0, 1, -1, 3, -3, 6, -10, 15, -25, 41, -65, 106, -172, 277, -449, 727, -1175, 1902, -3078, 4979, -8057, 13037, -21093, 34130, -55224, 89353, -144577, 233931, -378507, 612438, -990946, 1603383, -2594329, 4197713, -6792041, 10989754, -17781796, 28771549, -46553345

Offset: 0

Jean-François Alcover and Paul Curtz, Sep 18 2017

sign

Successive differences begin:
2,    0,   1,  -1,   3,   -3,    6,  -10,   15,   -25, ... = a(n)
-2,   1,  -2,   4,  -6,    9,  -16,   25,  -40,    66, ... = b(n)
3,   -3,   6, -10,  15,  -25,   41,  -65,  106,  -172, ... = a(n+4)
-6,   9, -16,  25, -40,   66, -106,  171, -278,   449, ... = b(n+4)
15, -25,  41, -65, 106, -172,  277, -449,  727, -1175, ... = a(n+8)
...
Main diagonal [2] 1, 6, 25, 106, 449, ... (omitting first term) is A048875 (Pellian numbers with second term 6).

Index entries for linear recurrences with constant coefficients, signature (0,1,-2,1).

Cf. A048875, A291660.

LinearRecurrence[{0, 1, -2, 1}, {2, 0, 1, -1}, 40]

x='x+O('x^99); Vec((2-x^2+3*x^3)/(1-x^2+2*x^3-x^4)) \\ Altug Alkan, Sep 18 2017

G.f.: (2 - x^2 + 3*x^3) / (1 - x^2 + 2*x^3 - x^4).
a(n) = A291660(-n) (negative indices computed using A291660 sequence function).
a(n) = (1/15)*2^(n-1)*((9*sqrt(5)+30)/(1+sqrt(5))^n + (30-9*sqrt(5))/(1- sqrt(5))^n - 5*sqrt(3)*2^(1-n)*sin(n*Pi/3)).

A292616 a(n) = 3*a(n-2) - a(n-4) for n > 3, with a(0)=4, a(1)=3, a(2)=a(3)=6, a sequence related to bisections of Fibonacci numbers.

Original entry on oeis.org

4, 3, 6, 6, 14, 15, 36, 39, 94, 102, 246, 267, 644, 699, 1686, 1830, 4414, 4791, 11556, 12543, 30254, 32838, 79206, 85971, 207364, 225075, 542886, 589254, 1421294, 1542687, 3720996, 4038807, 9741694, 10573734, 25504086, 27682395, 66770564, 72473451, 174807606, 189737958

Offset: 0

Jean-François Alcover and Paul Curtz, Sep 20 2017

nonn

The "Pisano" periods, i.e., the periods of this sequence mod n, begin: 1, 6, 8, 6, 20, 24, 16, 12, 24, 60, 5, 24, ..., a sequence which is not in the OEIS. Period 5 for mod 11 seems to be the only odd period > 1 (checked up to 1000 terms).

Eric Weisstein's World of Mathematics, Pisano Number
Wikipedia, Pisano period
Index entries for linear recurrences with constant coefficients, signature (0, 3, 0, -1).

Cf. A000045, A001519, A001906, A291660, A292521.

a := [4,3,6,6];; for n in [5..10^2] do a[n] := 3 * a[n-2] - a[n-4]; od; A292616 := a; #  Muniru A Asiru, Oct 31 2017

A292616 := gfun:-rectoproc({a(n) = 3 * a(n-2) - a(n-4), a(0) = 4,a(1) = 3,a(2) = 6, a(3) = 6}, a(n), remember):  map(A292616, [$0..10^3]);  # Muniru A Asiru, Oct 16 2017

LinearRecurrence[{0, 3, 0, -1}, {4, 3, 6, 6}, 40]
(* Or, recomputing from Fibonacci numbers: *)
Join[{4, -1, 3}, Flatten[Table[{Fibonacci[2*n], Fibonacci[2*n+6]}, {n, 0, 18} ]]] // Accumulate

G.f.: (4 + 3*x - 6*x^2 - 3*x^3)/(1 - 3*x^2 + x^4).
a(n) = A291660(n) + A292521(n).
a(n) = (1/20)*((25 - 13*sqrt(5))*((1/2)*(-1 - sqrt(5)))^n + (5 - 7*sqrt(5)) *((1/2)*(1 - sqrt(5)))^n + ((1/2)*(1 + sqrt(5)))^n*(5 + 7*sqrt(5)) + ((1/2)*(-1 + sqrt(5)))^n*(25 + 13*sqrt(5))).
a(2n+1) = 3*A001519(n+1).
a(2n+1) = a(2n) + A001906(n-1).

cp's OEIS Frontend

A292521 a(n) = a(n-2) - 2a(n-3) + a(n-4) for n>3, with a(0)=2, a(1)=0, a(2)=1, a(3)=-1, a sequence related to Pellian numbers.

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