A217444 -id:A217444 - cp's OEIS frontend

A217274 a(n) = 7a(n-1) - 14a(n-2) + 7*a(n-3) with a(0)=0, a(1)=1, a(2)=7.

0, 1, 7, 35, 154, 637, 2548, 9996, 38759, 149205, 571781, 2184910, 8333871, 31750824, 120875944, 459957169, 1749692735, 6654580387, 25306064602, 96226175941, 365880389868, 1391138718116, 5289228800247, 20109822277181, 76457523763621, 290689756066542

Offset: 0

Roman Witula, Sep 29 2012

This is the Berndt-type sequence number 18 for the argument 2*Pi/7 defined by the relation
a(n)*sqrt(7) = c(4)*s(1)^(2n+1) + c(2)*s(4)^(2n+1) + c(1)*s(2)^(2n+1) = (1/s(4))*s(1)^(2n+2) + (1/s(2))*s(4)^(2n+2) + (1/s(1))*s(2)^(2n+2), where c(j) := 2*cos(2*Pi*j/7) and s(j) := 2*sin(2*Pi*j/7) (for the sums of the respective even powers see A094429). For the proof of this formula see the Witula/Slota and Witula references.
The definitions of the other Berndt-type sequences for the argument 2*Pi/7 (with numbers from 1 to 17) are in the cross references.
We note that all numbers of the form a(n)*7^(-floor((n+1)/3)) = A217444(n) are integers.
It can be proved that Sum_{k=2..n}a(k) = 7*(a(n-1) - a(n-2)).

			Writing c(j) as cj and s(k) as sk,
we have 7*sqrt(7) = c4*s1^5 + c2*s4^5 + c1*s2^5
and c4*s1^13 + c2*s4^13 + c1*s2^13 = 4(c4*s1^11 + c2*s4^11 + c1*s2^11).
We note that a(9) = 87*a(3)*a(2)^2 and a(11) = 2*a(3)*a(5)*a(2)^2.

G. C. Greubel, Table of n, a(n) for n = 0..1000
R. Witula, Ramanujan type trigonometric formulas, Demonstratio Math., Vol. XLV, No. 4, 2012, pp. 789-796.
Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6.
Index entries for linear recurrences with constant coefficients, signature (7,-14,7).

Cf. A033304, A094429, A094430, A094648, A108716, A215007, A215008, A215143, A215493, A215494, A215510, A215512, A215575, A215694, A215695, A215794, A215817, A215828, A215877, A217444.

I:=[0,1,7]; [n le 3 select I[n] else 7*Self(n-1)-14*Self(n-2)+7*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 26 2015

LinearRecurrence[{7,-14,7}, {0,1,7}, 30]
CoefficientList[Series[x/(1 - 7*x + 14*x^2 - 7*x^3), {x,0,50}], x] (* G. C. Greubel, Apr 16 2017 *)

a[0]:0$
a[1]:1$
a[2]:7$
a[n]:=7*a[n-1] - 14*a[n-2] + 7*a[n-3];
makelist(a[n], n, 0, 25); /* Martin Ettl, Oct 11 2012 */

concat(0, Vec(x/(1-7*x+14*x^2-7*x^3) + O(x^40))) \\ Michel Marcus, Jul 25 2015

G.f.: x/(1-7*x+14*x^2-7*x^3).

A319512 a(n) = 7a(n-1) - 14a(n-2) + 7*a(n-3), a(0) = 1, a(1) = 3, a(2) = 11.

Original entry on oeis.org

1, 3, 11, 42, 161, 616, 2352, 8967, 34153, 129997, 494606, 1881355, 7154980, 27208132, 103456689, 393367835, 1495638123, 5686513994, 21620239081, 82199944512, 312521862408, 1188195487255, 4517461948657, 17175149855885, 65298950120782, 248262786503683

Offset: 0

Kai Wang, Dec 10 2018

Let {X,Y,Z} be the roots of the cubic equation
  t^3 + at^2 + bt + c = 0
where {a, b, c} are integers. Let {u, v, w} be three numbers such that {u + v + w, u*X + v*Y + w*Z, u*X^2 + v*Y^2 + w*Z^2} are integers. Then
{p(n) = u*X^n + v*Y^n + w*Z^n | n = 0, 1, 2, ...}
is an integer sequence with the recurrence relation:
p(n) = -a*p(n-1) - b*p(n-2) - c*p(n-3).
This sequence has (a, b, c) = (-7, 14, -7), (u, v, w) = (1/(sqrt(7)*tan(4*(Pi/7))), 1/(sqrt(7)*tan(8*(Pi/7))), 1/(sqrt(7)*tan(2*(Pi/7)))).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-14,7)

Cf. A215007, A215008, A215143, A215493, A215494, A215510, A217274, A217444, A094430.

LinearRecurrence[{7, -14, 7}, {1, 3, 11}, 30] (* Amiram Eldar, Dec 10 2018 *)
CoefficientList[Series[(1-2x)^2/(1-7x+14x^2-7x^3),{x,0,30}],x] (* Harvey P. Dale, Oct 08 2023 *)

Vec((1 - 2*x)^2 / (1 - 7*x + 14*x^2 - 7*x^3) + O(x^40)) \\ Colin Barker, Dec 11 2018

(X, Y, Z) = (4*sin^2(2*(Pi/7)), 4*sin^2(4*(Pi/7)), 4*sin^2(8*(Pi/7)));
a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3), a(0) = 1, a(1) = 3, a(2) = 11.
G.f.: (1 - 2*x)^2 / (1 - 7*x + 14*x^2 - 7*x^3). - Colin Barker, Dec 11 2018

More terms from Felix Fröhlich, Dec 10 2018

cp's OEIS Frontend

A217274 a(n) = 7a(n-1) - 14a(n-2) + 7*a(n-3) with a(0)=0, a(1)=1, a(2)=7.

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A319512 a(n) = 7a(n-1) - 14a(n-2) + 7*a(n-3), a(0) = 1, a(1) = 3, a(2) = 11.

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cp's OEIS Frontend

A217274 a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3) with a(0)=0, a(1)=1, a(2)=7.

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Programs

Magma

Mathematica

Maxima

PARI

Formula

A319512 a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3), a(0) = 1, a(1) = 3, a(2) = 11.

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Extensions

A217274 a(n) = 7a(n-1) - 14a(n-2) + 7*a(n-3) with a(0)=0, a(1)=1, a(2)=7.

A319512 a(n) = 7a(n-1) - 14a(n-2) + 7*a(n-3), a(0) = 1, a(1) = 3, a(2) = 11.