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.

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

Author

Roman Witula, Sep 29 2012

Keywords

• nonn
• easy

Comments

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)).

Examples

			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.
		

Links

• 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).

Crossrefs

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

Programs

Magma

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

Mathematica

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 *)

Maxima

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 */

PARI

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

Formula

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