

A217274


a(n) = 7*a(n1)  14*a(n2) + 7*a(n3) with a(0)=0, a(1)=1, a(2)=7.


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
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OFFSET

0,3


COMMENTS

This is the Berndttype 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 Berndttype 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(n1)  a(n2)).


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. 789796.
Roman Witula and Damian Slota, New RamanujanType Formulas and QuasiFibonacci 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).


FORMULA

G.f.: x/(17*x+14*x^27*x^3).


EXAMPLE

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.


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 *)


PROG

(Maxima)
a[0]:0$
a[1]:1$
a[2]:7$
a[n]:=7*a[n1]  14*a[n2] + 7*a[n3];
makelist(a[n], n, 0, 25); /* Martin Ettl, Oct 11 2012 */
(PARI) concat(0, Vec(x/(17*x+14*x^27*x^3) + O(x^40))) \\ Michel Marcus, Jul 25 2015
(MAGMA) I:=[0, 1, 7]; [n le 3 select I[n] else 7*Self(n1)14*Self(n2)+7*Self(n3): n in [1..30]]; // Vincenzo Librandi, Jul 26 2015


CROSSREFS

Cf. A033304, A094429, A094430, A094648, A108716, A215007, A215008, A215143, A215493, A215494, A215510, A215512, A215575, A215694, A215695, A215794, A215817, A215828, A215877, A217444.
Sequence in context: A240423 A094825 A022635 * A000588 A005285 A006095
Adjacent sequences: A217271 A217272 A217273 * A217275 A217276 A217277


KEYWORD

nonn,easy


AUTHOR

Roman Witula, Sep 29 2012


STATUS

approved



