A115146 -id:A115146 - cp's OEIS frontend

A215493 a(n) = 7a(n-1) - 14a(n-2) + 7*a(n-3) with a(0)=0, a(1)=1, a(2)=4.

0, 1, 4, 14, 49, 175, 637, 2352, 8771, 32928, 124166, 469567, 1779141, 6749211, 25623472, 97329337, 369821228, 1405502182, 5342323441, 20307982135, 77201862045, 293497548512, 1115812645899, 4242135876440, 16128056932078, 61317184775679, 233122447515741

Offset: 0

Roman Witula, Aug 13 2012

The Berndt-type sequence number 4 for the argument 2Pi/7 - see also A215007, A215008, A215143 and A215494.

We have a(n)=A079309(n) for n=1..6, and A079309(7)-a(7)=1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
B. C. Berndt, A. Zaharescu, Finite trigonometric sums and class numbers, Math. Ann. 330 (2004), 551-575.
B. C. Berndt, L.-C. Zhang, Ramanujan's identities for eta-functions, Math. Ann. 292 (1992), 561-573.
Z.-G. Liu, Some Eisenstein series identities related to modular equations of the seventh order, Pacific J. Math. 209 (2003), 103-130.
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).

I:=[0,1,4]; [n le 3 select I[n] else 7*Self(n-1) - 14*Self(n-2) +7*Self(n-3): n in [1..30]]; // G. C. Greubel, Apr 23 2018

LinearRecurrence[{7,-14,7}, {0,1,4}, 50]

x='x+O('x^30); concat([0], Vec(x*(1-3*x)/(1-7*x+14*x^2-7*x^3))) \\ G. C. Greubel, Apr 23 2018

a(n)*sqrt(7) = s(1)^(2n-1) + s(2)^(2n-1) + s(4)^(2n-1), where s(j) := 2*Sin(2*Pi*j/7) (for the sums of the respective even powers see A215494, see also A094429, A115146). For the proof of these formula see Witula-Slota's paper.
G.f.: x*(1-3*x)/(1-7*x+14*x^2-7*x^3).
a(n) = A275830(2*n-1)/(7^n). - Kai Wang, May 25 2017

A215494 a(n) = 7a(n-1) - 14a(n-2) + 7*a(n-3) with a(1)=7, a(2)=21, a(3)=70.

Original entry on oeis.org

7, 21, 70, 245, 882, 3234, 12005, 44933, 169099, 638666, 2417807, 9167018, 34790490, 132119827, 501941055, 1907443237, 7249766678, 27557748813, 104759610858, 398257159370, 1514069805269, 5756205681709, 21884262613787, 83201447389466, 316323894905207

Offset: 1

Roman Witula, Aug 13 2012

The Berndt-type sequence number 5 for the argument 2*Pi/7; see also A215007, A215008, A215143, A215493 and A215510.
We note that if we set:
  x(n) := s(2)*s(1)^n + s(4)*s(2)^n + s(1)*s(4)^n,
  y(n) := s(4)*s(1)^n + s(1)*s(2)^n + s(2)*s(4)^n,
  z(n) := s(1)^(n+1) + s(2)^(n+1) + s(4)^(n+1),
  for every n=0,1,..., where s(j) := 2*sin(2*Pi*j/7), then the following system of recurrence equations holds true:
  x(n+2)=2*x(n)-y(n), y(n+2)=2*y(n)-x(n)+z(n), z(n+2)=y(n)+3*z(n).
Moreover we have a(n)=z(2*n-1), A215493(n)=z(2*n-2), A094429(n)=y(2n-1)-x(2n-1)=-x(2*n+2)/sqrt(7), A094430(n)=-x(2*n+3), y(2*n-2)=sqrt(7)*A215143(n), y(2*n-1)=A215510(n) and x(11)=-(y(10)+z(10))/sqrt(7)=-1078.
We can also deduce the following relations:
  x(n-1) = c(1)*s(1)^n + c(2)*s(2)^n + c(4)*s(4)^n,
  -y(n-1)-z(n-1) = c(2)*s(1)^n + c(4)*s(2)^n + c(1)*s(4)^n,
  y(n-1)-x(n-1) = c(4)*s(1)^n + c(1)*s(2)^n + c(2)*s(4)^n,
  for every n=1,2,..., where x(0)=y(0)=z(0)=sqrt(7), and c(j) := 2*cos(2*Pi*j/7).
All these sequences satisfy the following recurrence equation: Z(n+6)-7*Z(n+4)+14*Z(n+2)-7*Z(n)=0. The characteristic polynomial of this equation (after rescaling) has the form (X-s(1)^2)*(X-s(2)^2)*(X-s(3)^2)=X^3-7*X^2+14*X-7 and was recognized by Johannes Kepler (1571-1630); see the Savio-Suryanarayan paper.
We also have the following decomposition: (X-s(1)^(n+1))*(X-s(2)^(n+1))*(X-s(4)^(n+1)) = X^3 - z(n)*X^2 + (1/2)*(z(n)^2-z(2n+1))*X - (-sqrt(7))^(n+1).
Further we have a(n)=A146533(n) for n=1,...,6, and A146533(7)-a(7)=7. We note that all numbers 7^(-1-floor(n/3))*a(n) are integers.

			We have a(3)=5*7^2 and a(6)=5*7^4, which implies that s(1)^12 + s(2)^12 + s(4)^12 = 49*(s(1)^6 + s(2)^6 + s(4)^6). We also have a(9) = (a(1) + a(3))*7^49.

G. C. Greubel, Table of n, a(n) for n = 1..1000
B. C. Berndt, A. Zaharescu, Finite trigonometric sums and class numbers, Math. Ann. 330 (2004), 551-575.
B. C. Berndt, L.-C. Zhang, Ramanujan's identities for eta-functions, Math. Ann. 292 (1992), 561-573.
L. Bankoff and J. Garfunkel, The Heptagonal Triangle, Math. Magazine, 46 (1973), 7-19.
Z.-G. Liu, Some Eisenstein series identities related to modular equations of the seventh order, Pacific J. Math. 209 (2003), 103-130.
D. Y. Savio and E. R. Suryanarayan, Chebyshev polynomials and regular polygons, Amer. Math. Monthly, 100 (1993), 657-661.
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).

See A122068.

I:=[7,21,70]; [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, Dec 01 2016

LinearRecurrence[{7,-14,7}, {7,21,70}, 50]

polsym(x^3 - 7*x^2 + 14*x - 7, 30) \\ (includes a(0)=3) Joerg Arndt, May 31 2017

x='x+O('x^30); Vec((7-28*x+21*x^2)/(1-7*x+14*x^2-7*x^3)) \\ G. C. Greubel, Apr 23 2018

Equals 7*A122068. - M. F. Hasler, Aug 25 2012
a(n) = s(1)^(2n) + s(2)^(2n) + s(4)^(2n), where s(j) := 2*Sin(2*Pi*j/7) (for the sums of the respective odd powers see A215493, see also A094429, A115146). For the proof of these formula see Witula-Slota's paper.
G.f.: (7 - 28*x + 21*x^2)/(1 - 7*x + 14*x^2 - 7*x^3) = -d(log(1 - 7*x + 14*x^2 - 7*x^3))/dx.

A115147 Eighth convolution of A115140.

Original entry on oeis.org

1, -8, 20, -16, 2, 0, 0, 0, -1, -8, -44, -208, -910, -3808, -15504, -62016, -245157, -961400, -3749460, -14567280, -56448210, -218349120, -843621600, -3257112960, -12570420330, -48507033744, -187187399448, -722477682080, -2789279908316, -10772391370048

Offset: 0

Wolfdieter Lang, Jan 13 2006

Seiichi Manyama, Table of n, a(n) for n = 0..1670

Cf. A115139 - A115146, A115148, A115149.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-8*x+20*x^2-16*x^3+2*x^4 +(1-6*x+10*x^2-4*x^3)*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019

CoefficientList[Series[(1-8*x+20*x^2-16*x^3+2*x^4 +(1-6*x+10*x^2-4*x^3) *Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)

my(x='x+O('x^30)); Vec((1-8*x+20*x^2-16*x^3+2*x^4 +(1-6*x+10*x^2 -4*x^3)*sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019

((1-8*x+20*x^2-16*x^3+2*x^4 +(1-6*x+10*x^2-4*x^3)*sqrt(1-4*x))/2 ).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

O.g.f.: 1/c(x)^8 = P(9, x) - x*P(8, x)*c(x) with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(9, x)= 1-7*x+15*x^2-10*x^3+x^4 and P(8, x)=1-6*x+10*x^2-4*x^3.

a(n) = -C8(n-8), n>=8, with C8(n) = A003518(n+3) (eighth convolution of Catalan numbers). a(0)=1, a(1)=-8, a(2)=30, a(3)=-16, a(4)=2, a(5)=a(6)=a(7)=0. [1, -8, 20, -16, 2] is row n=8 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.

A115145 Sixth convolution of A115140.

Original entry on oeis.org

1, -6, 9, -2, 0, 0, -1, -6, -27, -110, -429, -1638, -6188, -23256, -87210, -326876, -1225785, -4601610, -17298645, -65132550, -245642760, -927983760, -3511574910, -13309856820, -50528160150, -192113383644, -731508653106, -2789279908316, -10649977831752, -40715807302800

Offset: 0

Wolfdieter Lang, Jan 13 2006

Seiichi Manyama, Table of n, a(n) for n = 0..1669

Cf. A115139, A115140, A115141, A115142, A115143.

Cf. A115144, A115146, A115147, A115148, A115149.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2)*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019

CoefficientList[Series[(1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2)*Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)

my(x='x+O('x^30)); Vec((1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2) *sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019

((1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2)*sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

O.g.f.: 1/c(x)^6 = P(7, x) - x*P(6, x)*c(x) with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(7, x)=1-5*x+6*x^2-x^3 and P(6, x) = 1-4*x+3*x^2.
a(n) = -C6(n-6), n>=6, with C6(n) = A003517(n+2) (sixth convolution of Catalan numbers). a(0)=1, a(1)=-6, a(2)=9, a(3)=-2, a(4)=0=a(5). [1, -6, 9, -2] is row n=6 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
D-finite with recurrence +n*(n-6)*a(n) -2*(2*n-7)*(n-4)*a(n-1)=0. - R. J. Mathar, Sep 23 2021

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A215493 a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3) with a(0)=0, a(1)=1, a(2)=4.

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A215494 a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3) with a(1)=7, a(2)=21, a(3)=70.

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A115147 Eighth convolution of A115140.

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A115145 Sixth convolution of A115140.

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A215493 a(n) = 7a(n-1) - 14a(n-2) + 7*a(n-3) with a(0)=0, a(1)=1, a(2)=4.

A215494 a(n) = 7a(n-1) - 14a(n-2) + 7*a(n-3) with a(1)=7, a(2)=21, a(3)=70.