cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A215948 a(n) = 3^n*A(2*n), where A(n) = 3*A(n-1) + A(n-2) - A(n-3)/3 with A(0)=A(1)=3, A(2)=11.

Original entry on oeis.org

3, 33, 1035, 33273, 1070163, 34420113, 1107069147, 35607149289, 1145248319907, 36835122733569, 1184744167018155, 38105444942752473, 1225602095969542131, 39419576386041628017, 1267869080483024344443, 40779027899804588036553, 1311593714249667872790339
Offset: 0

Views

Author

Roman Witula, Aug 28 2012

Keywords

Comments

The Berndt-type sequence number 12 for the argument 2*Pi/9 defined by the first trigonometric relations from the section "Formula" below (it is the complement of the sequence A215945). For more information see comments to A215945. We note that all a(n)/3 and 3^(-1 + floor((n+3)/3))*A(n) = A216034(n) are integers.

Examples

			We have t(1)^4 + t(2)^4 + t(4)^4 = 1035 = (345/11)*(t(1)^2 + t(2)^2 + t(4)^2) and (1 - 4*s(1)/sqrt(3))^4 + (1 + 4*s(2)/sqrt(3))^4 + (1 - 4*s(4)/sqrt(3))^4 = 115. Moreover we get a(2)/a(1) = 31,(36), a(3)/a(1) = 1008,(27), a(4)/a(1) = 32429,(18).

References

  • D. Chmiela and R. Witula, Two parametric quasi-Fibonacci numbers of the ninth order, (submitted, 2012).
  • R. Witula, Ramanujan type formulas for arguments 2Pi/7 and 2Pi/9, Demonstratio Math. (in press, 2012).

Links

Crossrefs

Cf. A215945, A216034, A215829, A215794, A215575.

Programs

  • Mathematica
    LinearRecurrence[{33,-27,3}, {3,33,1035}, 50]

Formula

a(n) = t(1)^(2*n) + t(2)^(2*n) + t(4)^(2*n) = (-sqrt(3) + 4*s(1))^(2*n) + (sqrt(3) + 4*s(2))^(2*n) + (-sqrt(3) + 4*s(4))^(2*n), where t(j) := tan(2*Pi*j/9) and s(j) := sin(2*Pi*j/9). For the respective sums of odd powers - see A215945.
a(n) = 33*a(n-1) - 27*a(n-2) + 3*a(n-3).
G.f.: 3*(1-22*x+9*x^2)/(1-33*x+27*x^2-3*x^3).
a(n) = cot(Pi/18)^(2*n) + cot(5*Pi/18)^(2*n) + cot(7*Pi/18)^(2*n). - Greg Dresden, Oct 01 2020

A275195 Sum of n-th powers of the roots of x^3 - 7*x^2 - 49*x - 49.

Original entry on oeis.org

3, 7, 147, 1519, 18179, 208887, 2427411, 28118111, 326005379, 3778768231, 43803428627, 507757907279, 5885832996995, 68227336438359, 790877309377939, 9167686467977919, 106269932920844035, 1231859155536345287, 14279457438806692755, 165524527406049126063, 1918726204965152746499
Offset: 0

Views

Author

Kai Wang, Jul 19 2016

Keywords

Comments

a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial x^3 - 7*x^2 - 49*x -49.
x1 = -sqrt(7)*tan(Pi/7),
x2 = -sqrt(7)*tan(2*Pi/7),
x3 = -sqrt(7)*tan(4*Pi/7).
a(2n) = A108716(n)* (7^n),
a(2n+1) = -A215794(n)* 7^(n+1).

Links

Crossrefs

Cf. A108716, A215794.

Programs

  • Mathematica
    CoefficientList[Series[(-49 x^2 - 14 x + 3)/(-49 x^3 - 49 x^2 - 7 x + 1), {x, 0, 20}], x] (* Michael De Vlieger, Jul 19 2016 *)
LinearRecurrence[{7,49,49},{3,7,147},30] (* Harvey P. Dale, Jan 01 2023 *)
  • PARI
    terms(n) = my(a=3, b=7, c=147, d=0, i=0); while(i < n, if(i==0, print1(a, ", "); i++, if(i==1, print1(b, ", "); i++, if(i==2, print1(c, ", "); i++, while(i < n, d=7*c + 49*b + 49*a; print1(d, ", "); a=b; b=c; c=d; i++)))))
/* Call function as follows to print initial 10 terms: */
terms(10) \\ Felix Fröhlich, Jul 19 2016
  • PARI
    polsym(x^3 - 7*x^2 - 49*x - 49, 30) \\ Charles R Greathouse IV, Jul 20 2016
  • PARI
    Vec((1-7*x)*(3+7*x)/(1-7*x-49*x^2-49*x^3) + O(x^30)) \\ Colin Barker, Jul 23 2016

Formula

G.f.: (-49*x^2-14*x+3)/(-49*x^3-49*x^2-7*x+1).
a(n) = (-sqrt(7)*tan(Pi/7))^n + (-sqrt(7)*tan(2*Pi/7))^n + (-sqrt(7)*tan(4*Pi/7))^n.
a(0)=3, a(1)=7, a(2)=147; thereafter a(n) = 7*a(n-1) + 49*a(n-2) + 49*a(n-3).

A216034 a(n) = 3^(-1+floor((n+1)/3))*A(n), where A(n) = 3*A(n-1) + A(n-2) - A(n-3)/3 with A(0)=A(1)=3, A(2)=11.

Original entry on oeis.org

1, 1, 11, 35, 115, 1129, 3697, 12105, 118907, 389339, 1274819, 12522481, 41002561, 134255345, 1318783307, 4318113395, 14138868147, 138885370201, 454754601649, 1489010307001, 14626471197755, 47891689912619, 156812530628611, 1540361374197601
Offset: 0

Views

Author

Roman Witula, Aug 30 2012

Keywords

Comments

The Berndt-type sequence number 11a for the argument 2Pi/9 - see A215945 for more details.

Links

Crossrefs

Cf. A215945, A215948, A215829, A215794, A215575.

Programs

  • Magma
    i:=24; I:=[3,3,11]; A:=[m le 3 select I[m] else 3*Self(m-1)+Self(m-2)-Self(m-3)/3: m in [1..i]]; [3^(-1+Floor(n/3))*A[n]: n in [1..i]]; // Bruno Berselli, Oct 02 2012

Formula

G.f.: (1+x+11*x^2-70*x^3+10*x^4-26*x^5-11*x^6-3*x^7-x^8)/(1-105*x^3-33*x^6+x^9). [Bruno Berselli, Oct 02 2012]

A275830 a(n) = (2*sqrt(7)*sin(Pi/7))^n + (-2*sqrt(7)*sin(2*Pi/7))^n + (-2*sqrt(7)*sin(4*Pi/7))^n.

Original entry on oeis.org

3, -7, 49, -196, 1029, -4802, 24010, -117649, 588245, -2941225, 14823774, -74942413, 380476866, -1936973136, 9886633715, -50563069571, 259029803333, -1328763571296, 6823754590093, -35073821767334, 180407337377834, -928487386730281, 4780794440512601, -24625601552074341, 126883328914736618
Offset: 0

Views

Author

Kai Wang, Aug 11 2016

Keywords

Comments

2*sqrt(7)*sin(Pi/7), -2*sqrt(7)*sin(2*Pi/7) and -2*sqrt(7)*sin(4*Pi/7) are roots of polynomial x^3 + 7*x^2 - 49.

Links

Crossrefs

Cf. A108716, A215794, A275195, A274975, A275831.

Programs

  • Mathematica
    RecurrenceTable[{a[0] == 3, a[1] == -7, a[2] == 49, a[n] == -7 a[n - 1] + 49 a[n - 3]}, a, {n, 0, 30}] (* Bruno Berselli, Aug 11 2016 *)
  • PARI
    Vec((3 + 14*x)/(1 + 7*x - 49*x^3) + O(x^30)) \\ Colin Barker, Aug 30 2016

Formula

G.f.: (3 + 14*x)/(1 + 7*x - 49*x^3). - Bruno Berselli, Aug 11 2016
a(n) = -7*a(n-1) + 49*a(n-3) with n>2, a(0)=3, a(1)=-7, a(2)=49.
a(2*n-1) = 7^n*A215493(n). - Kai Wang, May 25 2017

A217444 a(n) = A(n)*7^(-floor(n+1)/3), where A(n) = 7*A(n-1) - 14*A(n-2) + 7*A(n-3) with A(0)=0, A(1)=1, A(2)=7.

Original entry on oeis.org

0, 1, 1, 5, 22, 13, 52, 204, 113, 435, 1667, 910, 3471, 13224, 7192, 27367, 104105, 56563, 215098, 817909, 444276, 1689212, 6422529, 3488381, 13262821, 50424942, 27387681, 104126704, 395884336, 215018609, 817488295, 3108041875, 1688083894, 6417991803, 24400809980
Offset: 0

Views

Author

Roman Witula, Oct 03 2012

Keywords

Comments

The Berndt-type sequence number 18a for the argument 2Pi/7, which is closely connected with the sequence A217274. Definitions other Berndt-type sequences for the argument 2Pi/7 like A215575, A215877, A033304 in sequences from Crossrefs are given.

Links

Crossrefs

Cf. A215007, A215008, A215143, A215493, A215494, A215510, A215512, A215575, A215694, A215695, A108716, A215794, A215828, A215817, A215877, A094429, A094430, A217274, A094648, A033304.

Programs

  • Magma
    i:=35; I:=[0, 1, 7]; A:=[m le 3 select I[m] else 7*Self(m-1)-14*Self(m-2)+7*Self(m-3): m in [1..i]]; [7^(-Floor(n/3))*A[n]: n in [1..i]]; // Bruno Berselli, Oct 03 2012
  • Mathematica
    CoefficientList[Series[x*(1+x+5*x^2+12*x^3+3*x^4+2*x^5+x^6)/(1 - 10*x^3 + 17*x^6 - x^9), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 15 2012 *)
LinearRecurrence[{0,0,10,0,0,-17,0,0,1}, {0, 1, 1, 5, 22, 13, 52, 204, 113}, 50] (* G. C. Greubel, Apr 23 2018 *)
  • PARI
    x='x+O('x^30); concat([0], Vec(x*(1+x+5*x^2+12*x^3+3*x^4 +2*x^5 +x^6)/(1- 10*x^3+17*x^6-x^9))) \\ G. C. Greubel, Apr 23 2018

Formula

G.f.: x*(1+x+5*x^2+12*x^3+3*x^4+2*x^5+x^6)/(1-10*x^3+17*x^6-x^9). - Bruno Berselli, Oct 03 2012
a(n) = 10*a(n-3) - 17*a(n-6) + a(n-9). - G. C. Greubel, Apr 23 2018
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