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.

Showing 1-4 of 4 results.

A163614 a(n) = ((1 + 3*sqrt(2))*(3 + sqrt(2))^n + (1 - 3*sqrt(2))*(3 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 9, 47, 219, 985, 4377, 19367, 85563, 377809, 1667913, 7362815, 32501499, 143469289, 633305241, 2795546423, 12340141851, 54472026145, 240451163913, 1061402800463, 4685258655387, 20681732329081, 91293583386777
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

Keywords

Comments

Binomial transform of A163613. Third binomial transform of A163864. Inverse binomial transform of A163615.

Links

Crossrefs

Cf. A163613, A163864, A163615.

Programs

  • Magma
    Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+3*r)*(3+r)^n+(1-3*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009
  • Mathematica
    LinearRecurrence[{6,-7},{1,9},30] (* Harvey P. Dale, Sep 24 2015 *)
  • PARI
    x='x+O('x^50); Vec((1+3*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Jul 30 2017

Formula

a(n) = 6*a(n-1)-7*a(n-2) for n > 1; a(0) = 1, a(1) = 9.
G.f.: (1+3*x)/(1-6*x+7*x^2).
E.g.f.: exp(3*x)*( cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 30 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 06 2009

A163615 a(n) = ((1 + 3*sqrt(2))*(4 + sqrt(2))^n + (1 - 3*sqrt(2))*(4 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 10, 66, 388, 2180, 12008, 65544, 356240, 1932304, 10471072, 56716320, 307135552, 1663055936, 9004549760, 48753614976, 263965223168, 1429171175680, 7737856281088, 41894453789184, 226825642378240, 1228082785977344
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

Keywords

Comments

Binomial transform of A163614. Fourth binomial transform of A163864. Inverse binomial transform of A163616.

Links

Crossrefs

Cf. A163614, A163864, A163616.

Programs

  • Magma
    Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+3*r)*(4+r)^n+(1-3*r)*(4-r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009
  • Mathematica
    LinearRecurrence[{8,-14},{1,10},30] (* Harvey P. Dale, Jun 11 2014 *)
  • PARI
    x='x+O('x^50); Vec((1+2*x)/(1-8*x+14*x^2)) \\ G. C. Greubel, Jul 30 2017

Formula

a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 1, a(1) = 10.
G.f.: (1+2*x)/(1-8*x+14*x^2).
E.g.f.: exp(4*x)*( cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 30 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 06 2009

A163613 a(n) = ((1 + 3*sqrt(2))*(2 + sqrt(2))^n + (1 - 3*sqrt(2))*(2 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 8, 30, 104, 356, 1216, 4152, 14176, 48400, 165248, 564192, 1926272, 6576704, 22454272, 76663680, 261746176, 893657344, 3051137024, 10417233408, 35566659584, 121432171520, 414595366912, 1415517124608, 4832877764608
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

Keywords

Comments

Binomial transform of A048694. Second binomial transform of A163864. Inverse binomial transform of A163614.

Links

Crossrefs

Cf. A048694, A163864 (1, 6, 2, 12, 4, 24, ...), A163614.

Programs

  • Magma
    Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+3*r)*(2+r)^n+(1-3*r)*(2-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009
  • Mathematica
    LinearRecurrence[{4, -2}, {1, 8}, 50] (* G. C. Greubel, Jul 30 2017 *)
  • PARI
    x='x+O('x^50); Vec((1+4*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Jul 30 2017

Formula

a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0) = 1, a(1) = 8.
G.f.: (1+4*x)/(1-4*x+2*x^2).
E.g.f.: exp(2*x)*( cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 30 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 06 2009

A163616 a(n) = ((1 + 3*sqrt(2))*(5 + sqrt(2))^n + (1 - 3*sqrt(2))*(5 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 11, 87, 617, 4169, 27499, 179103, 1158553, 7466161, 48014891, 308427207, 1979929577, 12705470009, 81516319819, 522937387983, 3354498523993, 21517425316321, 138020787111371, 885307088838327, 5678592784821737
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

Keywords

Comments

Binomial transform of A163615. Fifth binomial transform of A163864. Inverse binomial transform of A081183 without initial 0.

Links

Crossrefs

Cf. A163615, A163864, A081183.

Programs

  • Magma
    Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+3*r)*(5+r)^n+(1-3*r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009
  • Mathematica
    CoefficientList[Series[(1 + x)/(1 - 10 x + 23 x^2), {x, 0, 20}], x] (* Wesley Ivan Hurt, Jun 14 2014 *)
LinearRecurrence[{10, -23}, {1, 11}, 50] (* G. C. Greubel, Jul 30 2017 *)
  • PARI
    x='x+O('x^50); Vec((1+x)/(1-10*x+23*x^2)) \\ G. C. Greubel, Jul 30 2017

Formula

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 1, a(1) = 11.
G.f.: (1+x)/(1-10*x+23*x^2).
E.g.f.: exp(5*x)*( cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 30 2017
a(n) = A081182(n)+A081182(n+1). - R. J. Mathar, Jul 01 2022

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 06 2009
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.