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.

A164603 a(n) = ((1+4*sqrt(2))*(2+2*sqrt(2))^n + (1-4*sqrt(2))*(2-2*sqrt(2))^n)/2.

Original entry on oeis.org

1, 18, 76, 376, 1808, 8736, 42176, 203648, 983296, 4747776, 22924288, 110688256, 534450176, 2580553728, 12460015616, 60162277376, 290489171968, 1402605797376, 6772379877376, 32699942699008, 157889290305536
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

Keywords

Comments

Binomial transform of A164602. Second binomial transform of A164702. Inverse binomial transform of A164604.

Links

Crossrefs

Cf. A164602, A164702, A164604.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(2+2*r)^n+(1-4*r)*(2-2*r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 23 2009
  • Mathematica
    CoefficientList[Series[(-1-14 n)/(-1+4 n+4 n^2),{n,0,20}],n]  (* Harvey P. Dale, Feb 22 2011 *)
  • PARI
    Vec((1+14*x)/(1-4*x-4*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jun 12 2011

Formula

a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 18.
G.f.: (1+14*x)/(1-4*x-4*x^2).
E.g.f.: exp(2*x)*( cosh(2*sqrt(2)*x) + 4*sqrt(2)*sinh(2*sqrt(2)*x) ). - G. C. Greubel, Aug 11 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 23 2009

A164604 a(n) = ((1+4*sqrt(2))*(3+2*sqrt(2))^n + (1-4*sqrt(2))*(3-2*sqrt(2))^n)/2.

Original entry on oeis.org

1, 19, 113, 659, 3841, 22387, 130481, 760499, 4432513, 25834579, 150574961, 877615187, 5115116161, 29813081779, 173763374513, 1012767165299, 5902839617281, 34404270538387, 200522783613041, 1168732431139859
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

Keywords

Comments

Binomial transform of A164603. Third binomial transform of A164702. Inverse binomial transform of A164605.
From Klaus Purath, Mar 14 2024: (Start)
For any two consecutive terms (a(n), a(n+1)) = (x,y): x^2 - 6xy + y^2 = 248 = A028884(13). In general, the following applies to all recursive sequences (t) with constant coefficients (6,-1) and t(0) = 1 and two consecutive terms (x,y): x^2 - 6xy + y^2 = A028884(t(1)-6). This includes and interprets the Feb 04 2014 comment on A001541 by Colin Barker as well as the Mar 17 2021 comment on A054489 by John O. Oladokun.
By analogy to this, for three consecutive terms (x,y,z) of any recursive sequence (t) of form (6,-1) with t(0) = 1: y^2 - xz = A028884(t(1)-6). (End)

Links

Crossrefs

Cf. A164603, A164702, A164605.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(3+2*r)^n+(1-4*r)*(3-2*r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 23 2009
  • Mathematica
    LinearRecurrence[{6,-1}, {1,19}, 50] (* G. C. Greubel, Aug 11 2017 *)
  • PARI
    Vec((1+13*x)/(1-6*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Jun 12 2011

Formula

a(n) = 6*a(n-1) - a(n-2) for n > 1; a(0) = 1, a(1) = 19.
G.f.: (1+13*x)/(1-6*x+x^2).
E.g.f.: exp(3*x)*( cosh(2*sqrt(2)*x) + 4*sqrt(2)*sinh(2*sqrt(2)*x) ). - G. C. Greubel, Aug 11 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 23 2009

A164605 a(n) = ((1+4*sqrt(2))*(4+2*sqrt(2))^n + (1-4*sqrt(2))*(4-2*sqrt(2))^n)/2.

Original entry on oeis.org

1, 20, 152, 1056, 7232, 49408, 337408, 2304000, 15732736, 107429888, 733577216, 5009178624, 34204811264, 233565061120, 1594881998848, 10890535501824, 74365228023808, 507797540175872, 3467458497216512, 23677287656325120
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

Keywords

Comments

Binomial transform of A164604. Fourth binomial transform of A164702. Inverse binomial transform of A164606.

Links

Crossrefs

Cf. A164604, A164702, A164606.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(4+2*r)^n+(1-4*r)*(4-2*r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 23 2009
  • Mathematica
    LinearRecurrence[{8,-8},{1,20},30] (* Harvey P. Dale, Mar 24 2015 *)
  • PARI
    Vec((1+12*x)/(1-8*x+8*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jun 12 2011

Formula

a(n) = 8*a(n-1) - 8*a(n-2) for n > 1; a(0) = 1, a(1) = 20.
G.f.: (1+12*x)/(1-8*x+8*x^2).
E.g.f.: exp(4*x)*(cosh(2*sqrt(2)*x) + 4*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 10 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 23 2009

A164606 a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 21.

Original entry on oeis.org

1, 21, 193, 1573, 12449, 97749, 765857, 5996837, 46948801, 367541781, 2877288193, 22524671653, 176332817249, 1380408754389, 10806429650657, 84597347681957, 662264172758401, 5184486816990741, 40586377233014593, 317727496441303333
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

Keywords

Comments

Binomial transform of A164605. Fifth binomial transform of A164702.

Links

Crossrefs

Cf. A164605, A164702.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(5+2*r)^n+(1-4*r)*(5-2*r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 23 2009
  • Mathematica
    LinearRecurrence[{10,-17},{1,21},30] (* Harvey P. Dale, May 22 2013 *)
  • PARI
    x='x+O('x^50); Vec((1+11*x)/(1-10*x+17*x^2)) \\ G. C. Greubel, Aug 10 2017

Formula

a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 21.
a(n) = ((1+4*sqrt(2))*(5+2*sqrt(2))^n + (1-4*sqrt(2))*(5-2*sqrt(2))^n)/2.
G.f.: (1+11*x)/(1-10*x+17*x^2).
E.g.f.: exp(5*x)*(cosh(2*sqrt(2)*x) + 4*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 10 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 23 2009
Showing 1-4 of 4 results.

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