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.

Previous Showing 11-14 of 14 results.

A117865 Number of palindromes (in base 6) below 6^n.

Original entry on oeis.org

5, 10, 40, 70, 250, 430, 1510, 2590, 9070, 15550, 54430, 93310, 326590, 559870, 1959550, 3359230, 11757310, 20155390, 70543870, 120932350, 423263230, 725594110, 2539579390, 4353564670, 15237476350, 26121388030, 91424858110, 156728328190, 548549148670
Offset: 1

Views

Author

Martin Renner, May 02 2006

Keywords

Crossrefs

Cf. A050250.

Programs

  • Mathematica
    Table[If[OddQ[n], 7*6^((n - 1)/2) - 2, 2*6^(n/2) - 2], {n,25}] (* or *) LinearRecurrence[{1,6,-6},{5, 10, 40},25] (* G. C. Greubel, Oct 27 2016 *)

Formula

a(n) = 7*6^((n-1)/2)-2 (n odd), 2*6^(n/2)-2 (n even).
G.f.: 5*x*(x+1) / ((x-1)*(6*x^2-1)). - Colin Barker, Feb 15 2013

Extensions

More terms from Colin Barker, Feb 15 2013

A117866 Number of palindromes (in base 7) below 7^n.

Original entry on oeis.org

6, 12, 54, 96, 390, 684, 2742, 4800, 19206, 33612, 134454, 235296, 941190, 1647084, 6588342, 11529600, 46118406, 80707212, 322828854, 564950496, 2259801990, 3954653484, 15818613942, 27682574400, 110730297606, 193778020812, 775112083254, 1356446145696
Offset: 1

Views

Author

Martin Renner, May 02 2006

Keywords

Crossrefs

Cf. A050250.

Programs

  • Magma
    [IsOdd(n) select 8*7^((n-1) div 2)-2 else 2*7^(n div 2)-2: n in [1..30]]; // Vincenzo Librandi, Oct 29 2016
    
  • Mathematica
    Table[If[OddQ[n],8*7^((n-1)/2)-2,2*7^(n/2)-2],{n,30}] (* or *) LinearRecurrence[{1,7,-7},{6,12,54},30] (* Harvey P. Dale, Oct 31 2013 *)
    Rest@ CoefficientList[Series[6 x (x + 1)/((x - 1) (7 x^2 - 1)), {x, 0, 28}], x] (* Michael De Vlieger, Oct 31 2016 *)
  • PARI
    a(n)=([0,1,0; 0,0,1; -7,7,1]^(n-1)*[6;12;54])[1,1] \\ Charles R Greathouse IV, Oct 31 2016

Formula

a(n) = 8*7^((n-1)/2)-2 (n odd), 2*7^(n/2)-2 (n even).
G.f.: 6*x*(x+1) / ((x-1)*(7*x^2-1)). - Colin Barker, Feb 15 2013
From G. C. Greubel, Oct 27 2016: (Start)
a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3).
a(n) = (1/sqrt(7))*(-2*sqrt(7) + 7^((n+1)/2) + (-1)^n*7^((n+1)/2) + 4*7^(n/2) - 4*(-1)^n*7^(n/2)).
E.g.f.: (1/sqrt(7))*( (sqrt(7) - 4)*exp(-sqrt(7)*x) + (4 + sqrt(7))*exp(sqrt(7)*x) - 2*sqrt(7)*exp(x)). (End)

Extensions

More terms from Colin Barker, Feb 15 2013

A117867 Number of palindromes (in base 8) below 8^n.

Original entry on oeis.org

7, 14, 70, 126, 574, 1022, 4606, 8190, 36862, 65534, 294910, 524286, 2359294, 4194302, 18874366, 33554430, 150994942, 268435454, 1207959550, 2147483646, 9663676414, 17179869182, 77309411326, 137438953470, 618475290622, 1099511627774, 4947802324990
Offset: 1

Views

Author

Martin Renner, May 02 2006

Keywords

Crossrefs

Cf. A050250.

Programs

  • Mathematica
    Table[If[OddQ[n],9*8^((n-1)/2)-2, 2*8^(n/2)-2], {n,1,25}] (* or *) LinearRecurrence[{1,8,-8}, {7,14,70}, 25] (* G. C. Greubel, Oct 27 2016 *)
    Rest@ CoefficientList[Series[7 x (x + 1)/((x - 1) (8 x^2 - 1)), {x, 0, 27}], x] (* Michael De Vlieger, Oct 27 2016 *)

Formula

a(n) = 9*8^((n-1)/2)-2 (n odd), 2*8^(n/2)-2 (n even).
G.f.: 7*x*(x+1) / ((x-1)*(8*x^2-1)). - Colin Barker, Feb 15 2013
From G. C. Greubel, Oct 27 2016: (Start)
a(n) = a(n-1) + 8*a(n-2) - 8*a(n-3).
a(n) = (1/(4*sqrt(2)))*( -8*sqrt(2) + 9*(1 - (-1)^n)*2^(3*n/2) + (1 + (-1)^n)*2^((3*n+5)/2) ).
E.g.f.: 2*cosh(2*sqrt(2)*x) + (9/(2*sqrt(2)))*sinh(2*sqrt(2)*x) -2*cosh(x) - 2*sinh(x). (End)

Extensions

More terms from Colin Barker, Feb 15 2013

A117868 Number of palindromes (in base 9) below 9^n.

Original entry on oeis.org

8, 16, 88, 160, 808, 1456, 7288, 13120, 65608, 118096, 590488, 1062880, 5314408, 9565936, 47829688, 86093440, 430467208, 774840976, 3874204888, 6973568800, 34867844008, 62762119216, 313810596088, 564859072960, 2824295364808, 5083731656656, 25418658283288
Offset: 1

Views

Author

Martin Renner, May 02 2006

Keywords

Crossrefs

Cf. A050250.

Programs

  • Maple
    seq( 8 * 3^(n-1) + 2 * (-3)^(n-1) - 2, n=1..100); # Robert Israel, Apr 26 2015
  • Mathematica
    Table[If[OddQ[n],10*9^((n-1)/2)-2,2*9^(n/2)-2],{n,0,30}] (* or *) LinearRecurrence[ {1,9,-9},{0,8,16},30] (* Harvey P. Dale, Jul 17 2012 *)
  • PARI
    Vec(8*x*(x+1)/((x-1)*(3*x-1)*(3*x+1)) + O(x^100)) \\ Colin Barker, Apr 26 2015

Formula

a(n) = 10*9^((n-1)/2)-2 (n odd), 2*9^(n/2)-2 (n even).
a(1)=0, a(2)=8, a(3)=16, a(n)=a(n-1)+9*a(n-2)-9*a(n-3). - Harvey P. Dale, Jul 17 2012
G.f.: 8*x*(x+1) / ((x-1)*(3*x-1)*(3*x+1)). - Colin Barker, Apr 26 2015
a(n) = 8 * 3^(n-1) + 2 * (-3)^(n-1) - 2. - Robert Israel, Apr 26 2015

Extensions

More terms from Harvey P. Dale, Jul 17 2012
Previous Showing 11-14 of 14 results.