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 1-5 of 5 results.

A164539 a(n) = 2*a(n-1) + 7*a(n-2) for n > 1; a(0) = 1, a(1) = 13.

Original entry on oeis.org

1, 13, 33, 157, 545, 2189, 8193, 31709, 120769, 463501, 1772385, 6789277, 25985249, 99495437, 380887617, 1458243293, 5582699905, 21373102861, 81825105057, 313261930141, 1199299595681, 4591432702349, 17577962574465, 67295954065373
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009

Keywords

Comments

Binomial transform of A164675. Inverse binomial transform of A164540.
Pisano period lengths: 1, 1, 8, 1, 24, 8, 3, 2, 24, 24, 15, 8, 168, 3, 24, 2, 4, 24, 120, 24, ... - R. J. Mathar, Aug 10 2012

Links

Crossrefs

Cf. A164675, A164540.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+3*r)*(1+2*r)^n+(1-3*r)*(1-2*r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 20 2009
  • Mathematica
    LinearRecurrence[{2,7},{1,13},50] (* Harvey P. Dale, Oct 16 2011 *)

Formula

a(n) = 2*a(n-1) + 7*a(n-2) for n > 1; a(0) = 1, a(1) = 13.
G.f.: (1+11*x)/(1-2*x-7*x^2).
a(n) = ((1+3*sqrt(2))*(1+2*sqrt(2))^n + (1-3*sqrt(2))*(1-2*sqrt(2))^n)/2.
a(n) = 11*A015519(n) + A015519(n+1). - R. J. Mathar, Aug 10 2012

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 20 2009

A164540 a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 14.

Original entry on oeis.org

1, 14, 60, 296, 1424, 6880, 33216, 160384, 774400, 3739136, 18054144, 87173120, 420909056, 2032328704, 9812951040, 47381118976, 228776280064, 1104629596160, 5333623504896, 25753012404224, 124346543636480, 600398224162816
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009

Keywords

Comments

Binomial transform of A164539. Second binomial transform of A164675. Inverse binomial transform of A164541.

Links

Crossrefs

Cf. A164539, A164675, A164541.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+3*r)*(2+2*r)^n+(1-3*r)*(2-2*r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 20 2009
  • Mathematica
    LinearRecurrence[{4,4},{1,14},30] (* Harvey P. Dale, Jul 18 2024 *)

Formula

a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 14.
G.f.: (1+10*x)/(1-4*x-4*x^2).
a(n) = ((1+3*sqrt(2))*(2+2*sqrt(2))^n + (1-3*sqrt(2))*(2-2*sqrt(2))^n)/2.

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 20 2009

A164541 a(n) = 6*a(n-1) - a(n-2) for n > 1; a(0) = 1, a(1) = 15.

Original entry on oeis.org

1, 15, 89, 519, 3025, 17631, 102761, 598935, 3490849, 20346159, 118586105, 691170471, 4028436721, 23479449855, 136848262409, 797610124599, 4648812485185, 27095264786511, 157922776233881, 920441392616775
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009

Keywords

Comments

Binomial transform of A164540. Third binomial transform of A164675. Inverse binomial transform of A164542.

Links

Crossrefs

Cf. A164540, A164675, A164542.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+3*r)*(3+2*r)^n+(1-3*r)*(3-2*r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 20 2009
  • Mathematica
    LinearRecurrence[{6,-1},{1,15},20] (* Harvey P. Dale, Feb 04 2023 *)

Formula

a(n) = 6*a(n-1) - a(n-2) for n > 1; a(0) = 1, a(1) = 15.
G.f: (1+9*x)/(1-6*x+x^2).
a(n) = ((1+3*sqrt(2))*(3+2*sqrt(2))^n + (1-3*sqrt(2))*(3-2*sqrt(2))^n)/2.

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 20 2009

A164542 a(n) = 8*a(n-1) - 8*a(n-2) for n > 1; a(0) = 1, a(1) = 16.

Original entry on oeis.org

1, 16, 120, 832, 5696, 38912, 265728, 1814528, 12390400, 84606976, 577732608, 3945005056, 26938179584, 183945396224, 1256057733120, 8576898695168, 58566727696384, 399918632009728, 2730815234506752, 18647172819976192
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009

Keywords

Comments

Binomial transform of A164541. Fourth binomial transform of A164675. Inverse binomial transform of A164543.

Links

Crossrefs

Cf. A164541, A164675, A164543.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+3*r)*(4+2*r)^n+(1-3*r)*(4-2*r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 20 2009
  • Mathematica
    LinearRecurrence[{8,-8},{1,16},30] (* Harvey P. Dale, Jul 23 2018 *)

Formula

a(n) = 8*a(n-1) - 8*a(n-2) for n > 1; a(0) = 1, a(1) = 16.
G.f.: (1+8*x)/(1-8*x+8*x^2).
a(n) = ((1+3*sqrt(2))*(4+2*sqrt(2))^n + (1-3*sqrt(2))*(4-2*sqrt(2))^n)/2.

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 20 2009

A164543 a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 17.

Original entry on oeis.org

1, 17, 153, 1241, 9809, 76993, 603177, 4722889, 36974881, 289459697, 2266023993, 17739425081, 138871842929, 1087148202913, 8510660699337, 66625087543849, 521569643549761, 4083069947252177, 31964015532175833, 250227966218471321
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009

Keywords

Comments

Binomial transform of A164542. Fifth binomial transform of A164675.

Links

Crossrefs

Cf. A164542, A164675.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+3*r)*(5+2*r)^n+(1-3*r)*(5-2*r)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 20 2009

Formula

a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 17.
G.f.: (1+7*x)/(1-10*x+17*x^2).
a(n) = ((1+3*sqrt(2))*(5+2*sqrt(2))^n + (1-3*sqrt(2))*(5-2*sqrt(2))^n)/2.

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 20 2009
Showing 1-5 of 5 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.