The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A048635

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A048635

Number of rational points of Klein curve over GF(2^n).
(history; published version)

#18 by Jon E. Schoenfield at Sat Mar 07 00:49:23 EST 2020

STATUS

proposed

approved

#17 by Jon E. Schoenfield at Sat Mar 07 00:07:11 EST 2020

STATUS

editing

proposed

#16 by Jon E. Schoenfield at Sat Mar 07 00:07:09 EST 2020

FORMULA

From Colin Barker, Aug 01 2013: (Start)

a(n) = 4*a(n-1) - 7*a(n-2) + 8*a(n-3) - 4*a(n-4). G.f.: 2*x^2*(8*x^2-16*x+7) / ((x-1)*(2*x-1)*(2*x^2-x+1)). - _Colin Barker_, Aug 01 2013

G.f.: 2*x^2*(8*x^2-16*x+7) / ((x-1)*(2*x-1)*(2*x^2-x+1)). (End)

AUTHOR

N. J. A. Sloane.

STATUS

approved

editing

#15 by Harvey P. Dale at Tue May 09 14:57:53 EDT 2017

STATUS

editing

approved

#14 by Harvey P. Dale at Tue May 09 14:57:49 EDT 2017

MATHEMATICA

LinearRecurrence[{4, -7, 8, -4}, {0, 14, 24, 14}, 40] (* Harvey P. Dale, May 09 2017 *)

STATUS

approved

editing

#13 by Charles R Greathouse IV at Sat Jun 13 00:50:00 EDT 2015

LINKS

<a href="/index/Rec">Index to sequences with entries for linear recurrences with constant coefficients</a>, signature (4,-7,8,-4).

Discussion

Sat Jun 13

00:50

OEIS Server: https://oeis.org/edit/global/2439

#12 by Charles R Greathouse IV at Fri Jun 12 15:24:46 EDT 2015

LINKS

<a href="/index/Rea#recLCCRec">Index to sequences with linear recurrences with constant coefficients</a>, signature (4,-7,8,-4).

Discussion

Fri Jun 12

15:24

OEIS Server: https://oeis.org/edit/global/2436

#11 by Michael Somos at Sun Nov 09 10:50:20 EST 2014

STATUS

editing

approved

#10 by Michael Somos at Sun Nov 09 10:49:49 EST 2014

REFERENCES

N. Elkies, The Klein quartic in number theory, pp. 51-101 of S. Levy, ed., The Eightfold Way, Cambridge Univ. Press, 1999. See p. 77 eq. (3.13), (3.14).

LINKS

<a href="/index/Rea#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (4,-7,8,-4).

FORMULA

a(n) = 2^n + 1 - 3*(a^n + b^n), where a, b are roots of X^2 - X + 2 = 0.

Conjecture: a(n) = 4*a(n-1) - 7*a(n-2) + 8*a(n-3) - 4*a(n-4). G.f.: 2*x^2*(8*x^2-16*x+7) / ((x-1)*(2*x-1)*(2*x^2-x+1)). - Colin Barker, Aug 01 2013

EXAMPLE

G.f. = 14*x^2 + 24*x^3 + 14*x^4 + 38*x^6 + 168*x^7 + 350*x^8 + 528*x^9 + ...

PROG

(PARI) {a(n) = if( n<1, 0, 2^n + 1 - 3 * polsym(x^2 - x + 2, n)[n+1])}; /* Michael Somos, Nov 09 2014 */

CROSSREFS

Cf. A002249.

STATUS

approved

editing

Discussion

Sun Nov 09

10:50

Michael Somos: Added more info. Light edits. The conjecture is true.

#9 by T. D. Noe at Thu Aug 01 16:23:13 EDT 2013

STATUS

editing

approved