A002249 -id:A002249 - cp's OEIS frontend

A105580 a(n+3) = a(n) - a(n+1) - a(n+2); a(0) = -5, a(1) = 6, a(2) = 0.

-5, 6, 0, -11, 17, -6, -22, 45, -29, -38, 112, -103, -47, 262, -318, 9, 571, -898, 336, 1133, -2367, 1570, 1930, -5867, 5507, 2290, -13664, 16881, -927, -29618, 47426, -18735, -58309, 124470, -84896, -97883, 307249, -294262, -110870, 712381, -895773, 72522, 1535632, -2503927, 1040817, 2998742

Offset: 0

Creighton Dement, Apr 14 2005

Floretion Algebra Multiplication Program, FAMP Code: 2tesforseq[.5'j + .5'k + .5j' + .5k' + .5'ii' + .5e], 1vesforseq = A000004, ForType: 1A.

			This sequence was generated using the same floretion which generated the sequences A105577, A105578, A105579, etc.. However, in this case a force transform was applied. [Specifically, (a(n)) may be seen as the result of a tesfor-transform of the zero-sequence A000004 with respect to the floretion given in the program code.]

Index entries for linear recurrences with constant coefficients, signature (-1,-1,1).

Cf. A002249, A014551, A078020, A105577, A105578, A105581.

Transpose[NestList[Join[Rest[#],ListCorrelate[ {1,-1,-1}, #]]&,{-5,6,0},50]][[1]]  (* Harvey P. Dale, Mar 14 2011 *)
CoefficientList[Series[(5-x-x^2)/(x^3-x^2-x-1),{x,0,50}],x]  (* Harvey P. Dale, Mar 14 2011 *)

G.f. (5-x-x^2)/(x^3-x^2-x-1)
a(n) = A078046(n-1) - A073145(n+3).
a(n) = -5*A057597(n+2) + A057597(n+1)+A057597(n). - R. J. Mathar, Oct 25 2022

A105225 a(n+3) = 2a(n+2) - 3a(n+1) + 2a(n); a(0) = 1, a(1) = -1, a(2) = -2.

Original entry on oeis.org

1, -1, -2, 1, 6, 5, -6, -15, -2, 29, 34, -23, -90, -43, 138, 225, -50, -499, -398, 601, 1398, 197, -2598, -2991, 2206, 8189, 3778, -12599, -20154, 5045, 45354, 35265, -55442, -125971, -15086, 236857, 267030, -206683, -740742, -327375, 1154110, 1808861, -499358, -4117079, -3118362, 5115797

Offset: 0

Creighton Dement, Apr 14 2005

Floretion Algebra Multiplication Program, FAMP Code: 2tesseq[.5'j + .5'k + .5j' + .5k' + .5'ii' + .5e]

Robert Munafo, Sequences Related to Floretions
Index entries for linear recurrences with constant coefficients, signature (2, -3, 2).

Cf. A002249, A014551.

LinearRecurrence[{2,-3,2},{1,-1,-2},50] (* or *) CoefficientList[ Series[ (-3*x^2+3*x-1)/(2*x^3-3*x^2+2*x-1),{x,0,50}],x] (* Harvey P. Dale, Jul 23 2012 *)

a(n) - a(n+1) = A002249(n).
a(n) = (A002249(n+1) + 1)/2.
From Harvey P. Dale, Jul 23 2012: (Start)
G.f.: -(3*x^2-3*x+1)/((x-1)*(2*x^2-x+1)).
a(n)=1/2*(1+(1/2*(1-I*Sqrt[7]))^n+(1/2*(1+I*Sqrt[7]))^n). (End)

A048635 Number of rational points of Klein curve over GF(2^n).

Original entry on oeis.org

0, 14, 24, 14, 0, 38, 168, 350, 528, 854, 1848, 4238, 8736, 16646, 31944, 64190, 131376, 265142, 526680, 1044974, 2088768, 4193126, 8404200, 16795166, 33541200, 67059734, 134195064, 268511054, 536991840, 1073711558, 2147211528

Offset: 1

N. J. A. Sloane

nonn

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

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).

Index entries for linear recurrences with constant coefficients, signature (4,-7,8,-4).

Cf. A002249.

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

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

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

A307988 T(n, k) the number of A-polynomials in F_2^k[T] of degree n, array read by descending antidiagonals.

Original entry on oeis.org

1, 2, 1, 1, 2, 0, 4, 7, 4, 1, 11, 36, 42, 18, 2, 14, 121, 344, 259, 48, 2, 29, 518, 2750, 4068, 1652, 172, 4, 72, 2059, 21924, 65461, 52368, 10962, 588, 9, 127, 8136, 174986, 1048950, 1677940, 699288, 74998, 2034, 14, 242, 32893, 1398576, 16778791, 53686584, 44738782, 9587880, 524475, 7308, 24

Offset: 1

Michel Marcus, May 22 2019

			Array begins:
1   2     1       4         11           14             29
1   2     7      36        121          518           2059
0   4    42     344       2750        21924         174986
1  18   259    4068      65461      1048950       16778791
2  48  1652   52368    1677940     53686584     1717985404
2 172 10962  699288   44738782   2863291620   183251786538
4 588 74998 9587880 1227132434 157072960476 20105353937606

Alp Bassa, Ricardo Menares, Enumeration of a special class of irreducible polynomials in characteristic 2, arXiv:1905.08345 [math.NT], 2019.
Harald Niederreiter, An enumeration formula for certain irreducible polynomials with an application to the construction of irreducible polynomials over the binary field, Applicable Algebra in Engineering, Communication and Computing, vol.1, no.2, pp.119-124, (September-1990).

Cf. A175390 (1st column).

Cf. A002249 or A077021 (sequences related to alpha).

f(n) = 2 * real(((-1 + quadgen(-28)) / 2)^n);
a(n, r) = {my(k = valuation(n, 2), m = n/2^k, q = 2^r); sumdiv(m, d, moebius(m/d)*(q^(2^k*d)+1-f(r*2^k*d)))/(4*n);}

T(n, k) = Sum_{d|n} moebius(m/d)*q^(2^k*d) + 1 - alpha^(r*2^k*d) - alphabar^(r*2^k*d), where n = 2^k*m, m odd, alpha = (-1+sqrt(-7))/2 and alphabar = (-1-sqrt(-7))/2 is the conjugate of alpha.

cp's OEIS Frontend

A105580 a(n+3) = a(n) - a(n+1) - a(n+2); a(0) = -5, a(1) = 6, a(2) = 0.

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A105225 a(n+3) = 2a(n+2) - 3a(n+1) + 2a(n); a(0) = 1, a(1) = -1, a(2) = -2.

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A048635 Number of rational points of Klein curve over GF(2^n).

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A307988 T(n, k) the number of A-polynomials in F_2^k[T] of degree n, array read by descending antidiagonals.

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Author

Keywords

Examples

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Programs

PARI

Formula