A100051 -id:A100051 - cp's OEIS frontend

A036952 Numbers whose binary expansion is a decimal prime.

3, 5, 23, 47, 89, 101, 149, 157, 163, 173, 179, 185, 199, 229, 247, 253, 295, 313, 329, 331, 355, 367, 379, 383, 405, 425, 443, 453, 457, 471, 523, 533, 539, 565, 583, 587, 595, 631, 643, 647, 653, 659, 671, 675, 689, 703, 709, 755, 781, 785, 815, 841, 855

Offset: 1

Patrick De Geest, Jan 04 1999

A100051(f(a(n))) = 1 with f(x) = if x<2 then x else 10*f(floor(x/2)) + x mod 2. - Reinhard Zumkeller, Mar 31 2010

Primes in A007088. - N. J. A. Sloane, Feb 17 2023

			1 = 1_2 is not a prime.
2 = 10_2 is not OK because 10 = 2*5 is not a prime.
3 = 11_2 is OK because 11 is a prime.
4 = 100_2 is not OK because 100 = 4*25 is not a prime.
5 = 101_2 is OK because 101 is a prime.
7 = 111_2 is not OK because 111 = 3*37.
11 = 1011_2 is not OK because 1011 = 3*337.
313 = 100111001_2 is OK because 100111001 is prime.

Cf. A007088, A020449, A036953-A036964.

Union of A156059 and A065720.

A007088 := proc(n)
dgs := convert(n,base,2) ;
add(op(i,dgs)*10^(i-1),i=1..nops(dgs)) ;
end proc:
isA036952 := proc(n)
isprime( A007088(n)) :
end proc:
A036952 := proc(n)
if n =1 then
3;
else
for a from procname(n-1)+1 do
if isA036952(a) then
return a ;
end if;
end do:
end if;
end proc:
seq(A036952(n),n=1..80) ;
# R. J. Mathar, Mar 12 2010
A036952 := proc() if isprime(convert(n,binary)) then RETURN (n); fi; end: seq(A036952(), n=1..1000);  # K. D. Bajpai, Jul 04 2014

f[n_,k_]:=FromDigits[IntegerDigits[n,k]];lst={};Do[If[PrimeQ[f[n,2]],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 12 2010 *)
NestList[NestWhile[# + 2 &, #, ! PrimeQ[FromDigits[IntegerDigits[#2, 2]]] &, 2] &, 3, 52] (* Jan Mangaldan, Jul 02 2020 *)

is(n)=my(v=binary(n));isprime(sum(i=1,#v,v[i]*10^(#v-i))) \\ Charles R Greathouse IV, Jun 28 2013

Entry revised by R. J. Mathar and N. J. A. Sloane, Mar 12 2010

A099837 Expansion of (1 - x^2) / (1 + x + x^2) in powers of x.

Original entry on oeis.org

1, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1

Offset: 0

Paul Barry, Oct 27 2004

A transform of (-1)^n.
Row sums of Riordan array ((1-x)/(1+x), x/(1+x)^2), A110162.
Let b(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)(-1)^(n-2k). Then a(n) = b(n) - b(n-2) = A049347(n) - A049347(n-2) (n > 0). The g.f. 1/(1+x) of (-1)^n is transformed to (1-x^2)/(1+x+x^2) under the mapping G(x)->((1-x^2)/(1+x^2))G(x/(1+x^2)). Partial sums of A099838.
A(n) = a(n+3) (or a(n) if a(0) is replaced by 2) appears, together with B(n) = A049347(n) in the formula 2*exp(2*Pi*n*i/3) = A(n) + B(n)*sqrt(3)*i, n >= 0, with i = sqrt(-1). See A164116 for the case N=5. - Wolfdieter Lang, Feb 27 2014

			G.f. = 1 - x - x^2 + 2*x^3 - x^4 - x^5 + 2*x^6 - x^7 - x^8 + 2*x^9 - x^10 + ...

Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (-1,-1).

Cf. A061347, A100051, A100063, A098554, A110162.

A099837 := proc(n)
    option remember;
    if n <=2 then
        op(n+1,[1,-1,-1]) ;
    else
        -procname(n-1)-procname(n-2) ;
    end if;
end proc:
seq(A099837(n),n=0..80) ; # R. J. Mathar, Apr 26 2022

a[0] = 1; a[n_] := Mod[n+2, 3] - Mod[n, 3]; A099837 = Table[a[n], {n, 0, 71}](* Jean-François Alcover, Feb 15 2012, after Michael Somos *)
LinearRecurrence[{-1, -1}, {1, -1, -1}, 50] (* G. C. Greubel, Aug 08 2017 *)

A099837(n) := block(
        if n = 0 then 1 else [2,-1,-1][1+mod(n,3)]
)$ /* R. J. Mathar, Mar 19 2012 */

{a(n) = [2, -1, -1][n%3 + 1] - (n == 0)}; /* Michael Somos, Jan 19 2012 */

Vec((1-x^2)/(1+x+x^2) + O(x^20)) \\ Felix Fröhlich, Aug 08 2017

G.f.: (1-x^2)/(1+x+x^2).
Euler transform of length 3 sequence [-1, -1, 1]. - Michael Somos, Mar 21 2011
Moebius transform is length 3 sequence [-1, 0, 3]. - Michael Somos, Mar 22 2011
a(n) = -b(n) where b(n) = A061347(n) is multiplicative with b(3^e) = -2 if e > 0, b(p^e) = 1 otherwise. - Michael Somos, Jan 19 2012
a(n) = a(-n). a(n) = c_3(n) if n > 1 where c_k(n) is Ramanujan's sum. - Michael Somos, Mar 21 2011
G.f.: (1 - x) * (1 - x^2) / (1 - x^3). a(n) = -a(n-1) - a(n-2) unless n = 0, 1, 2. - Michael Somos, Jan 19 2012
Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s)*(3^(1-s)-1). - R. J. Mathar, Apr 11 2011
a(n+3) = R(n,-1) for n >= 0, with the monic Chebyshev T-polynomials R with coefficient table A127672. - Wolfdieter Lang, Feb 27 2014
For n > 0, a(n) = 2*cos(n*Pi/3)*cos(n*Pi). - Wesley Ivan Hurt, Sep 25 2017
From Peter Bala, Apr 20 2024: (Start)
a(n) is equal to the n-th order Taylor polynomial (centered at 0) of 1/c(x)^(2*n) evaluated at x = 1, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108. Cf. A333093.
Row sums of the Riordan array A110162. (End)

A054535 Square array giving Ramanujan sum T(n,k) = c_n(k) = Sum_{m=1..n, (m,n)=1} exp(2 Pi i m k / n), read by antidiagonals upwards (n >= 1, k >= 1).

Original entry on oeis.org

1, -1, 1, -1, 1, 1, 0, -1, -1, 1, -1, -2, 2, 1, 1, 1, -1, 0, -1, -1, 1, -1, -1, -1, 2, -1, 1, 1, 0, -1, -2, -1, 0, 2, -1, 1, 0, 0, -1, -1, 4, -2, -1, 1, 1, 1, 0, 0, -1, 1, -1, 0, -1, -1, 1, -1, -1, -3, -4, -1, 2, -1, 2, 2, 1, 1, 0, -1, 1, 0, 0, -1, 1, -1, 0, -1, -1, 1, -1, 2, -1, -1, 0, 0, 6, -1, -1, -2, -1, 1, 1, 1, -1

Offset: 1

N. J. A. Sloane, Apr 09 2000

Replace the first column in A077049 with any k-th column in A177121 to get a new array. Then the matrix inverse of the new array will have the k-th column of A054535 (this array) as its first column. - Mats Granvik, May 03 2010
We have T(n, k) = c_n(k) = Sum_{m=1..n, (m,n)=1} exp(2 Pi i m k / n) and
A054534(n, k) = c_k(n) = Sum_{m=1..k, (m,k)=1} exp(2 Pi i m n / k). That is, the current array is the transpose of array A054534. Dirichlet g.f.'s for these two arrays are given below by R. J. Mathar and  Mats Granvik. - Petros Hadjicostas, Jul 27 2019

			Square array T(n,k) = c_n(k) (with rows n >= 1 and columns k >= 1) starts as follows:
   1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  -1,  1, -1,  1, -1,  1, -1,  1, -1,  1, -1,  1, -1, ...
  -1, -1,  2, -1, -1,  2, -1, -1,  2, -1, -1,  2, -1, ...
   0, -2,  0,  2,  0, -2,  0,  2,  0, -2,  0,  2,  0, ...
  -1, -1, -1, -1,  4, -1, -1, -1, -1,  4, -1, -1, -1, ...
   1, -1, -2, -1,  1,  2,  1, -1, -2, -1,  1,  2,  1, ...
  -1, -1, -1, -1, -1, -1,  6, -1, -1, -1, -1, -1, -1, ...
   0,  0,  0, -4,  0,  0,  0,  4,  0,  0,  0, -4,  0, ...
   ... [example edited by _Petros Hadjicostas_, Jul 27 2019]

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 160.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Oxford Science Publications, Clarendon Press, Oxford, 2003.
E. C. Titchmarsh and D. R. Heath-Brown, The theory of the Riemann zeta-function, 2nd ed., 1986.

Robert Israel, Table of n, a(n) for n = 1..10011 (T(n,k) for n+k <= 142).
Tom M. Apostol, Arithmetical properties of generalized Ramanujan sums, Pacific J. Math. 41 (1972), 281-293.
Eckford Cohen, A class of arithmetic functions, Proc. Natl. Acad. Sci. USA 41 (1955), 939-944.
A. Elashvili, M. Jibladze, and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), 173-188.
M. L. Fredman, A symmetry relationship for a class of partitions, J. Combinatorial Theory Ser. A 18 (1975), 199-202.
Emiliano Gagliardo, Le funzioni simmetriche semplici delle radici n-esime primitive dell'unità, Bollettino dell'Unione Matematica Italiana Serie 3, 8(3) (1953), 269-273.
Otto Hölder, Zur Theorie der Kreisteilungsgleichung K_m(x)=0, Prace mat.-fiz. 43 (1936), 13-23.
Peter H. van der Kamp, On the Fourier transform of the greatest common divisor, Integers 13 (2013), #A24. [See Section 3 for historical remarks.]
J. C. Kluyver, Some formulae concerning the integers less than n and prime to n, in: KNAW, Proceedings, 9 I, 1906, Amsterdam, 1906, pp. 408-414; see p. 410.
C. A. Nicol, On restricted partitions and a generalization of the Euler phi number and the Moebius function, Proc. Natl. Acad. Sci. USA 39(9) (1953), 963-968.
C. A. Nicol and H. S. Vandiver, A von Sterneck arithmetical function and restricted partitions with respect to a modulus, Proc. Natl. Acad. Sci. USA 40(9) (1954), 825-835.
K. G. Ramanathan, Some applications of Ramanujan's trigonometrical sum C_m(n), Proc. Indian Acad. Sci., Sect. A 20 (1944), 62-69.
Srinivasa Ramanujan, On certain trigonometric sums and their applications in the theory of numbers, Trans. Camb. Phil. Soc. 22 (1918), 259-276.
R. D. von Sterneck, Ein Analogon zur additiven Zahlentheorie, Sitzungsber. Akad. Wiss. Sapientiae Math.-Naturwiss. Kl. 111 (1902), 1567-1601 (Abt. IIa). [It may not be universally accessible.]
R. D. von Sterneck, Über ein Analogon zur additiven Zahlentheorie, Jahresbericht der Deutschen Mathematiker-Vereinigung 12 (1903), 110-113. [Summary of the 1902 paper.]
Wikipedia, Ramanujan's sum.
Wikipedia, Robert Daublebsky von Sterneck der Jüngere.
Aurel Wintner, On a statistics of the Ramanujan sums, Amer. J. Math., 64(1) (1942), 106-114.

Transpose of array in A054534. Cf. A054532, A054533, A282634.

Cf. A086831=c_n(2) (2nd column), A085097=c_n(3) (3rd column), A085384=c_n(4) (4th column), A085639=c_n(5) (fifth column), A085906=c_n(6) (sixth column), A099837=c_3(n) (third row), A176742=c_4(n) (fourth row), A100051=c_6(n) (sixth row).

with(numtheory): c:=(n,k)->phi(n)*mobius(n/gcd(n,k))/phi(n/gcd(n,k)): for n from 1 to 13 do seq(c(n+1-j,j),j=1..n) od; # gives the sequence in triangular form # Emeric Deutsch
# to get the example above
for n to 8 do
    seq(c(n, k), k = 1 .. 13);
end do
# Petros Hadjicostas, Jul 27 2019

nmax = 14; t[n_, k_] := EulerPhi[n]*(MoebiusMu[n / GCD[n, k]] / EulerPhi[n / GCD[n, k]]); Flatten[ Table[t[n - k + 1, k], {n, 1, nmax}, {k, 1, n}]] (* Jean-François Alcover, Nov 10 2011, after Emeric Deutsch *)
(* To get the example above in table format *)
TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 13}]]
(* Petros Hadjicostas, Jul 27 2019 *)

T(n,k) = c_n(k) = phi(n) * Moebius(n/gcd(n, k))/phi(n/gcd(n, k)). - Emeric Deutsch, Dec 23 2004 [The r.h.s. of this formula is known as the von Sterneck function, and it was introduced by him around 1900. - Petros Hadjicostas, Jul 20 2019]
Dirichlet series: Sum_{n>=1} c_n(k)/n^s = sigma_{1-s}(k)/zeta(s) where sigma is the sum-of-divisors function. Sum_{n>=1} c_k(n)/n^s = zeta(s)*Sum_{d|k} mu(k/d)*d^(1-s). [Hardy & Wright, Titchmarsh] - R. J. Mathar, Apr 01 2012 [We have sigma_{1-s}(k) = Sum_{d|k} d^{1-s} = Sum_{d|k} (k/d)^{1-s} = sigma_{s-1}(k) / k^{s-1}. - Petros Hadjicostas, Jul 27 2019]
From Mats Granvik, Oct 10 2016: (Start)
For n >= 1 and k >= 1 let
A(n,k) := if n mod k = 0 then k^r, otherwise 0;
B(n,k) := if n mod k = 0 then k/n^s, otherwise 0.
Then the Ramanujan's sum matrix equals
inverse(A).transpose(B) evaluated at s=0 and r=0.
Equals inverse(A051731).transpose(A127093).
Dirichlet g.f.: Sum_{n>=1} Sum_{k>=1} T(n,k)/(n^r*k^s) = zeta(s)*zeta(s + r - 1)/zeta(r) as in Wikipedia. (End)
T(n,k) = c_n(k) = Sum_{s | gcd(n,k)} s * Moebius(n/s). - Petros Hadjicostas, Jul 27 2019
Lambert series and a consequence: Sum_{n >= 1} c_n(k) * z^n / (1 - z^n) = Sum_{s|k} s * z^s and -Sum_{n >= 1} (c_n(k) / n) * log(1 - z^n) = Sum_{s|k} z^s for |z| < 1 (using the principal value of the logarithm). - Petros Hadjicostas, Aug 15 2019

Name edited by Petros Hadjicostas, Jul 27 2019

A054534 Square array giving Ramanujan sum T(n,k) = c_k(n) = Sum_{m=1..k, (m,k)=1} exp(2 Pi i m n / k), read by antidiagonals upwards (n >= 1, k >= 1).

Original entry on oeis.org

1, 1, -1, 1, 1, -1, 1, -1, -1, 0, 1, 1, 2, -2, -1, 1, -1, -1, 0, -1, 1, 1, 1, -1, 2, -1, -1, -1, 1, -1, 2, 0, -1, -2, -1, 0, 1, 1, -1, -2, 4, -1, -1, 0, 0, 1, -1, -1, 0, -1, 1, -1, 0, 0, 1, 1, 1, 2, 2, -1, 2, -1, -4, -3, -1, -1, 1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, 1, 1, -1, -2, -1, -1, 6, 0, 0, -1, -1, 2, -1, 1, -1, 2, 0, 4, -2, -1, 0, -3, -4, -1, 0, -1, 1

Offset: 1

N. J. A. Sloane, Apr 09 2000

The Ramanujan sum is also known as the von Sterneck arithmetic function. Robert Daublebsky von Sterneck introduced it around 1900. - Petros Hadjicostas, Jul 20 2019

T(n, k) = c_k(n) is the sum of the n-th powers of the k-th primitive roots of unity. - Petros Hadjicostas, Jul 27 2019

			Array T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  1, -1, -1,  0, -1,  1, -1,  0,  0,  1, -1, ...
  1,  1, -1, -2, -1, -1, -1,  0,  0, -1, -1, ...
  1, -1,  2,  0, -1, -2, -1,  0, -3,  1, -1, ...
  1,  1, -1,  2, -1, -1, -1, -4,  0, -1, -1, ...
  1, -1, -1,  0,  4,  1, -1,  0,  0, -4, -1, ...
  1,  1,  2, -2, -1,  2, -1,  0, -3, -1, -1, ...
  1, -1, -1,  0, -1,  1,  6,  0,  0,  1, -1, ...
  ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 160.
H. Rademacher, Collected Papers of Hans Rademacher, vol. II, MIT Press, 1974, p. 435.
S. Ramanujan, On Certain Trigonometrical Sums and their Applications in the Theory of Numbers, pp. 179-199 of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea Publishing 2000.
R. D. von Sterneck, Ein Analogon zur additiven Zahlentheorie, Sitzungsber. Acad. Wiss. Sapientiae Math.-Naturwiss. Kl. 111 (1902), 1567-1601 (Abt. IIa).

Seiichi Manyama, Antidiagonals n = 1..140, flattened
Tom M. Apostol, Arithmetical properties of generalized Ramanujan sums, Pacific J. Math. 41 (1972), 281-293.
Austrian Biographical Encyclopedia from 1815 onwards, Daublebsky von Sterneck, Robert.
Eckford Cohen, A class of arithmetic functions, Proc. Natl. Acad. Sci. USA 41 (1955), 939-944.
A. Elashvili, M. Jibladze, and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), 173-188.
M. L. Fredman, A symmetry relationship for a class of partitions, J. Combinatorial Theory Ser. A 18 (1975), 199-202.
Emiliano Gagliardo, Le funzioni simmetriche semplici delle radici n-esime primitive dell'unità, Bollettino dell'Unione Matematica Italiana Serie 3, 8(3) (1953), 269-273.
Otto Hölder, Zur Theorie der Kreisteilungsgleichung K_m(x)=0, Prace mat.-fiz. 43 (1936), 13-23.
Peter H. van der Kamp, On the Fourier transform of the greatest common divisor, Integers 13 (2013), #A24. [See Section 3 for historical remarks.]
C. A. Nicol, On restricted partitions and a generalization of the Euler phi number and the Moebius function, Proc. Natl. Acad. Sci. USA 39(9) (1953), 963-968.
C. A. Nicol and H. S. Vandiver, A von Sterneck arithmetical function and restricted partitions with respect to a modulus, Proc. Natl. Acad. Sci. USA 40(9) (1954), 825-835.
K. G. Ramanathan, Some applications of Ramanujan's trigonometrical sum C_m(n), Proc. Indian Acad. Sci., Sect. A 20 (1944), 62-69.
Srinivasa Ramanujan, On certain trigonometric sums and their applications in the theory of numbers, Trans. Camb. Phil. Soc. 22 (1918), 259-276.
M. V. Subbarao, The Brauer-Rademacher identity, Amer. Math. Monthly 72 (1965), 135-138.
Wikipedia, Ramanujan's sum.
Wikipedia, Robert Daublebsky von Sterneck der Jüngere.
Aurel Wintner, On a statistics of the Ramanujan sums, Amer. J. Math., 64(1) (1942), 106-114.

Cf. A000010, A033999, A054532, A054533, A054535, A062570, A085097, A058384, A085639, A085906, A099837, A100051, A176742, A282634.

nmax = 14; mu[n_Integer] = MoebiusMu[n]; mu[] = 0; t[n, k_] := Total[ #*mu[k/#]& /@ Divisors[n]]; Flatten[ Table[ t[n-k+1, k], {n, 1, nmax}, {k, 1, n}]] (* Jean-François Alcover, Nov 14 2011, after Pari *)
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 11}]] (* to print a table like the one in the example - Petros Hadjicostas, Jul 27 2019 *)

{T(n, k) = if( n<1 || k<1, 0, sumdiv( n, d, if( k%d==0, d * moebius(k / d))))} /* Michael Somos, Dec 05 2002 */

{T(n, k) = if( n<1 || k<1, 0, polsym( polcyclo( k), n) [n + 1])} /* Michael Somos, Mar 21 2011 */

/*To get an array like in the example above using Michael Somos' programs:*/
{for (n=1, 20, for (k=1, 40, print1(T(n,k), ","); ); print(); ); } /* Petros Hadjicostas, Jul 27 2019 */

T(n, 1) = c_1(n) = 1. T(n, 2) = c_2(n) = A033999(n). T(n, 3) = c_3(n) = A099837(n) if n>1. T(n, 4) = c_4(n) = A176742(n) if n>1. T(n, 6) = c_6(n) = A100051(n) if n>1. - Michael Somos, Mar 21 2011
T(1, n) = c_n(1) = A008683(n). T(2, n) = c_n(2) = A086831(n). T(3, n) = c_n(3) = A085097(n). T(4, n) = c_n(4) = A085384(n). T(5, n) = c_n(5) = A085639(n). T(6, n) = c_n(6) = A085906(n). - Michael Somos, Mar 21 2011
T(n, n) = T(k * n, n) = A000010(n), T(n, 2*n) = -A062570(n). - Michael Somos, Mar 21 2011
Lambert series and a consequence: Sum_{k >= 1} c_k(n) * z^k / (1 - z^k) = Sum_{s|n} s * z^s and -Sum_{k >= 1} (c_k(n) / k) * log(1 - z^k) = Sum_{s|n} z^s for |z| < 1 (using the principal value of the logarithm). - Petros Hadjicostas, Aug 15 2019

A087204 Period 6: repeat [2, 1, -1, -2, -1, 1].

Original entry on oeis.org

2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 19 2003

Satisfies (a(n))^2 = a(2n) + 2. Shifted differences of itself.
Moebius transform is length 6 sequence [1, -2, -3, 0, 0, 6]. - Michael Somos, Oct 22 2006
Twice the real part of x^n, where x is either of the primitive 6th roots of unity. For the root with positive imaginary part, the imaginary part of x^n is i*A128834(n)*sqrt(3)/2. - Peter Munn, Apr 25 2022

			a(2) = -1 = a(1) - a(0) = 1 - 2 = ((1+sqrt(-3))/2)^2 + ((1-sqrt(-3))/2)^2 = -1 = -2/4 + 2*sqrt(-3)/4 - 2/4 -2 sqrt(-3)/4 = -1.
G.f. = 2 + x - x^2 - 2*x^3 - x^4 + x^5 + 2*x^6 + x^7 - x^8 - 2*x^9 - x^10 + ...

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 176.
Paulo Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory, Springer-Verlag, NY, 2000, p. 6.

Tanya Khovanova, Recursive Sequences.
Wikipedia, Lucas sequence.
Index entries for linear recurrences with constant coefficients, signature (1,-1).
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2).

Essentially the same as A057079 and A100051. Pairwise sums of A010892.

Cf. A128834.

&cat[[2, 1, -1, -2, -1, 1]^^20]; // Wesley Ivan Hurt, Jun 19 2016

A087204:=n->[2, 1, -1, -2, -1, 1][(n mod 6)+1]: seq(A087204(n), n=0..100); # Wesley Ivan Hurt, Jun 19 2016

PadLeft[{}, 108, {2,1,-1,-2,-1,1}] (* Harvey P. Dale, Sep 11 2011 *)
a[ n_] := {1, -1, -2, -1, 1, 2}[[Mod[n, 6, 1]]]; (* Michael Somos, Jan 29 2015 *)
a[ n_] := 2 Re[ Exp[ Pi I n / 3]]; (* Michael Somos, Mar 29 2015 *)

{a(n) = [2, 1, -1, -2, -1, 1][n%6 + 1]}; /* Michael Somos, Oct 22 2006 */

A087204(n) = if(0==n, 2, my(f = factor(n)); prod(k=1, #f~, if(f[k, 1]<=3, 1-f[k, 1], 1))); \\ (After David W. Wilson's multiplicative formula) - Antti Karttunen, Apr 28 2022

[lucas_number2(n,1,1) for n in range(0, 102)] # Zerinvary Lajos, Apr 30 2009

a(n) = a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 1.
G.f.: (2-x)/(1-x+x^2).
a(n) = Sum_{k>=0} (-1)^k*n/(n-k)*C(n-k, k).
a(n) = (1/2)*((-1)^floor(n/3) + 2*(-1)^floor((n+1)/3) + (-1)^floor((n+2)/3)).
Multiplicative with a(2^e) = -1, a(3^e) = -2, a(p^e) = 1 otherwise. - David W. Wilson, Jun 12 2005
a(n) = a(-n) = -a(n-3) for all n in Z. - Michael Somos, Oct 22 2006
E.g.f.: 2*exp(x/2)*cos(sqrt(3)*x/2). - Sergei N. Gladkovskii, Aug 12 2012
a(n) = r^n + s^n, with r=(1+i*sqrt(3))/2 and s=(1-i*sqrt(3))/2 the roots of 1-x+x^2. - Ralf Stephan, Jul 19 2013
a(n) = 2*cos(n*Pi/3). - Wesley Ivan Hurt, Jun 19 2016
Dirichlet g.f.: zeta(s)*(1-2^(1-s)-3^(1-s)+6^(1-s)). - Amiram Eldar, Jan 01 2023

Edited by Ralf Stephan, Feb 04 2005

A100063 A Chebyshev transform of Jacobsthal numbers.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1

Offset: 0

Paul Barry, Nov 02 2004

A Chebyshev transform of A001045(n+1): if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))*A(x/(1+x^2)).
Also decimal expansion of 1111/9990. - Elmo R. Oliveira, Feb 18 2024
Also partial quotients of the continued fraction expansion of sqrt(5/2). - Hugo Pfoertner, Jan 10 2025

			G.f. = 1 + x + x^2 + 2*x^3 + x^4 + x^5 + 2*x^6 + x^7 + x^8 + 2*x^9 + ... - _Michael Somos_, Feb 20 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A000012, A001045, A100051, A061347, A057079, A154272.

Cf. A000032, A000045.

PadRight[{1},120,{2,1,1}] (* or *) LinearRecurrence[{0,0,1},{1,1,1,2},120] (* Harvey P. Dale, Jul 08 2015 *)
a[ n_] := If[n<1, Boole[n==0], {2, 1, 1}[[1+Mod[n, 3]]]]; (* Michael Somos, Feb 20 2024 *)

my(x='x+O('x^50)); Vec((1+x)(1+x^2)/(1-x^3)) \\ G. C. Greubel, May 03 2017

{a(n) = if(n<1, n==0, [2, 1, 1][n%3+1])}; /* Michael Somos, Feb 20 2024 */

contfrac(sqrt(5/2),,80) \\ Hugo Pfoertner, Jan 10 2025

G.f.: (1+x)(1+x^2)/(1-x^3).
a(n) = n*Sum_{k=0..floor(n/2)} binomial(n-k, k)(-1)^k*A001045(n-2k+1)/(n-k).
Multiplicative with a(3^e) = 2, a(p^e) = 1 otherwise. - David W. Wilson, Jun 11 2005
Dirichlet g.f.: zeta(s)*(1+1/3^s). Dirichlet convolution of A154272 and A000012. - R. J. Mathar, Feb 07 2011
a(n) = 2 if n == 0 (mod 3) and n > 0, and a(n) = 1 otherwise. - Amiram Eldar, Nov 01 2022
a(n) = gcd(Fibonacci(n), Lucas(n)) = gcd(A000045(n), A000032(n)), for n >= 1. - Amiram Eldar, Jul 10 2023

A132367 Period 6: repeat [1, 1, 2, -1, -1, -2].

Original entry on oeis.org

1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2

Offset: 0

Paul Curtz, Nov 09 2007

Nonsimple continued fraction expansion of 1+1/sqrt(3) = 1 + A020760. - R. J. Mathar, Mar 08 2012

Index entries for linear recurrences with constant coefficients, signature (0,0,-1).

Cf. A020760, A061347, A100051, A100063, A122876.

&cat[[1, 1, 2, -1, -1, -2]^^20]; // Wesley Ivan Hurt, Jun 19 2016

A132367:=n->[1, 1, 2, -1, -1, -2][(n mod 6)+1]: seq(A132367(n), n=0..100); # Wesley Ivan Hurt, Jun 19 2016

PadRight[{}, 120, {1,1,2,-1,-1,-2}] (* Harvey P. Dale, Jul 28 2012 *)

a(n)=[1,1,2,-1,-1,-2][n%6+1] \\ Charles R Greathouse IV, Jun 02 2011

a(n) = cos(Pi*n/3)/3+sqrt(3)*sin(Pi*n/3)+2*(-1)^n/3. - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Jun 19 2016: (Start)
G.f.: (1+x+2*x^2)/(1+x^3).
a(n) + a(n-3) = 0 for n>2. (End)

A100050 A Chebyshev transform of n.

Original entry on oeis.org

0, 1, 2, 0, -4, -5, 0, 7, 8, 0, -10, -11, 0, 13, 14, 0, -16, -17, 0, 19, 20, 0, -22, -23, 0, 25, 26, 0, -28, -29, 0, 31, 32, 0, -34, -35, 0, 37, 38, 0, -40, -41, 0, 43, 44, 0, -46, -47, 0, 49, 50, 0, -52, -53, 0, 55, 56, 0, -58, -59, 0, 61, 62, 0, -64, -65, 0, 67, 68, 0, -70, -71, 0, 73, 74, 0, -76, -77, 0, 79, 80, 0, -82, -83, 0, 85, 86, 0

Offset: 0

Paul Barry, Oct 31 2004

A Chebyshev transform of x/(1-x)^2: if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)).

			x + 2*x^2 - 4*x^4 - 5*x^5 + 7*x^7 + 8*x^8 - 10*x^10 - 11*x^11 + 13*x^13 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (2,-3,2,-1).

Cf. A165202 (partial sums).

Cf. A099837, A099443, A011655, A100047, A100048, A100051, A091684.

LinearRecurrence[{2, -3, 2, -1}, {0, 1, 2, 0},50] (* G. C. Greubel, Aug 08 2017 *)

{a(n) = n * (-1)^(n\3) * sign( n%3)} /* Michael Somos, Mar 19 2011 */

{a(n) = local(A, p, e); if( abs(n)<1, 0, A = factor(abs(n)); prod( k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p==2, -(-2)^e, (kronecker( -12, p) * p)^e))))} /* Michael Somos, Mar 19 2011 */

[lucas_number1(n,2,1)*lucas_number1(n,1,1) for n in range(0,88)] # Zerinvary Lajos, Jul 06 2008

Euler transform of length 6 sequence [ 2, -3, -2, 0, 0, 2]. - Michael Somos, Mar 19 2011
a(n) is multiplicative with a(2^e) = -(-2)^e if e>0, a(3^e) = 0^e, a(p^e) = p^e if p == 1 (mod 6), a(p^e) = (-p)^e if p == 5 (mod 6). - Michael Somos, Mar 19 2011
G.f.: x*(1 - x^2)^3 *(1 - x^3)^2 / ((1 - x)^2 *(1 - x^6)^2) = x *(1 + x)^2 *(1 - x^2) / (1 + x^3)^2. - Michael Somos, Mar 19 2011
a(3*n) = 0, a(3*n + 1) = (-1)^n * (3*n + 1), a(3*n + 2) = (-1)^n * (3*n + 2). a(-n) = a(n). - Michael Somos, Mar 19 2011
G.f.: x(1-x^2)/(1-x+x^2)^2.
a(n) = 2*a(n-1) -3*a(n-2) +2*a(n-3) -a(n-4).
a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k,k)*(n-2k)/(n-k).

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A036952 Numbers whose binary expansion is a decimal prime.

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A099837 Expansion of (1 - x^2) / (1 + x + x^2) in powers of x.

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A054535 Square array giving Ramanujan sum T(n,k) = c_n(k) = Sum_{m=1..n, (m,n)=1} exp(2 Pi i m k / n), read by antidiagonals upwards (n >= 1, k >= 1).

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A054534 Square array giving Ramanujan sum T(n,k) = c_k(n) = Sum_{m=1..k, (m,k)=1} exp(2 Pi i m n / k), read by antidiagonals upwards (n >= 1, k >= 1).

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A087204 Period 6: repeat [2, 1, -1, -2, -1, 1].

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A100063 A Chebyshev transform of Jacobsthal numbers.

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