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 1-10 of 12 results. Next

A131855 Numbers m such that A131852(m) = 0.

Original entry on oeis.org

0, 1, 4, 5, 10, 11, 14, 15, 16, 17, 20, 21, 26, 27, 30, 31, 40, 41, 44, 45, 56, 57, 60, 61, 64, 65, 68, 69, 74, 75, 78, 79, 80, 81, 84, 85, 90, 91, 94, 95, 104, 105, 108, 109, 120, 121, 124, 125, 130, 131, 134, 135, 146, 147, 150, 151, 160, 161, 164, 165, 170, 171, 174
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 22 2007

Keywords

Links

Crossrefs

Cf. A131852, A131853, A131854, A131857, A131862, A131864.

A131857 Numbers m such that A131852(m) = 1.

Original entry on oeis.org

2, 3, 6, 7, 18, 19, 22, 23, 32, 33, 36, 37, 42, 43, 46, 47, 48, 49, 52, 53, 58, 59, 62, 63, 66, 67, 70, 71, 82, 83, 86, 87, 96, 97, 100, 101, 106, 107, 110, 111, 112, 113, 116, 117, 122, 123, 126, 127, 162, 163, 166, 167, 178, 179, 182, 183, 226, 227, 230, 231, 242, 243
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 22 2007

Keywords

Links

Crossrefs

Cf. A131856, A131854, A131855, A131862, A131864.

Programs

  • Maple
    Z:= proc(n) option remember;
I*procname(floor(n/2))+(n mod 2)
end proc:
Z(0):= 0:
select(Im@Z=1, [$0..1000]); # Robert Israel, Dec 18 2017
  • Mathematica
    z[0] = 0; z[n_] := z[n] = z[Floor[n/2]]*I + Mod[n, 2]; Select[Range[0, 250], Im[z[#]] == 1&] (* Jean-François Alcover, Jan 31 2018 *)
  • PARI
    isok(n) = {d = Vecrev(binary(n)); imag(sum(k=1, #d, d[k]*I^(k-1))) == 1;} \\ Michel Marcus, Jan 31 2018

A131862 Numbers m such that A131852(m) > 0.

Original entry on oeis.org

2, 3, 6, 7, 18, 19, 22, 23, 32, 33, 34, 35, 36, 37, 38, 39, 42, 43, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 58, 59, 62, 63, 66, 67, 70, 71, 82, 83, 86, 87, 96, 97, 98, 99, 100, 101, 102, 103, 106, 107, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122, 123, 126, 127
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 22 2007

Keywords

Crossrefs

Cf. A131861, A131855, A131857, A131864.

A131864 Numbers m such that A131852(m) < 0.

Original entry on oeis.org

8, 9, 12, 13, 24, 25, 28, 29, 72, 73, 76, 77, 88, 89, 92, 93, 128, 129, 132, 133, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 148, 149, 152, 153, 154, 155, 156, 157, 158, 159, 168, 169, 172, 173, 184, 185, 188, 189, 192, 193, 196, 197, 200, 201, 202, 203
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 22 2007

Keywords

Crossrefs

Cf. A131863, A131855, A131857, A131862.

A056594 Period 4: repeat [1,0,-1,0]; expansion of 1/(1 + x^2).

Original entry on oeis.org

1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0
Offset: 0

Views

Author

Wolfdieter Lang, Aug 04 2000

Keywords

Comments

G.f. is inverse of cyclotomic(4,x). Unsigned: A000035(n+1).
Real part of i^n and imaginary part of i^(n+1), i=sqrt(-1). - Reinhard Zumkeller, Jul 22 2007
The BINOMIAL transform generates A009116(n); the inverse BINOMIAL transform generates (-1)^n*A009116(n). - R. J. Mathar, Apr 07 2008
a(n-1), n >= 1, is the nontrivial Dirichlet character modulo 4, called Chi_2(4;n) (the trivial one is Chi_1(4;n) given by periodic(1,0) = A000035(n)). See the Apostol reference, p. 139, the k = 4, phi(k) = 2 table. - Wolfdieter Lang, Jun 21 2011
a(n-1), n >= 1, is the character of the Dirichlet beta function. - Daniel Forgues, Sep 15 2012
a(n-1), n >= 1, is also the (strongly) multiplicative function h(n) of Theorem 5.12, p. 150, of the Niven-Zuckerman reference. See the formula section. This function h(n) can be employed to count the integer solutions to n = x^2 + y^2. See A002654 for a comment with the formula. - Wolfdieter Lang, Apr 19 2013
This sequence is duplicated in A101455 but with offset 1. - Gary Detlefs, Oct 04 2013
For n >= 2 this gives the determinant of the bipartite graph with 2*n nodes and the adjacency matrix A(n) with elements A(n;1,2) = 1 = A(n;n,n-1), and for 1 < i < n A(n;i,i+1) = 1 = A(n;i,i-1), otherwise 0. - Wolfdieter Lang, Jun 25 2023

Examples

			With a(n-1) = h(n) of Niven-Zuckerman: a(62) = h(63) = h(3^2*7^1) = (-1)^(2*1)*(-1)^(1*3) = -1 = h(3)^2*h(7) = a(2)^2*a(6) = (-1)^2*(-1) = -1. - _Wolfdieter Lang_, Apr 19 2013

References

  • T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986.
  • I. S. Gradstein and I. M. Ryshik, Tables of series, products, and integrals, Volume 1, Verlag Harri Deutsch, 1981.
  • Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, New York: John Wiley (1980), p. 150.
  • Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 32, equation 32:6:1 at page 300.

Links

Crossrefs

Cf. A049310, A074661, A131852, A002654, A146559 (binomial transform).

Programs

  • Magma
    &cat[ [1, 0, -1, 0]: n in [0..23] ]; // Bruno Berselli, Feb 08 2011
  • Maple
    A056594 := n->(1-irem(n,2))*(-1)^iquo(n,2); # Peter Luschny, Jul 27 2011
  • Mathematica
    CoefficientList[Series[1/(1 + x^2), {x, 0, 50}], x]
a[n_]:= KroneckerSymbol[-4,n+1];Table[a[n],{n,0,93}] (* Thanks to Jean-François Alcover. - Wolfdieter Lang, May 31 2013 *)
CoefficientList[Series[1/Cyclotomic[4, x], {x, 0, 100}], x] (* Vincenzo Librandi, Apr 03 2014 *)
  • Maxima
    A056594(n) := block(
        [1,0,-1,0][1+mod(n,4)]
)$ /* R. J. Mathar, Mar 19 2012 */
  • PARI
    {a(n) = real( I^n )}
  • PARI
    {a(n) = kronecker(-4, n+1) }
  • Python
    def A056594(n): return (1,0,-1,0)[n&3] # Chai Wah Wu, Sep 23 2023

Formula

G.f.: 1/(1+x^2).
E.g.f.: cos(x).
a(n) = (1/2)*((-i)^n + i^n), where i = sqrt(-1). - Mitch Harris, Apr 19 2005
a(n) = (1/2)*((-1)^(n+floor(n/2)) + (-1)^floor(n/2)).
Recurrence: a(n)=a(n-4), a(0)=1, a(1)=0, a(2)=-1, a(3)=0.
a(n) = T(n, 0) = A053120(n, 0); T(n, x) Chebyshev polynomials of the first kind. - Wolfdieter Lang, Aug 21 2009
a(n) = S(n, 0) = A049310(n, 0); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind.
Sum_{k>=0} a(k)/(k+1) = Pi/4. - Jaume Oliver Lafont, Mar 30 2010
a(n) = Sum_{k=0..n} A101950(n,k)*(-1)^k. - Philippe Deléham, Feb 10 2012
a(n) = (1/2)*(1 + (-1)^n)*(-1)^(n/2). - Bruno Berselli, Mar 13 2012
a(0) = 1, a(n-1) = 0 if n is even, a(n-1) = Product_{j=1..m} (-1)^(e_j*(p_j-1)/2) if the odd n-1 = p_1^(e_1)*p_2^(e_2)*...*p_m^(e_m) with distinct odd primes p_j, j=1..m. See the function h(n) of Theorem 5.12 of the Niven-Zuckerman reference. - Wolfdieter Lang, Apr 19 2013
a(n) = (-4/(n+1)), n >= 0, where (k/n) is the Kronecker symbol. See the Eric Weisstein and Wikipedia links. Thanks to Wesley Ivan Hurt. - Wolfdieter Lang, May 31 2013
a(n) = R(n,0)/2 with the row polynomials R of A127672. This follows from the product of the zeros of R, and the formula Product_{k=0..n-1} 2*cos((2*k+1)*Pi/(2*n)) = (1 + (-1)^n)*(-1)^(n/2), n >= 1 (see the Gradstein and Ryshik reference, p. 63, 1.396 4., with x = sqrt(-1)). - Wolfdieter Lang, Oct 21 2013
a(n) = Sum_{k=0..n} i^(k*(k+1)), where i=sqrt(-1). - Bruno Berselli, Mar 11 2015
Dirichlet g.f. of a(n) shifted right: L(chi_2(4),s) = beta(s) = (1-2^(-s))*(d.g.f. of A034947), see comments by Lang and Forgues. - Ralf Stephan, Mar 27 2015
a(n) = cos(n*Pi/2). - Ridouane Oudra, Sep 29 2024

A131851 Real part of the function z(n)=Sum(d(k)*i^k: d as in n=Sum(d(k)*2^k), i=sqrt(-1)).

Original entry on oeis.org

0, 1, 0, 1, -1, 0, -1, 0, 0, 1, 0, 1, -1, 0, -1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 0, 1, 0, 1, 0, 1, 0, 1, -1, 0, -1, 0, 0, 1, 0, 1, -1, 0, -1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 0, 1, 0, 1, -1, 0, -1, 0, -2, -1, -2, -1, -1, 0, -1, 0, -2, -1, -2, -1, 0, 1, 0, 1, -1, 0, -1, 0, 0, 1, 0, 1, -1, 0, -1, 0, -1, 0, -1, 0, -2, -1, -2, -1, -1, 0, -1, 0, -2, -1, -2
Offset: 0

Views

Author

Reinhard Zumkeller, Jul 22 2007

Keywords

Comments

A131852(n) = Im(z(n));
z(A000079(n))=(A056594(n),A056594(n+3)); a(A000079(n))=A056594(n);
a(A131854(n))=0; a(A131861(n))>0; a(A131859(n))=1; a(A131863(n))<0;
z(A131853(n))=(0,0); z(A131856(n))=(0,1); z(A131858(n))=(1,0); z(A131860(n))=(1,1);
for n>0: a(A131865(n))=n and ABS(a(m))A131865(n).

Links

Crossrefs

Cf. A007088.

Programs

  • Mathematica
    z[0] = 0; z[n_] := z[n] = z[Floor[n/2]]*I + Mod[n, 2]; Table[z[n] // Re, {n, 0, 110}] (* Jean-François Alcover, Jul 03 2013 *)

Formula

z(n) = if n=0 then (0, 0) else z(floor(n/2))*(0, 1) + (n mod 2, 0), complex multiplication.

A131853 Numbers m such that z(m)=(0,0) with z as defined in A131851.

Original entry on oeis.org

0, 5, 10, 15, 20, 30, 40, 45, 60, 65, 75, 80, 85, 90, 95, 105, 120, 125, 130, 135, 150, 160, 165, 170, 175, 180, 190, 195, 210, 215, 225, 235, 240, 245, 250, 255, 260, 270, 300, 320, 325, 330, 335, 340, 350, 360, 365, 380, 390, 420, 430, 450, 455, 470, 480, 485
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 22 2007

Keywords

Comments

Intersection of A131854 and A131855: A131851(a(n))=0, A131852(a(n))=0;
conjecture: a(n) mod 5 = 0.

Links

Crossrefs

Cf. A131851, A131856, A131858, A131860, A008587.

Programs

  • Mathematica
    z[n_] := z[n] = If[n == 0, 0, z[Floor[n/2]] I + Mod[n, 2]];
Flatten[Position[Table[z[n], {n, 0, 500}], 0] - 1] (* Jean-François Alcover, Oct 12 2021 *)

A131856 Numbers m such that z(m)=(0,1) with z as defined in A131851.

Original entry on oeis.org

2, 7, 22, 32, 37, 42, 47, 52, 62, 67, 82, 87, 97, 107, 112, 117, 122, 127, 162, 167, 182, 227, 242, 247, 262, 292, 302, 322, 327, 342, 352, 357, 362, 367, 372, 382, 422, 482, 487, 502, 512, 517, 522, 527, 532, 542, 552, 557, 572, 577, 587, 592, 597, 602, 607
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 22 2007

Keywords

Comments

Intersection of A131854 and A131857: A131851(a(n))=0, A131852(a(n))=1.

Links

Crossrefs

Cf. A131851, A131853, A131858, A131860.

A131858 Numbers m such that z(m)=(1,0) with z as defined in A131851.

Original entry on oeis.org

1, 11, 16, 21, 26, 31, 41, 56, 61, 81, 91, 121, 131, 146, 151, 161, 171, 176, 181, 186, 191, 211, 241, 251, 256, 261, 266, 271, 276, 286, 296, 301, 316, 321, 331, 336, 341, 346, 351, 361, 376, 381, 386, 391, 406, 416, 421, 426, 431, 436, 446, 451, 466, 471
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 22 2007

Keywords

Comments

Intersection of A131855 and A131859: A131851(a(n))=1, A131852(a(n))=0.

Links

Crossrefs

Cf. A131851, A131853, A131856, A131860.

A131860 Numbers m such that z(m)=(1,1) with z as defined in A131851.

Original entry on oeis.org

3, 18, 23, 33, 43, 48, 53, 58, 63, 83, 113, 123, 163, 178, 183, 243, 258, 263, 278, 288, 293, 298, 303, 308, 318, 323, 338, 343, 353, 363, 368, 373, 378, 383, 418, 423, 438, 483, 498, 503, 513, 523, 528, 533, 538, 543, 553, 568, 573, 593, 603, 633, 643, 658
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 22 2007

Keywords

Comments

Intersection of A131857 and A131859: A131851(a(n))=1, A131852(a(n))=1.

Links

Crossrefs

Cf. A131851, A131853, A131856, A131858.
Showing 1-10 of 12 results. Next

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