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 1-7 of 7 results.

A132723 Binomial transform of A132429.

Original entry on oeis.org

3, 4, 4, 0, -8, -16, -16, 0, 32, 64, 64, 0, -128, -256, -256, 0, 512, 1024, 1024, 0, -2048, -4096, -4096, 0, 8192, 16384, 16384, 0, -32768, -65536, -65536, 0, 131072, 262144, 262144, 0, -524288, -1048576, -1048576
Offset: 0

Views

Author

Paul Curtz, Nov 16 2007

Keywords

Comments

Sequence is identical to its fourth differences.

Links

Programs

  • Magma
    [3] cat [n le 2 select 4 else 2*(Self(n-1) - Self(n-2)): n in [1..40]]; // G. C. Greubel, Feb 14 2021
  • Mathematica
    Join[{3},LinearRecurrence[{2,-2},{4,4},50]] (* Harvey P. Dale, Mar 06 2014 *)
Table[If[n<2, n+3, 2*((1+I)^(n-1) + (1-I)^(n-1))], {n,0,40}] (* G. C. Greubel, Feb 14 2021 *)
  • Sage
    def A132723(n): return n+3 if (n<2) else 2*( (1+i)^(n-1) + (1-i)^(n-1) )
[A132723(n) for n in (0..40)] # G. C. Greubel, Feb 14 2021

Formula

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3), n positive. For a(0)=3,a(1)=a(2)=4,a(3)=0.
From R. J. Mathar, Apr 02 2008: (Start)
O.g.f.: 1 + 2/(1 -2*x +2*x^2).
a(n) = 2*a(n-1) - 2*a(n-2) if n>2. (End)
E.g.f.: 1 + 2*sqrt(2)*exp(x)*sin(x + Pi/4). - G. C. Greubel, Feb 14 2021

Extensions

More terms from R. J. Mathar, Apr 02 2008

A152134 Maximal length of rook tour on an n X n+3 board.

Original entry on oeis.org

8, 24, 54, 102, 174, 270, 396, 556, 756, 996, 1282, 1618, 2010, 2458, 2968, 3544, 4192, 4912, 5710, 6590, 7558, 8614, 9764, 11012, 12364, 13820, 15386, 17066, 18866, 20786, 22832, 25008, 27320, 29768, 32358, 35094, 37982, 41022, 44220, 47580
Offset: 1

Views

Author

R. J. Mathar, Mar 22 2009

Keywords

References

  • M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 76.

Links

Crossrefs

Cf. A006071, A152132, A152133, A152135.

Programs

  • Magma
    I:=[8, 24, 54, 102, 174, 270, 396]; [n le 7 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3)+Self(n-4)-3*Self(n-5)+3*Self(n-6)-Self(n-7): n in [1..40]];// Vincenzo Librandi, Dec 19 2012
  • Mathematica
    CoefficientList[Series[-2*(- 4 - 3*x^2 - 2*x^3 + x^4)/(1+x)/(x^2+1)/(x-1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 18 2012 *)

Formula

G.f.: -2*x*(-4-3*x^2-2*x^3+x^4)/(1+x)/(x^2+1)/(x-1)^4.
a(n) = 17*n/6+3/4+2*n^3/3+3*n^2+A132429(n+3)/4. - R. J. Mathar, Sep 27 2009
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-4) - 3*a(n-5) + 3*a(n-6) - a(n-7). - Vincenzo Librandi, Dec 19 2012

Extensions

More terms from R. J. Mathar, Sep 27 2009

A159966 Lodumo_4 of A102370 (sloping binary numbers).

Original entry on oeis.org

0, 3, 2, 1, 4, 7, 6, 5, 8, 11, 10, 9, 12, 15, 14, 13, 16, 19, 18, 17, 20, 23, 22, 21, 24, 27, 26, 25, 28, 31, 30, 29, 32, 35, 34, 33, 36, 39, 38, 37, 40, 43, 42, 41, 44, 47, 46, 45, 48, 51, 50, 49, 52, 55, 54, 53, 56, 59, 58, 57, 60, 63, 62, 61, 64, 67, 66, 65, 68, 71, 70, 69, 72
Offset: 0

Views

Author

Philippe Deléham, Apr 28 2009

Keywords

Comments

A permutation of the nonnegative integers.
A092486 preceded by a zero. - Philippe Deléham, May 05 2009
Fixed points are the even numbers. - Wesley Ivan Hurt, Oct 16 2015

Links

Crossrefs

Cf. A002162, A005843, A102370, A132429, A158459, A166549.

Programs

  • Magma
    [n-(1-(-1)^n)*(-1)^((2*n+1-(-1)^n) div 4) : n in [0..100]]; // Wesley Ivan Hurt, Oct 16 2015
  • Maple
    A159966:=n->n-(1-(-1)^n)*(-1)^((2*n+1-(-1)^n)/4): seq(A159966(n), n=0..100); # Wesley Ivan Hurt, Oct 16 2015
  • Mathematica
    Table[n - (1 - (-1)^n) (-1)^((2 n + 1 - (-1)^n)/4), {n, 0, 40}] (* or *) CoefficientList[Series[(3 x - 4 x^2 + 3 x^3)/((x - 1)^2 (1 + x^2)), {x, 0, 100}], x] (* Wesley Ivan Hurt, Oct 16 2015 *)
LinearRecurrence[{2,-2,2,-1},{0,3,2,1},80] (* Harvey P. Dale, Jul 02 2022 *)
  • PARI
    concat(0, Vec((3*x-4*x^2+3*x^3)/((x-1)^2*(1+x^2)) + O(x^100))) \\ Altug Alkan, Oct 17 2015

Formula

a(n) = lod_4 (A102370(n)).
From Wesley Ivan Hurt, Oct 16 2015: (Start)
G.f.: (3*x-4*x^2+3*x^3)/((x-1)^2*(1+x^2)).
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4), n>3.
a(n) = n-(1-(-1)^n)*(-1)^((2*n+1-(-1)^n)/4).
a(2n) = A005843(n); a(2n+1) = A166549(n).
a(n+1) - a(n) = A132429(n)*(-1)^n. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) (A002162). - Amiram Eldar, Nov 28 2023

A214865 n such that n XOR 9 = n - 9.

Original entry on oeis.org

9, 11, 13, 15, 25, 27, 29, 31, 41, 43, 45, 47, 57, 59, 61, 63, 73, 75, 77, 79, 89, 91, 93, 95, 105, 107, 109, 111, 121, 123, 125, 127, 137, 139, 141, 143, 153, 155, 157, 159, 169, 171, 173, 175, 185, 187, 189, 191, 201, 203, 205, 207, 217, 219, 221, 223, 233, 235, 237, 239, 249, 251
Offset: 1

Views

Author

Brad Clardy, Mar 09 2013

Keywords

Links

Crossrefs

Cf. A214863, A016825, A132429.

Programs

  • Magma
    XOR := func;
m:=9;
for n in [1 .. 500] do
      if (XOR(n, m) eq n-m) then n; end if;
end for;
  • Mathematica
    CoefficientList[Series[x*(9 + 2*x + 2*x^2 + 2*x^3 + x^4)/((1 + x)*(x^2 + 1)*(x - 1)^2), {x,0,50}], x] (* G. C. Greubel, Feb 22 2017 *)
  • PARI
    x='x+O('x^50); Vec(x*(9+2*x+2*x^2+2*x^3+x^4) / ( (1+x)*(x^2+1)*(x-1)^2 )) \\ G. C. Greubel, Feb 22 2017

Formula

a(n) = 4*n + 6 + (-1)^n + 2*(-1)^((2*n+(-1)^n-1)/4) for n>=0.
a(n) = A016825(n+1) + A132429(n) for n>=0.
G.f. x*(9+2*x+2*x^2+2*x^3+x^4) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Mar 10 2013
a(n+4) = a(n) + 16. - Alexander R. Povolotsky, Mar 15 2013

A139814 a(n)=a(n-1)+a(n-2)+a(n-3)+2a(n-4); a(0)=0,a(1)=1,a(2)=3,a(3)=7.

Original entry on oeis.org

0, 1, 3, 7, 11, 23, 47, 95, 187, 375, 751, 1503, 3003, 6007, 12015, 24031, 48059, 96119, 192239, 384479, 768955, 1537911, 3075823, 6151647, 12303291, 24606583, 49213167, 98426335, 196852667, 393705335, 787410671, 1574821343, 3149642683
Offset: 0

Views

Author

Paul Curtz, May 23 2008

Keywords

Links

Crossrefs

Cf. A132429.

Programs

  • Mathematica
    LinearRecurrence[{1,1,1,2},{0,1,3,7},35]  (* Harvey P. Dale, Apr 20 2011 *)

Formula

O.g.f.: -x(1+2x+3x^2)/((2x-1)(1+x)(x^2+1)). a(n) = (-1)^(n+1)/3 +11*2^n/15 -2*(-1)^[n/2]*A000034(n)/5 . - R. J. Mathar, May 24 2008

Extensions

More terms from R. J. Mathar, May 24 2008

A263426 Permutation of the nonnegative integers: [4k+2, 4k+1, 4k, 4k+3, ...].

Original entry on oeis.org

2, 1, 0, 3, 6, 5, 4, 7, 10, 9, 8, 11, 14, 13, 12, 15, 18, 17, 16, 19, 22, 21, 20, 23, 26, 25, 24, 27, 30, 29, 28, 31, 34, 33, 32, 35, 38, 37, 36, 39, 42, 41, 40, 43, 46, 45, 44, 47, 50, 49, 48, 51, 54, 53, 52, 55, 58, 57, 56, 59, 62, 61, 60, 63, 66, 65, 64
Offset: 0

Views

Author

Wesley Ivan Hurt, Oct 17 2015

Keywords

Comments

Fixed points are the odd numbers (A005408).

Links

Crossrefs

Cf. A004442, A005408, A092486, A132429, A176742, A244009, A263449.

Programs

  • Magma
    [n+(1+(-1)^n)*(-1)^(n*(n+1) div 2) : n in [0..80]];
  • Magma
    /* By definition: */ &cat[[4*k+2,4*k+1,4*k,4*k+3]: k in [0..20]]; // Bruno Berselli, Nov 08 2015
  • Maple
    A263426:=n->n + (1 + (-1)^n)*(-1)^(n*(n + 1)/2): seq(A263426(n), n=0..80);
  • Mathematica
    Table[n + (1 + (-1)^n)*(-1)^(n*(n + 1)/2), {n, 0, 80}]
  • PARI
    Vec((2-3*x+2*x^2+x^3)/((x-1)^2*(1+x^2)) + O(x^100)) \\ Altug Alkan, Oct 19 2015

Formula

G.f.: (2 - 3*x + 2*x^2 + x^3)/((x - 1)^2*(1 + x^2)).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>3.
a(n) = n + (1 + (-1)^n)*(-1)^(n*(n+1)/2).
a(n) = 4*floor((n+1)/4) - (n mod 4) + 2.
a(n) = A092486(n) - 1.
a(n) = n + A176742(n) for n>0.
a(2n) = 2*A004442(n), a(2n+1) = A005408(n).
a(-n-1) = -A263449(n).
a(n+1) = a(n) - A132429(n+1)*(-1)^n.
Sum_{n>=0, n!=2} (-1)^(n+1)/a(n) = 1 - log(2) (A244009). - Amiram Eldar, Dec 25 2023

A319078 Expansion of phi(-q) * phi(q)^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, -4, -8, 6, 8, -8, 0, 12, 10, -8, -24, 8, 8, -16, 0, 6, 16, -12, -24, 24, 16, -8, 0, 24, 10, -24, -32, 0, 24, -16, 0, 12, 16, -16, -48, 30, 8, -24, 0, 24, 32, -16, -24, 24, 24, -16, 0, 8, 18, -28, -48, 24, 24, -32, 0, 48, 16, -8, -72, 0, 24, -32, 0, 6, 32
Offset: 0

Views

Author

Michael Somos, Sep 09 2018

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 2*x - 4*x^2 - 8*x^3 + 6*x^4 + 8*x^5 - 8*x^6 + 12*x^8 + ...

Links

Crossrefs

Cf. A004015, A004024, A005887, A008443, A045834, A083703, A080965, A132429, A212885, A213624.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(16), 3/2), 66); A[1] + 2*A[2] - 4*A[3] - 8*A[4];
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q]^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^2]^2, {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^9 / (eta(x + A)^2 * eta(x^4 + A)^4), n))};

Formula

Expansion of eta(q^2)^9 / (eta(q)^2 * eta(q^4)^4) in powers of q.
Expansion of phi(q) * phi(-q^2)^2 = phi(-q^2)^4 / phi(-q) in powers of q.
Euler transform of period 4 sequence [2, -7, 2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(11/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A045834.
G.f. Product_{k>0} (1 - x^k)^3 * (1 + x^k)^5 / (1 + x^(2*k))^4.
a(n) = (-1)^n * A212885(n) = A083703(2*n) = A080965(2*n).
a(4*n) = a(n) * -A132429(n + 2) where A132429 is a period 4 sequence.
a(4*n) = A005875(n). a(4*n + 1) = 2 * A045834(n). a(4*n + 2) = -4 * A045828(n).
a(8*n) = A004015(n). a(8*n + 1) = 2 * A213022(n). a(8*n + 2) = -4 * A213625(n). a(8*n + 3) = -8 * A008443(n). a(8*n + 4) = A005887(n). a(8*n + 5) = 2 * A004024(n). a(8*n + 6) = -8 * A213624(n). a(8*n + 7) = 0.
Showing 1-7 of 7 results.

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