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 1-4 of 4 results.

A276390 Bisection of A115716.

Original entry on oeis.org

1, 3, 3, 11, 3, 11, 3, 43, 3, 11, 3, 43, 3, 11, 3, 171, 3, 11, 3, 43, 3, 11, 3, 171, 3, 11, 3, 43, 3, 11, 3, 683, 3, 11, 3, 43, 3, 11, 3, 171, 3, 11, 3, 43, 3, 11, 3, 683, 3, 11, 3, 43, 3, 11, 3, 171, 3, 11, 3, 43, 3, 11, 3, 2731, 3, 11, 3, 43, 3, 11, 3, 171
Offset: 0

Views

Author

N. J. A. Sloane, Sep 07 2016

Keywords

Links

Crossrefs

Cf. A115716.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1,
      `if`(n::odd, 1, 4*b(n/2-1)-1))
    end:
a:= n-> b(2*n):
seq(a(n), n=0..100);  # Alois P. Heinz, Sep 07 2016
  • Mathematica
    b[n_] := b[n] = If[n == 0, 1, If[OddQ[n], 1, 4*b[n/2 - 1] - 1]];
a[n_] := b[2*n];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 12 2018, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Sep 07 2016

A115717 A divide-and-conquer triangle related to A007583.

Original entry on oeis.org

1, 0, 1, 3, -1, 1, 0, 0, 0, 1, 0, 4, -1, -1, 1, 0, 0, 0, 0, 0, 1, 12, -4, 4, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 16, -4, -4, 4, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 48, -16, 16, 0, -4, -4, 4, 0, 0, 0, 0, 0, -1, -1, 1
Offset: 0

Views

Author

Paul Barry, Jan 29 2006

Keywords

Comments

Product of (1-x, x), which is A167374, and number triangle A115715.

Examples

			Triangle begins
   1;
   0,   1;
   3,  -1,  1;
   0,   0,  0,  1;
   0,   4, -1, -1,  1;
   0,   0,  0,  0,  0,  1;
  12,  -4,  4,  0, -1, -1,  1;
   0,   0,  0,  0,  0,  0,  0,  1;
   0,   0,  0,  4,  0,  0, -1, -1,  1;
   0,   0,  0,  0,  0,  0,  0,  0,  0,  1;
   0,  16, -4, -4,  4,  0,  0,  0, -1, -1,  1;
   0,   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
   0,   0,  0,  0,  0,  4,  0,  0,  0,  0, -1, -1,  1;
   0,   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
  48, -16, 16,  0, -4, -4,  4,  0,  0,  0,  0,  0, -1, -1,  1;

Links

Crossrefs

Cf. A007583, A115715, A115716 (row sums), A167374.

Programs

  • Maple
    A115717 := proc(n,k)
    add( A167374(n,j)*A115715(j,k),j=k..n) ;
end proc: # R. J. Mathar, Sep 07 2016
  • Mathematica
    A167374[n_, k_]:= If[k>n-2, (-1)^(n-k), 0];
g[n_, k_]:= g[n, k]= If[k==n, 1, If[k==n-1, -Mod[n, 2], If[n==2*k+2, -4, 0]]]; (* g = A115713 *)
f[n_, k_]:= f[n, k]= If[k==n, 1, -Sum[f[n,j]*g[j,k], {j,k+1,n}]]; (* f=A115715 *)
A115717[n_, k_]:= A115717[n, k]= Sum[A167374[n,j]*f[j,k], {j,k,n}];
Table[A115717[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 23 2021 *)
  • Sage
    @cached_function
def A115717(n,k):
    def A167374(n, k):
        if (k>n-2): return (-1)^(n-k)
        else: return 0
    def A115713(n,k):
        if (k==n): return 1
        elif (k==n-1): return -(n%2)
        elif (n==2*k+2): return -4
        else: return 0
    def A115715(n,k):
        if (k==0): return 4^(floor(log(n+2, 2)) -1)
        elif (k==n): return 1
        elif (k==n-1): return (n%2)
        else: return (-1)*sum( A115715(n,j+k+1)*A115713(j+k+1,k) for j in (0..n-k-1) )
    return sum( A167374(n, j+k)*A115715(j+k, k) for j in (0..n-k) )
flatten([[A115717(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Nov 23 2021

Formula

Sum_{k=0..n} T(n, k) = A115716(n).
T(n ,k) = Sum_{j=k..n} A167374(n, j)*A115715(j, k). - R. J. Mathar, Sep 07 2016

A276391 G.f. satisfies A(x) - 4*A(x^2) = x/(1+x).

Original entry on oeis.org

1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, 171, 1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, 683, 1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, 171, 1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, 2731, 1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, 171, 1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, 683, 1, 3, 1, 11, 1, 3
Offset: 1

Views

Author

Bill Gosper, Sep 07 2016

Keywords

Comments

Describes one of the two patterns of spacings of preimages of quadruple points of the Hilbert curve, H(t), 0 <= t <= 1. If H fills the complex unit square [0,1] X [0,i], H(0)=0, H(1)=1, then 1/2 + i/4 is a quadruple point with preimages t in {5/48, 7/48, 41/48, 43/48}. If we can characterize the rest of the quadruple points along the vertical bisector 1/2 + iy, all the rest are generated recursively by the to-quadrant maps (H/i + i)/2, (H + i)/2, (H + i + 1)/2, and (i H + 2)/2. Julian Ziegler Hunts has privately observed that H = 1/2 + ir is a quadruple point for all dyadic rational r in (0,1/2). E.g., the 31 r with denominator 64, i.e., 1/64, 3/64, ..., 31/64 generate preimage 4-tuples
{{1025, 1027, 11261, 11263}, {1037, 1039, 11249, 11251},
{1073, 1075, 11213, 11215}, {1085, 1087, 11201, 11203},
{1217, 1219, 11069, 11071}, {1229, 1231, 11057, 11059},
{1265, 1267, 11021, 11023}, {1277, 1279, 11009, 11011},
{1793, 1795, 10493, 10495}, {1805, 1807, 10481, 10483},
{1841, 1843, 10445, 10447}, {1853, 1855, 10433, 10435},
{1985, 1987, 10301, 10303}, {1997, 1999, 10289, 10291},
{2033, 2035, 10253, 10255}, {2045, 2047, 10241, 10243}}/12288
with differences
{{1, 1, -1, -1}, {3, 3, -3, -3}, {1, 1, -1, -1}, {11, 11, -11, -11},
{1, 1, -1, -1}, {3, 3, -3, -3}, {1, 1, -1, -1}, {43, 43, -43, -43},
{1, 1, -1, -1}, {3, 3, -3, -3}, {1, 1, -1, -1}, {11, 11, -11, -11},
{1, 1, -1, -1}, {3, 3, -3, -3}, {1, 1, -1, -1}}/1024
But the r in (1/2,1) are 1/6th as dense. The relevant quadruple points with denominator 2^n are 1/2 + i (6k - mod(5^n, 12))/2^n, 1 <= k < 2^n/6. E.g., if n = 6, then r is in {37/64, 43/64, 49/64, 55/64, 61/64} and the preimage 4-tuples of 1/2 + ir have differences {{-11, -11, 11, 11}, {-1, -1, 1, 1}, {-3, -3, 3, 3}, {-1, -1, 1, 1}}5/1024 (the reverse of) probably just -5*(this sequence).

Examples

			A(4) = 11. Thus
Table[unbert[1/2 + (2*4+1) I/2^n] - unbert[1/2 + (2*4-1) I/2^n], {n, 5, 9}]
{{11/256, 11/256, -11/256, -11/256},
{11/1024, 11/1024, -11/1024, -11/1024},
{11/4096, 11/4096, -11/4096, -11/4096},
{11/16384, 11/16384, -11/16384, -11/16384},
{11/65536, 11/65536, -11/65536, -11/65536}}
where unbert(H(t)) = {t}, the multivalued inverse Hilbert function (with I = sqrt(-1). See the definition of unbert[] in the MATHEMATICA section.
Note that this table must have n > 4, lest (2*4+1)/2^n > 1/2.

Links

Crossrefs

Cf. A000007, A000695, A001511, A007583, A115716, A163361, A260482.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 0,
      `if`(n::odd, 1, 4*a(n/2)-1))
    end:
seq(a(n), n=1..100); # Alois P. Heinz, Sep 07 2016
  • Mathematica
    (* Cf. the numerators of Out[339], below*)
hilbert[t_] :=
piecewiserecursivefractal[t, Identity, {Min[4, 1 + Floor[4*#]]} &,
    {1 - 4*# &, 4*# - 1 &, 4*# - 2 &, 4 - 4*# &},
    {I*(1 - #)/2 &, (I + #)/2 &, (I + 1 + #)/2 &, 1 + #*I/2 &}]
(* E.g., hilbert[1/2] {1/2 + I/2} *)
unbert[z_] :=
piecewiserecursivefractal[z, Identity,
     If[0 <= Re[#] <= 1 && 0 <= Im[#] <= 1,
   Range[4], {}] &,
    {1 - 2*#/I &, 2*# - I &, 2*# - I - 1 &, (# - 1)*2/I &},
    {(1 - #)/4 &, (# + 1)/4 &, (# + 2)/4 &, 1 - #/4 &}]
(* unbert[1/2 + I/2] {1/6, 1/2, 5/6} a triple point: hilbert/@% {{1/2 + I/2}, {1/2 + I/2}, {1/2 + I/2}} *)
ClearAll[piecewiserecursivefractal];
piecewiserecursivefractal[x_, f_, which_, iters_, fns_] :=
CheckAbort[
  Check[piecewiserecursivefractal[x, g_, which, iters,
     fns] = ((piecewiserecursivefractal[x, h_, which, iters, fns] :=
       Block[{y}, y /. Solve[f[y] == h[y], y]]);
     Union @@ ((fns[[#]] /@
           piecewiserecursivefractal[iters[[#]][x],
            Composition[f, fns[[#]]], which, iters, fns]) & /@
        which[x])),
   Abort[], {$RecursionLimit::reclim, $RecursionLimit::reclim2}],
  piecewiserecursivefractal[x, g_, which, iters, fns] =.; Abort[]]
(* For a simpler but less bulletproof version, see the MATHEMATICA section of A260482 *)
In[338]:= unbert /@ (1/2 + I Range[1/32, 15/32, 1/16])
Out[338]= {{257/3072, 259/3072, 2813/3072, 2815/3072},
             {269/3072, 271/3072, 2801/3072, 2803/3072},
             {305/3072, 307/3072, 2765/3072, 2767/3072},
             {317/3072, 319/3072, 2753/3072, 2755/3072},
             {449/3072, 451/3072, 2621/3072, 2623/3072},
             {461/3072, 463/3072, 2609/3072, 2611/3072},
             {497/3072, 499/3072, 2573/3072, 2575/3072},
             {509/3072, 511/3072, 2561/3072, 2563/3072}}
In[339]:= Differences@%
Out[339]= {{1/256, 1/256, -1/256, -1/256},
             {3/256, 3/256, -3/256, -3/256},
             {1/256, 1/256, -1/256, -1/256},
             {11/256, 11/256, -11/256, -11/256},
             {1/256, 1/256, -1/256, -1/256},
             {3/256, 3/256, -3/256, -3/256},
             {1/256, 1/256, -1/256, -1/256}}
(* Check that %338[[1]] is a quadruple point *)
In[340]:= hilbert /@ %%[[1]]
Out[340]= {{1/2 + I/32}, {1/2 + I/32}, {1/2 + I/32}, {1/2 + I/32}}
In[341]:= Select[Range[0, 1, 1/512], Length[unbert[# + I/2] > 3] &]
Out[341]= {}
(* I.e., there aren't any quadruple points on the horizontal bisector of the unit square! Other such horizontal and vertical lines of dyadic rationals intersect a dense set of quadruple points. *)
a[n_] := (2^(2*IntegerExponent[n, 2]+1) + 1)/3; Array[a, 100] (* Amiram Eldar, Dec 18 2023 *)
  • PARI
    a(n)= fromdigits(binary(n), 4)-fromdigits(binary(n-1), 4) \\ Bill McEachen, Dec 20 2024

Formula

a(n) = (2 + 4^A001511(n))/6.
G.f.: A(x) - 4*A(x^2) = x/(1+x).
From Alois P. Heinz, Sep 07 2016: (Start)
a(2^n) = A007583(n).
a(2^n+n) = a(n) + A000007(n).
(a(2*n)+1)/4 = a(n) for n>0. (End)
a(n) = A000695(n) - A000695(n-1). - Bill McEachen, Oct 30 2020
G.f.: Sum_{k>=0} 4^k * x^(2^k) / (1 + x^(2^k)). - Ilya Gutkovskiy, Dec 14 2020

Extensions

Keyword:mult added by Andrew Howroyd, Aug 06 2018

A115718 Inverse of number triangle A115717; a divide-and-conquer related triangle.

Original entry on oeis.org

1, 0, 1, -3, 1, 1, 0, 0, 0, 1, -3, -3, 1, 1, 1, 0, 0, 0, 0, 0, 1, -3, -3, -3, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, -3, -3, -3, -3, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -3, -3, -3, -3, -3, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -3, -3, -3, -3, -3, -3, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Paul Barry, Jan 29 2006

Keywords

Comments

Product of A115713 and (1/(1-x), x).
Row sums are 1,1,-1,1,-3,1,-5,1,-7,1, ... with g.f. (1+x-3*x^2-x^3)/(1-x^2)^2.
Row sums of inverse are A115716.

Examples

			Triangle begins
   1;
   0,  1;
  -3,  1,  1;
   0,  0,  0,  1;
  -3, -3,  1,  1,  1;
   0,  0,  0,  0,  0,  1;
  -3, -3, -3,  1,  1,  1,  1;
   0,  0,  0,  0,  0,  0,  0,  1;
  -3, -3, -3, -3,  1,  1,  1,  1,  1;
   0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
  -3, -3, -3, -3, -3,  1,  1,  1,  1,  1,  1;
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
  -3, -3, -3, -3, -3, -3,  1,  1,  1,  1,  1,  1,  1;
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
  -3, -3, -3, -3, -3, -3, -3,  1,  1,  1,  1,  1,  1,  1,  1;

Links

Crossrefs

Cf. A115713, A115716, A115717, A237420.

Programs

  • Mathematica
    T[n_, k_]:= If[OddQ[n], If[kG. C. Greubel, Nov 29 2021 *)
  • Sage
    def A115718(n,k):
    if (n%2==0): return 0 if (kA115718(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Nov 29 2021

Formula

From G. C. Greubel, Nov 29 2021: (Start)
T(2*n, k) = -3 if (k < n/2) otherwise 1.
T(2*n+1, k) = 0 if (k < n) otherwise 1.
Sum_{k=0..n} T(n, k) = (1/2)*(2 + (1 + (-1)^n)*n) = 1 + A237420(n). (End)
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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