A131858 -id:A131858 - cp's OEIS frontend

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)).

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

Reinhard Zumkeller, Jul 22 2007

sign

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).

R. Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A007088.

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 *)

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

Reinhard Zumkeller, Jul 22 2007

nonn

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

conjecture: a(n) mod 5 = 0.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

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

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

Reinhard Zumkeller, Jul 22 2007

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A131851, A131853, A131858, A131860.

A131859 Numbers m such that A131851(m) = 1.

Original entry on oeis.org

1, 3, 9, 11, 16, 18, 21, 23, 24, 26, 29, 31, 33, 35, 41, 43, 48, 50, 53, 55, 56, 58, 61, 63, 81, 83, 89, 91, 113, 115, 121, 123, 129, 131, 137, 139, 144, 146, 149, 151, 152, 154, 157, 159, 161, 163, 169, 171, 176, 178, 181, 183, 184, 186, 189, 191, 209, 211, 217

Offset: 1

Reinhard Zumkeller, Jul 22 2007

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A131851, A131858, A131857, A131859, A131861, A131863.

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

Reinhard Zumkeller, Jul 22 2007

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A131851, A131853, A131856, A131858.

A330714 For any n >= 0 with binary expansion Sum_{k=0..w} b_k * 2^k, let h(n) = Sum_{k=0..w} b_k * i^k (where i denotes the imaginary unit); a(n) is the square of the modulus of h(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 2, 1, 1, 2, 0, 1, 2, 1, 1, 0, 1, 4, 2, 5, 0, 1, 1, 2, 2, 5, 1, 4, 1, 2, 0, 1, 1, 2, 4, 5, 2, 1, 5, 4, 0, 1, 1, 2, 1, 0, 2, 1, 2, 5, 5, 8, 1, 2, 4, 5, 1, 4, 2, 5, 0, 1, 1, 2, 1, 0, 2, 1, 4, 1, 5, 2, 2, 1, 1, 0, 5, 2, 4, 1, 0, 1, 1, 2, 1

Offset: 0

Seiichi Manyama, Dec 27 2019

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Cf. A007088, A131851, A131852, A131853, A131856, A131858, A131860, A290886, A318479.

a[0] = 0; a[n_] := a[n] = a[Floor[n/2]]*I + Mod[n, 2]; Table[Abs[a[n]]^2, {n, 0, 100}] (* Amiram Eldar, May 06 2021, after Jean-François Alcover at A131851 *)

{a(n) = my(d=Vecrev(digits(n, 2))); norm(sum(k=1, #d, d[k]*I^k))}

a(n) = A131851(n)^2 + A131852(n)^2.

cp's OEIS Frontend

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)).

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

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

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A131859 Numbers m such that A131851(m) = 1.

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

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A330714 For any n >= 0 with binary expansion Sum_{k=0..w} b_k * 2^k, let h(n) = Sum_{k=0..w} b_k * i^k (where i denotes the imaginary unit); a(n) is the square of the modulus of h(n).

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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)).