A131851 -id:A131851 - cp's OEIS frontend

A131852 Imaginary part of the function z defined in A131851.

0, 0, 1, 1, 0, 0, 1, 1, -1, -1, 0, 0, -1, -1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, -1, -1, 0, 0, -1, -1, 0, 0, 1, 1, 2, 2, 1, 1, 2, 2, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, -1, -1, 0, 0, -1, -1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, -1, -1, 0, 0, -1, -1, 0, 0, 1, 1, 2, 2, 1, 1, 2, 2, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 0

Offset: 0

Reinhard Zumkeller, Jul 22 2007

sign

a(A000079(n)) = A056594(n+3);
a(A131855(n)) = 0; a(A131862(n)) > 0; a(A131857(n)) = 1; a(A131864(n)) < 0;
for n > 0: a(A098704(n+1)) = n and abs(a(m)) < n for m < A098704(n+1).

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

z[0] = 0; z[n_] := z[n] = z[Floor[n/2]]*I + Mod[n, 2]; Table[z[n] // Im, {n, 0, 120}] (* Jean-François Alcover, Mar 02 2019 *)

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 *)

A131854 Numbers m such that A131851(m) = 0.

Original entry on oeis.org

0, 2, 5, 7, 8, 10, 13, 15, 20, 22, 28, 30, 32, 34, 37, 39, 40, 42, 45, 47, 52, 54, 60, 62, 65, 67, 73, 75, 80, 82, 85, 87, 88, 90, 93, 95, 97, 99, 105, 107, 112, 114, 117, 119, 120, 122, 125, 127, 128, 130, 133, 135, 136, 138, 141, 143, 148, 150, 156, 158, 160, 162, 165

Offset: 1

Reinhard Zumkeller, Jul 22 2007

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A131853, A131855, A131859, A131861, A131863.

Z:= proc(n) option remember;
I*procname(floor(n/2))+(n mod 2)
end proc:
Z(0):= 0:
select(Re@Z=0, [$0..1000]); # Robert Israel, Dec 18 2017

z[0] = 0; z[n_] := z[n] = z[Floor[n/2]]*I + Mod[n, 2]; Select[Range[0, 200], Re[z[#]] == 0&] (* Jean-François Alcover, Jan 31 2018 *)

isok(n) = {d = Vecrev(binary(n)); real(sum(k=1, #d, d[k]*I^(k-1))) == 0;} \\ Michel Marcus, Jan 31 2018

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.

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

Reinhard Zumkeller, Jul 22 2007

nonn

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

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

Cf. A131851, A131853, A131856, 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.

A131861 Numbers m such that A131851(m) > 0.

Original entry on oeis.org

1, 3, 9, 11, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 29, 31, 33, 35, 41, 43, 48, 49, 50, 51, 53, 55, 56, 57, 58, 59, 61, 63, 81, 83, 89, 91, 113, 115, 121, 123, 129, 131, 137, 139, 144, 145, 146, 147, 149, 151, 152, 153, 154, 155, 157, 159, 161, 163, 169, 171, 176

Offset: 1

Reinhard Zumkeller, Jul 22 2007

nonn

Cf. A131862, A131854, A131859, A131863.

A131863 Numbers m such that A131851(m) < 0.

Original entry on oeis.org

4, 6, 12, 14, 36, 38, 44, 46, 64, 66, 68, 69, 70, 71, 72, 74, 76, 77, 78, 79, 84, 86, 92, 94, 96, 98, 100, 101, 102, 103, 104, 106, 108, 109, 110, 111, 116, 118, 124, 126, 132, 134, 140, 142, 164, 166, 172, 174, 192, 194, 196, 197, 198, 199, 200, 202, 204, 205, 206

Offset: 1

Reinhard Zumkeller, Jul 22 2007

nonn

Cf. A131864, A131854, A131859, A131861.

A131865 Partial sums of powers of 16.

Original entry on oeis.org

1, 17, 273, 4369, 69905, 1118481, 17895697, 286331153, 4581298449, 73300775185, 1172812402961, 18764998447377, 300239975158033, 4803839602528529, 76861433640456465, 1229782938247303441, 19676527011956855057, 314824432191309680913, 5037190915060954894609

Offset: 0

Reinhard Zumkeller, Jul 22 2007

16 = 2^4 is the growth measure for the Jacobsthal spiral (compare with phi^4 for the Fibonacci spiral). - Paul Barry, Mar 07 2008
Second quadrisection of A115451. - Paul Curtz, May 21 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=16, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 1, a(n-1) = det(A). - Milan Janjic, Feb 21 2010
Partial sums are in A014899. Also, the sequence is related to A014931 by A014931(n+1) = (n+1)*a(n) - Sum_{i=0..n-1} a(i) for n>0. - Bruno Berselli, Nov 07 2012
a(n) is the total number of holes in a certain box fractal (start with 16 boxes, 1 hole) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 28 2015
Except for 1 and 17, all terms are Brazilian repunits numbers in base 16, and so belong to A125134. All terms >= 273 are composite because a(n) = ((4^(n+1) + 1) * (4^(n+1) - 1))/15. - Bernard Schott, Jun 06 2017
The sequence in binary is 1, 10001, 100010001, 1000100010001, 10001000100010001, ... cf. Plouffe link, A330135. - Frank Ellermann, Mar 05 2020

			a(3) = 1 + 16 + 256 + 4096 = 4369 = in binary: 1000100010001.
a(4) = (16^5 - 1)/15 = (4^5 + 1) * (4^5 - 1)/15 = 1025 * 1023/15 = 205 * 341 = 69905 = 11111_16. - _Bernard Schott_, Jun 06 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..800
A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
Kival Ngaokrajang, Illustration of initial terms
Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. See Table 1.
Simon Plouffe, Identities and approximations inspired from Ramanujan notebooks, III, 2009.
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (17,-16).

Cf. A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723. - M. F. Hasler, Nov 05 2012

[(16^(n+1)-1)/15: n in [0..20]]; // Vincenzo Librandi, Sep 17 2011

A131865:=n->(16^(n+1)-1)/15: seq(A131865(n), n=0..30); # Wesley Ivan Hurt, Apr 29 2017

Table[(2^(4 n) - 1)/15, {n, 16}] (* Robert G. Wilson v, Aug 22 2007 *)
Accumulate[16^Range[0,20]] (* or *) LinearRecurrence[{17,-16},{1,17},20] (* Harvey P. Dale, Jul 19 2019 *)

a[0]:0$
a[n]:=16*a[n-1]+1$
A131865(n):=a[n]$
makelist(A131865(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

A131865(n)=16^n\15  \\ M. F. Hasler, Nov 05 2012

def A131865(n): return (1<<(n+1<<2))//15 # Chai Wah Wu, Nov 10 2022

[gaussian_binomial(n,1,16) for n in range(1,18)] # Zerinvary Lajos, May 28 2009

a(n) = if n=0 then 1 else a(n-1) + A001025(n).
for n > 0: A131851(a(n)) = n and abs(A131851(m)) < n for m < a(n).
a(n) = A098704(n+2)/2.
a(n) = (16^(n+1) - 1)/15. - Bernard Schott, Jun 06 2017
a(n) = (A001025(n+1) - 1)/15.
a(n) = 16*a(n-1) + 1. - Paul Curtz, May 20 2008
G.f.: 1 / ( (16*x-1)*(x-1) ). - R. J. Mathar, Feb 06 2011
E.g.f.: exp(x)*(16*exp(15*x) - 1)/15. - Stefano Spezia, Mar 06 2020

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A131852 Imaginary part of the function z defined in A131851.

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

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A131854 Numbers m such that A131851(m) = 0.

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

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A131858 Numbers m such that z(m)=(1,0) 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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A131861 Numbers m such that A131851(m) > 0.

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A131863 Numbers m such that A131851(m) < 0.

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A131865 Partial sums of powers of 16.

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