A195156 -id:A195156 - cp's OEIS frontend

A144864 a(n) = (4*16^(n-1)-1)/3.

1, 21, 341, 5461, 87381, 1398101, 22369621, 357913941, 5726623061, 91625968981, 1466015503701, 23456248059221, 375299968947541, 6004799503160661, 96076792050570581, 1537228672809129301, 24595658764946068821, 393530540239137101141, 6296488643826193618261, 100743818301219097892181

Offset: 1

Artur Jasinski, Sep 23 2008

Old name was: A144863, read as binary numbers, converted to base 10.
All numbers in this sequence for n>1 are congruent to 5 mod 16. - Artur Jasinski, Sep 25 2008
From Omar E. Pol, Sep 10 2011: (Start)
It appears that this is a bisection of A002450.
It appears that this is a bisection of A084241.
It appears that this is a bisection of A153497.
It appears that this is a bisection of A088556, if n>=2.
(End)
All of the above is trivially true. - Joerg Arndt, Aug 19 2014
The aerated sequence (b(n))n>=1 = [1, 0, 21, 0, 341, 0, 5461, 0, 87381, ...] is a fourth-order linear divisibility sequence; that is, a(n) divides a(m) whenever n divides m. It is the case P1 = 0, P2 = -9, Q = -4 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Aug 26 2022

Vincenzo Librandi, Table of n, a(n) for n = 1..500
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
Index entries for linear recurrences with constant coefficients, signature (17,-16).

Cf. A001025, A002450, A013776, A056830, A084241, A088556, A094028, A098704, A131865, A135576, A144863, A153497.

Third quadrisection of Jacobsthal numbers A001045; the other quadrisections are A195156 (first), A139792 (second), and A141060 (fourth).

[16^n/12-1/3: n in [1..20]]; // Vincenzo Librandi, Aug 03 2011

Table[1/3 (-1 + 16^(n - 1)) + 16^(n - 1), {n, 1, 17}] (* Artur Jasinski, Sep 25 2008 *)
LinearRecurrence[{17,-16},{1,21},20] (* Harvey P. Dale, Jun 29 2022 *)

vector(66,n,(4*16^(n-1)-1)/3) \\ Joerg Arndt, Aug 19 2014

a(n) = 16^n/12 - 1/3; a(n) = 16*a(n-1) + 5, a(1)=1. - Artur Jasinski, Sep 25 2008
G.f.: x*(1+4*x) / ( (16*x-1)*(x-1) ). - R. J. Mathar, Jan 06 2011
a(n)=b such that Integral_{x=-Pi/2..Pi/2} (-1)^(n+1)*2^(2*n-3)*(cos((2*n-1)*x))/(5/4+sin(x)) dx = c+b*log(3). - Francesco Daddi, Aug 02 2011
a(n) = (2^(4*n-2)-1)/3. - Klaus Purath, Jan 31 2021
From Jianing Song, Aug 30 2022: (Start)
a(n) = A001045(4*n-2).
a(n+1) - a(n) = 10*A013776(n-1) = 20*A001025(n-1) for n >= 1.
a(n) = 10*A098704(n) + 1 = 20*A131865(n-2) + 1 for n >= 2. (End)
E.g.f.: (exp(16*x) - 4*exp(x) + 3)/12. - Stefano Spezia, Apr 18 2024

New name from Joerg Arndt, Aug 19 2014

A141060 Fourth quadrisection of Jacobsthal numbers A001045: a(n)=16a(n-1)-5.

Original entry on oeis.org

3, 43, 683, 10923, 174763, 2796203, 44739243, 715827883, 11453246123, 183251937963, 2932031007403, 46912496118443, 750599937895083, 12009599006321323, 192153584101141163, 3074457345618258603, 49191317529892137643

Offset: 0

Paul Curtz, Jul 30 2008

Jacobsthal numbers ending with the decimal digit 3. - Jianing Song, Aug 30 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (17,-16).

The other quadrisections of A001045 are A195156 (first), A139792 (second), and A144864 (third).

[(1/3)*(1+8*16^n): n in [0..25]]; // Vincenzo Librandi, May 25 2011

LinearRecurrence[{17,-16},{3,43},30] (* Harvey P. Dale, Mar 16 2015 *)

a(n)=8*16^n\3+1 \\ Charles R Greathouse IV, May 25 2011

def A141060(n): return ((1<<(n<<2)+3)|1)//3 # Chai Wah Wu, Apr 18 2025

a(n) = A139792(n) + A013776(n).
a(n+1) - a(n) = 10*A013709(n) = 40*A001025(n).
G.f.: (3-8*x)/((1-x)*(1-16*x)). [Colin Barker, Apr 05 2012]
a(0)=3, a(1)=43, a(n)=17*a(n-1)-16*a(n-2). - Harvey P. Dale, Mar 16 2015
From Jianing Song, Aug 30 2022: (Start)
a(n) = A001045(4*n+3).
a(n) = 10*A141032(n) + 3 = 20*A098704(n+1) + 1 = 40*A131865(n-1) + 1 for n >= 1. (End)

A139792 First quadrisection of A139763 (1, 2, 3, 4, 11, ...).

Original entry on oeis.org

1, 11, 171, 2731, 43691, 699051, 11184811, 178956971, 2863311531, 45812984491, 733007751851, 11728124029611, 187649984473771, 3002399751580331, 48038396025285291, 768614336404564651, 12297829382473034411, 196765270119568550571, 3148244321913096809131

Offset: 0

Paul Curtz, May 21 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (17,-16)

Second quadrisection of Jacobsthal numbers A001045; the other quadrisections are A195156 (first), A144864 (third), and A141060 (fourth).

[(1+2*16^n)/3: n in [0..20]]; // Vincenzo Librandi, Aug 09 2011

Table[(1 + 2^(4*n+1))/3, {n,0,20}] (* G. C. Greubel, Nov 03 2018 *)

vector(20, n, n--; (1 + 2^(4*n+1))/3) \\ G. C. Greubel, Nov 03 2018

a(n) = 16*a(n-1) - 5.
a(n) = 10*A131865(n) + 1.
G.f.: ( 1-6*x ) / ( (16*x-1)*(x-1) ). - R. J. Mathar, Feb 06 2011
E.g.f.: (exp(x) + 2*exp(16*x))/3. - G. C. Greubel, Nov 03 2018
From Jianing Song, Aug 30 2022: (Start)
a(n) = A001045(4*n+1).
a(n+1) - a(n) = 10*A001025(n). (End)

A300867 a(n) is the least positive k such that k * n is a Fibbinary number (A003714).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 3, 3, 5, 3, 11, 1, 1, 1, 7, 1, 1, 3, 3, 3, 13, 5, 3, 3, 5, 11, 11, 1, 1, 1, 39, 1, 1, 7, 7, 1, 1, 1, 3, 3, 13, 3, 7, 3, 21, 13, 23, 5, 5, 3, 3, 3, 9, 5, 11, 11, 9, 11, 43, 1, 1, 1, 35, 1, 1, 39, 15, 1, 1, 1, 31, 7, 57, 7, 7, 1

Offset: 0

Rémy Sigrist, Mar 14 2018

This sequence is well defined: for any positive n, according to the pigeonhole principle, A195156(i) mod n = A195156(j) mod n for some distinct i and j, hence n divides f = abs(A195156(i) - A195156(j)), and as f is a Fibbinary number, a(n) <= f/n.

All terms are odd.

			The first terms, alongside the binary representation of n * a(n), are:
  n  a(n)   bin(n * a(n))
  -- ----   -------------
   0    1           0
   1    1           1
   2    1          10
   3    3        1001
   4    1         100
   5    1         101
   6    3       10010
   7    3       10101
   8    1        1000
   9    1        1001
  10    1        1010
  11    3      100001
  12    3      100100
  13    5     1000001
  14    3      101010
  15   11    10100101
  16    1       10000
  17    1       10001
  18    1       10010
  19    7    10000101
  20    1       10100

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Colored logarithmic scatterplot of the first 1000000 terms (where the color is function of A070939(n * a(n)))

Cf. A003714, A070939, A195156, A300889.

a(n) = forstep (k=1, oo, 2, if (bitand(k*n, 2*k*n)==0, return (k)))

a(n) = A300889(n) / n for any n > 0.
a(2*n) = a(n).
a(n) = 1 iff n belongs to A003714.

A377412 a(n) is the least k > 0 such that k*n belongs to A126684.

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 7, 3, 1, 9, 1, 31, 7, 5, 3, 91, 1, 1, 9, 55, 1, 1, 31, 3, 7, 13, 5, 3, 3, 9, 91, 11, 1, 33, 1, 39, 9, 113, 55, 7, 1, 25, 1, 127, 31, 121, 3, 443, 7, 21, 13, 87, 5, 97, 3, 19, 3, 73, 9, 1199, 91, 21, 11, 1387, 1, 1, 33, 983, 1, 1, 39, 19, 9

Offset: 0

Rémy Sigrist, Oct 27 2024

This sequence is well defined: for any positive integer n, according to the pigeonhole principle, A195156(i) mod n = A195156(j) mod n for some distinct i and j, hence n divides b = abs(A195156(i) - A195156(j)), and as b belongs to A126684, a(n) <= b/n.

			The first terms, alongside the binary expansion of a(n)*n, are:
  n   a(n)  bin(a(n)*n)
  --  ----  -----------
   0     1            0
   1     1            1
   2     1           10
   3     7        10101
   4     1          100
   5     1          101
   6     7       101010
   7     3        10101
   8     1         1000
   9     9      1010001
  10     1         1010
  11    31    101010101
  12     7      1010100

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, C++ program

See A300867 for a similar sequence.

Cf. A032937, A126684, A195156, A377413.

(* Increase nmax for n>92 in A377412 *) nmax = 1000; b[n_] := FromDigits[IntegerDigits[n, 2], 4];A126684=Union[A000695 = b /@ Range[0, nmax], 2 A000695][[1 ;; nmax+1]] ;A377412={};Do[k=0;Until[MemberQ[A126684,k*n],k++]; AppendTo[A377412,k],{n,0,72}];A377412 (* James C. McMahon, Oct 29 2024 *)

a(n) >= A300867(n).

a(n) = 1 iff n belongs to A126684.

cp's OEIS Frontend

A144864 a(n) = (4*16^(n-1)-1)/3.

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A141060 Fourth quadrisection of Jacobsthal numbers A001045: a(n)=16a(n-1)-5.

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A139792 First quadrisection of A139763 (1, 2, 3, 4, 11, ...).

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A300867 a(n) is the least positive k such that k * n is a Fibbinary number (A003714).

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A377412 a(n) is the least k > 0 such that k*n belongs to A126684.

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