A112627 -id:A112627 - cp's OEIS frontend

A114014 Expansion of g.f. (1 + 2x)^4/((1 + x)(1 - 16*x^2)).

1, 7, 33, 127, 529, 2031, 8465, 32495, 135441, 519919, 2167057, 8318703, 34672913, 133099247, 554766609, 2129587951, 8876265745, 34073407215, 142020251921, 545174515439, 2272324030737, 8722792247023, 36357184491793

Offset: 0

Roger L. Bagula, Jan 31 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,16,16).

Cf. A112627.

[1,7] cat [(1/30)*(4^(n-1)*(243 + 5*(-1)^n) - 2*(-1)^n): n in [2..40]]; // G. C. Greubel, Jul 07 2021

CoefficientList[Series[(1+2*x)^4/((1+x)*(1-16*x^2)), {x, 0, 40}], x]
a[n_]:= a[n]= If[n<2, 7^n, If[n==2, 33, 4*a[n-1] +(-1)^n*(4^(n-1) -1)/3]];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Jul 07 2021 *)
LinearRecurrence[{-1,16,16},{1,7,33,127,529},30] (* Harvey P. Dale, Aug 07 2023 *)

[1,7]+[(1/30)*(4^(n-1)*(243 + 5*(-1)^n) - 2*(-1)^n) for n in (2..40)] # G. C. Greubel, Jul 07 2021

G.f.: (-1)*(x + 1/2)^4/((x - 1/4)*(x + 1/4)*(x + 1)).
From Colin Barker, Dec 03 2012: (Start)
a(n) = (5*(-4)^n - 8*(-1)^n + 243*4^n)/120 for n>1.
G.f.: (1 +8*x +24*x^2 +32*x^3 +16*x^4)/((1+x)*(1-4*x)*(1+4*x)). (End)
From G. C. Greubel, Jul 07 2021: (Start)
a(n) = 4*a(n-1) + (-1)^n*(4^(n-1) -1)/3, n>2, with a(0) = 1, a(1) = 7, and a(2) = 33.
E.g.f.: (1/120)(243*exp(4*x) + 5*exp(-4*x) - 8*exp(-x) - 120*(1 + x)). (End)

New name and edited by G. C. Greubel, Jul 07 2021

A115243 G.f.: (4x^2 + 2x)/(4x^3 - x^2 - 4x + 1).

Original entry on oeis.org

0, 2, 12, 50, 204, 818, 3276, 13106, 52428, 209714, 838860, 3355442, 13421772, 53687090, 214748364, 858993458, 3435973836, 13743895346, 54975581388, 219902325554, 879609302220, 3518437208882, 14073748835532, 56294995342130, 225179981368524, 900719925474098

Offset: 0

Roger L. Bagula, Mar 04 2006

Inverse Z-transform of polynomial in A112627.

a(n) is also the number of corners in the n-th approximation of the Hilbert Curve. The 1st Hilbert Curve approximation has 2 corners. To find a(n) given a(n - 1), look at how the n-th Hilbert Curve approximation is constructed: duplicate the (n-1)-th approximation 4 times and connect the duplicates with 3 line segments. a(n) will always be 4 * a(n - 1) corners from the 4 duplicates plus 4 new corners if n is even or 2 new corners if n is odd. - Mikel Mcdaniel, Jan 10 2019

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (4,1,-4).

Cf. A112627.

[(4^(n+1)+(-1)^n)/5 - 1: n in [0..25]]; // Vincenzo Librandi, Jan 10 2019

seq((4^(n+1)+(-1)^n)/5 - 1, n=0..100); # Robert Israel, Mar 09 2016

Table[InverseZTransform[(1 + 2*x)/(1 - x - 16*x^2 + 16*x^3), x, n]*2^( 2*n), {n, 1, 25}]
LinearRecurrence[{4, 1, -4}, {0, 2, 12}, 50] (* G. C. Greubel, Feb 07 2016 *)

a(n) = (bitneg(0,2*n+2)-1)\5; \\ Kevin Ryde, May 05 2023

a(n) = InverseZTransform[(1 + 2*x)/(1 - x - 16*x^2 + 16*x^3), x, n] * 2^(2*n).
a(n) = 5*a(n-1)-4*a(n-2) +2*(-1)^n.
a(n) = 4*a(n-1)+a(n-2)-4*a(n-3). - Gary Detlefs Dec 17 2010
a(n) = (4^(n+1)+(-1)^n)/5 - 1. - Robert Israel, Mar 09 2016
a(n) = 4*a(n-1)+3+(-1)^n. - Mikel Mcdaniel, Jan 10 2019

Entry revised by N. J. A. Sloane, Dec 18 2010

A182460 a(n) = (3/5)*2^(4n+1) - (1/5).

Original entry on oeis.org

1, 19, 307, 4915, 78643, 1258291, 20132659, 322122547, 5153960755, 82463372083, 1319413953331, 21110623253299, 337769972052787, 5404319552844595, 86469112845513523, 1383505805528216371, 22136092888451461939, 354177486215223391027, 5666839779443574256435, 90669436471097188102963, 1450710983537555009647411

Offset: 0

Brad Clardy, Apr 30 2012

Bisection of A112627.

Index entries for linear recurrences with constant coefficients, signature (17,-16).

Cf. A112627, A016029, A153893.

[(3/5)*2^(4*n+1) - (1/5): n in [0..20]];

(3*2^(4*Range[0,20]+1)-1)/5 (* or *) LinearRecurrence[{17,-16},{1,19},30] (* Harvey P. Dale, Jul 21 2021 *)

a(n) = (3/5)*2^(4*n+1) - (1/5).
a(n) = 16*a(n-1) + 3 for n > 0.
a(n) = (1/5)*A153893(4*n+1).
a(n) = A016029(4*n+2).
a(n) = A112627(2*n+1).
G.f.: (1+2*x)/((1-x)*(1-16*x)). - Colin Barker, May 06 2012

A273180 Numbers n such that ror(n) + rol(n) is a power of 2, where ror(n)=A038572(n) is n rotated one binary place to the right, rol(n)=A006257(n) is n rotated one binary place to the left.

Original entry on oeis.org

1, 2, 6, 19, 38, 102, 307, 614, 1638, 4915, 9830, 26214, 78643, 157286, 419430, 1258291, 2516582, 6710886, 20132659, 40265318, 107374182, 322122547, 644245094, 1717986918, 5153960755, 10307921510, 27487790694, 82463372083, 164926744166, 439804651110

Offset: 1

Alex Ratushnyak, May 17 2016

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,17,0,0,-16).

Cf. A006257, A038572, A088161, A088163, A112627, A273105, A273106.

#include 
int main(int argc, char** argv)
{
  unsigned long long x, n, BL=0;
  for (n=1; n>0; ++n) {
    if ((n & (n-1))==0)  ++BL;
    x = (n>>1) + ((n&1) << (BL-1));   // A038572(n)
    x+= (n*2) - (1ull<A006257(n)  for n>0
    if ((x & (x-1))==0)  printf("%lld, ", n);
  }
}

Select[Range[10^6], IntegerQ@ Log2[FromDigits[RotateRight@ #, 2] + FromDigits[RotateLeft@ #, 2]] &@ IntegerDigits[#, 2] &] (* or *)
Rest@ CoefficientList[Series[x (1 + 2 x + 6 x^2 + 2 x^3 + 4 x^4)/((1 - x) (1 + x + x^2) (1 - 16 x^3)), {x, 0, 30}], x] (* Michael De Vlieger, May 19 2016 *)

Vec(x*(1+2*x+6*x^2+2*x^3+4*x^4)/((1-x)*(1+x+x^2)*(1-16*x^3)) + O(x^50)) \\ Colin Barker, May 19 2016

From Colin Barker, May 19 2016: (Start)
a(n) = 17*a(n-3) - 16*a(n-6) for n>6.
G.f.: x*(1+2*x+6*x^2+2*x^3+4*x^4) / ((1-x)*(1+x+x^2)*(1-16*x^3)).
(End)

A309709 Number of binary digits that change when n is multiplied by 4.

Original entry on oeis.org

0, 2, 2, 4, 2, 2, 4, 4, 2, 4, 2, 4, 4, 4, 4, 4, 2, 4, 4, 6, 2, 2, 4, 4, 4, 6, 4, 6, 4, 4, 4, 4, 2, 4, 4, 6, 4, 4, 6, 6, 2, 4, 2, 4, 4, 4, 4, 4, 4, 6, 6, 8, 4, 4, 6, 6, 4, 6, 4, 6, 4, 4, 4, 4, 2, 4, 4, 6, 4, 4, 6, 6, 4, 6, 4, 6, 6, 6, 6, 6, 2, 4, 4, 6, 2, 2, 4, 4

Offset: 0

Ali Sada, Aug 14 2019

All terms are even.

			00101_2 * 100_2 = 10100_2: 2 bits changed, so a(5) = 2.

David A. Corneth, Table of n, a(n) for n = 0..9999
Joerg Arndt, Matters Computational (The Fxtbook)
Index entries for sequences related to binary expansion of n

Cf. A000120, A048725, A112627.

Cf. also A007302, A069010.

a:= n-> add(i, i=Bits[Split](Bits[Xor](n*4,n))):
seq(a(n), n=0..120);  # Alois P. Heinz, Aug 23 2019

a[n_] := Total@ IntegerDigits[BitXor[n, 4 n], 2]; Array[a, 88, 0] (* Giovanni Resta, Sep 19 2019 *)

A309709(n) = hammingweight(bitxor(n, n<<2)); \\ Antti Karttunen, Aug 22 2019

def A309709(n):
    s = ""
    while n > 0:
        s, n = str(n%2)+s,n//2
    s, s4, i, j = "00"+s, s+"00", 0, 0
    while i < len(s):
        if s[i] != s4[i]:
            j = j+1
        i = i+1
    return j # A.H.M. Smeets, Aug 23 2019

a(n) = A000120(A048725(n)). - Antti Karttunen, Aug 22 2019

a(A112627(n)) = 2*n and A112627(n) is the first position where 2*n occurs in this sequence. - David A. Corneth, Sep 19 2019

A113968 a(0) = 0 and a(n) = (5(-4)^n + 16(-1)^n + 9*4^n)/240 for n >= 1.

Original entry on oeis.org

0, 0, 1, 1, 15, 17, 239, 273, 3823, 4369, 61167, 69905, 978671, 1118481, 15658735, 17895697, 250539759, 286331153, 4008636143, 4581298449, 64138178287, 73300775185, 1026210852591, 1172812402961, 16419373641455, 18764998447377

Offset: 0

Roger L. Bagula, Jan 31 2006

Index entries for linear recurrences with constant coefficients, signature (-1,16,16).

Cf. A112627.

a[n_] := If[n == 0, 1, (5(-4)^n + 16(-1)^n + 9*4^n) / 240];
a /@ Range[0, 25] (* Jean-François Alcover, Mar 30 2021 *)

From Colin Barker, Dec 03 2012: (Start)
a(n) = (5*(-4)^n + 16*(-1)^n + 9*4^n)/240 for n>0.
G.f.: -x^2*(2*x+1) / ((x+1)*(4*x-1)*(4*x+1)). (End)

New name (using Colin Barker's formula) from Joerg Arndt, Aug 30 2022

cp's OEIS Frontend

A114014 Expansion of g.f. (1 + 2*x)^4/((1 + x)*(1 - 16*x^2)).

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A115243 G.f.: (4*x^2 + 2*x)/(4*x^3 - x^2 - 4*x + 1).

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A182460 a(n) = (3/5)*2^(4n+1) - (1/5).

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A273180 Numbers n such that ror(n) + rol(n) is a power of 2, where ror(n)=A038572(n) is n rotated one binary place to the right, rol(n)=A006257(n) is n rotated one binary place to the left.

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A309709 Number of binary digits that change when n is multiplied by 4.

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A113968 a(0) = 0 and a(n) = (5*(-4)^n + 16*(-1)^n + 9*4^n)/240 for n >= 1.

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A114014 Expansion of g.f. (1 + 2x)^4/((1 + x)(1 - 16*x^2)).

A115243 G.f.: (4x^2 + 2x)/(4x^3 - x^2 - 4x + 1).

A113968 a(0) = 0 and a(n) = (5(-4)^n + 16(-1)^n + 9*4^n)/240 for n >= 1.