A066321 -id:A066321 - cp's OEIS frontend

A009545 Expansion of e.g.f. sin(x)*exp(x).

0, 1, 2, 2, 0, -4, -8, -8, 0, 16, 32, 32, 0, -64, -128, -128, 0, 256, 512, 512, 0, -1024, -2048, -2048, 0, 4096, 8192, 8192, 0, -16384, -32768, -32768, 0, 65536, 131072, 131072, 0, -262144, -524288, -524288, 0, 1048576, 2097152, 2097152, 0, -4194304, -8388608, -8388608, 0, 16777216, 33554432

Offset: 0

R. H. Hardin

Also first of the two associated sequences a(n) and b(n) built from a(0)=0 and b(0)=1 with the formulas a(n) = a(n-1) + b(n-1) and b(n) = -a(n-1) + b(n-1). The initial terms of the second sequence b(n) are 1, 1, 0, -2, -4, -4, 0, 8, 16, 16, 0, -32, -64, -64, 0, 128, 256, ... The points Mn(a(n)+b(n)*I) of the complex plane are located on the spiral logarithmic rho = 2*(1/2)^(2*theta)/Pi) and on the straight lines drawn from the origin with slopes: infinity, 1/2, 0, -1/2. - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 30 2007
A000225: (1, 3, 7, 15, 31, ...) = 2^n - 1 = INVERT transform of A009545 starting (1, 2, 2, 0, -4, -8, ...). (Cf. comments in A144081). - Gary W. Adamson, Sep 10 2008
Pisano period lengths: 1, 1, 8, 1, 4, 8, 24, 1, 24, 4, 40, 8, 12, 24, 8, 1, 16, 24, 72, 4, ... - R. J. Mathar, Aug 10 2012
The variant 0, 1, -2, 2, 0, -4, 8, -8, 0, 16, -32, 32, 0, -64, (with different signs) is the Lucas U(-2,2) sequence. - R. J. Mathar, Jan 08 2013
(1+i)^n = A146559(n) + a(n)*i where i = sqrt(-1). - Philippe Deléham, Feb 13 2013
This is the Lucas U(2,2) sequence. - Raphie Frank, Nov 28 2015
{A146559, A009545} are the difference analogs of {cos(x),sin(x)} (cf. [Shevelev] link). - Vladimir Shevelev, Jun 08 2017

N. J. A. Sloane, Table of n, a(n) for n = 0..2000, Apr 09 2016 (first 100 terms from T. D. Noe)
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=2, q=-2.
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs. (38) and (45), lhs, m=2.
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
N. J. A. Sloane, Table of n, (I-1)^n for n = 0..100
Wikipedia, Lucas sequence.
Index entries for linear recurrences with constant coefficients, signature (2,-2).
Index entries for sequences related to Chebyshev polynomials
Index entries for Lucas sequences

Cf. A009116. For minor variants of this sequence see A108520, A084102, A099087.
a(2*n) = A056594(n)*2^n, n >= 1, a(2*n+1) = A057077(n)*2^n.
This is the next term in the sequence A015518, A002605, A000129, A000079, A001477.
Cf. A000225, A144081. - Gary W. Adamson, Sep 10 2008
Cf. A146559.

I:=[0,1,2,2]; [n le 4 select I[n] else -4*Self(n-4): n in [1..60]]; // Vincenzo Librandi, Nov 29 2015

t1 := sum(n*x^n, n=0..100): F := series(t1/(1+x*t1), x, 100): for i from 0 to 50 do printf(`%d, `, coeff(F, x, i)) od: # Zerinvary Lajos, Mar 22 2009
G(x):=exp(x)*sin(x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..50 ); # Zerinvary Lajos, Apr 05 2009
A009545 := n -> `if`(n<2, n, 2^(n-1)*hypergeom([1-n/2, (1-n)/2], [1-n], 2)):
seq(simplify(A009545(n)), n=0..50); # Peter Luschny, Dec 17 2015

nn=104; Range[0,nn-1]! CoefficientList[Series[Sin[x]Exp[x], {x,0,nn}], x] (* T. D. Noe, May 26 2007 *)
Join[{a=0,b=1},Table[c=2*b-2*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)
f[n_] := (1 + I)^(n - 2) + (1 - I)^(n - 2); Array[f, 51, 0] (* Robert G. Wilson v, May 30 2011 *)
LinearRecurrence[{2,-2},{0,1},110] (* Harvey P. Dale, Oct 13 2011 *)

x='x+O('x^66); Vec(serlaplace(exp(x)*sin(x))) /* Joerg Arndt, Apr 24 2011 */

x='x+O('x^100); concat(0, Vec(x/(1-2*x+2*x^2))) \\ Altug Alkan, Dec 04 2015

def A009545(n): return ((0, 1, 2, 2)[n&3]<<((n>>1)&-2))*(-1 if n&4 else 1) # Chai Wah Wu, Feb 16 2024

[lucas_number1(n,2,2) for n in range(0, 51)] # Zerinvary Lajos, Apr 23 2009

def A146559():
    x, y = 0, -1
    while True:
        yield x
        x, y = x - y, x + y
a = A146559(); [next(a) for i in range(40)]  # Peter Luschny, Jul 11 2013

a(0)=0; a(1)=1; a(2)=2; a(3)=2; a(n) = -4*a(n-4), n>3. - Larry Reeves (larryr(AT)acm.org), Aug 24 2000
Imaginary part of (1+i)^n. - Marc LeBrun
G.f.: x/(1 - 2*x + 2*x^2).
E.g.f.: sin(x)*exp(x).
a(n) = S(n-1, sqrt(2))*(sqrt(2))^(n-1) with S(n, x)= U(n, x/2) Chebyshev's polynomials of the 2nd kind, Cf. A049310, S(-1, x) := 0.
a(n) = ((1+i)^n - (1-i)^n)/(2*i) = 2*a(n-1) - 2*a(n-2) (with a(0)=0 and a(1)=1). - Henry Bottomley, May 10 2001
a(n) = (1+i)^(n-2) + (1-i)^(n-2). - Benoit Cloitre, Oct 28 2002
a(n) = Sum_{k=0..n-1} (-1)^floor(k/2)*binomial(n-1, k). - Benoit Cloitre, Jan 31 2003
a(n) = 2^(n/2)sin(Pi*n/4). - Paul Barry, Sep 17 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k+1)*(-1)^k. - Paul Barry, Sep 20 2003
a(n+1) = Sum_{k=0..n} 2^k*A109466(n,k). - Philippe Deléham, Nov 13 2006
a(n) = 2*((1/2)^(2*theta(n)/Pi))*cos(theta(n)) where theta(4*p+1) = p*Pi + Pi/2, theta(4*p+2) = p*Pi + Pi/4, theta(4*p+3) = p*Pi - Pi/4, theta(4*p+4) = p*Pi - Pi/2, or a(0)=0, a(1)=1, a(2)=2, a(3)=2, and for n>3 a(n)=-4*a(n-4). Same formulas for the second sequence replacing cosines with sines. For example: a(0) = 0, b(0) = 1; a(1) = 0+1 = 1, b(1) = -0+1 = 1; a(2) = 1+1 = 2, b(2) = -1+1 = 0; a(3) = 2+0 = 2, b(3) = -2+0 = -2. - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 30 2007
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3), n > 3, which implies the sequence is identical to its fourth differences. Binomial transform of 0, 1, 0, -1. - Paul Curtz, Dec 21 2007
Logarithm g.f. arctan(x/(1-x)) = Sum_{n>0} a(n)/n*x^n. - Vladimir Kruchinin, Aug 11 2010
a(n) = A046978(n) * A016116(n). - Paul Curtz, Apr 24 2011
E.g.f.: exp(x) * sin(x) = x + x^2/(G(0)-x); G(k) = 2k + 1 + x - x*(2k+1)/(4k+3+x+x^2*(4k+3)/( (2k+2)*(4k+5) - x^2 - x*(2k+2)*(4k+5)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 15 2011
a(n) = Im( (1+i)^n ) where i=sqrt(-1). - Stanislav Sykora, Jun 11 2012
G.f.: x*U(0) where U(k) = 1 + x*(k+3) - x*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 10 2012
G.f.: G(0)*x/(2*(1-x)), where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013
G.f.: x + x^2*W(0), where W(k) = 1 + 1/(1 - x*(k+1)/( x*(k+2) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013
G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 - 2*x)/( x*(4*k+4 - 2*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 06 2013
a(n) = (A^n - B^n)/(A - B), where A = 1 + i and B = 1 - i; A and B are solutions of x^2 - 2*x + 2 = 0. - Raphie Frank, Nov 28 2015
a(n) = 2^(n-1)*hypergeom([1-n/2, (1-n)/2], [1-n], 2) for n >= 2. - Peter Luschny, Dec 17 2015
a(k+m) = a(k)*A146559(m) + a(m)*A146559(k). - Vladimir Shevelev, Jun 08 2017

Extended with signs by Olivier Gérard, Mar 15 1997
More terms from Larry Reeves (larryr(AT)acm.org), Aug 24 2000
Definition corrected by Joerg Arndt, Apr 24 2011

A009116 Expansion of e.g.f. cos(x) / exp(x).

Original entry on oeis.org

1, -1, 0, 2, -4, 4, 0, -8, 16, -16, 0, 32, -64, 64, 0, -128, 256, -256, 0, 512, -1024, 1024, 0, -2048, 4096, -4096, 0, 8192, -16384, 16384, 0, -32768, 65536, -65536, 0, 131072, -262144, 262144, 0, -524288, 1048576, -1048576, 0, 2097152, -4194304

Offset: 0

R. H. Hardin

Apart from signs, generated by 1,1 position of H_2^n = [1,1;-1,1]^n; and a(n) = 2^(n/2)*cos(Pi*n/2). - Paul Barry, Feb 18 2004
Equals binomial transform of "Period 4, repeat [1, 0, -1, 0]". - Gary W. Adamson, Mar 25 2009
Pisano period lengths: 1, 1, 8, 1, 4, 8, 24, 1, 24, 4, 40, 8, 12, 24, 8, 1, 16, 24, 72, 4, ... - R. J. Mathar, Aug 10 2012

			G.f. = 1 - x + 2*x^3 - 4*x^4 + 4*x^5 - 8*x^7 + 16*x^8 - 16*x^9 + 32*x^11 - 64*x^12 + ...

N. J. A. Sloane, Table of n, a(n) for n = 0..2000
Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
N. J. A. Sloane, Table of n, (I-1)^n for n=0..100
Index entries for linear recurrences with constant coefficients, signature (-2,-2).

Cf. A009545, A016116, A075553, A090132, A098158, A099087, A146559.
(With different signs) row sums of triangle A104597.
Also related to A066321 and A271472.

R:=PowerSeriesRing(Rationals(), 50); Coefficients(R!(Laplace( Exp(-x)*Cos(x) ))); // G. C. Greubel, Jul 22 2018; Apr 17 2023

A009116 := n->add((-1)^j*binomial(n,2*j),j=0..floor(n/2));

n = 50; (* n = 2 mod 4 *) (CoefficientList[ Series[ Cos[x]/Exp[x], {x, 0, n}], x]* Table[k!, {k,0,n-1}] )[[1 ;; 45]] (* Jean-François Alcover, May 18 2011 *)
Table[(1/2)*((-1-I)^n + (-1+I)^n), {n,0,50}] (* Jean-François Alcover, Jan 31 2018 *)

{a(n) = if( n<0, 0, polcoeff( (1 + x) / (1 + 2*x + 2*x^2) + x * O(x^n), n))} /* Michael Somos, Nov 17 2002 */

def A009116(n): return 2^(n/2)*chebyshev_T(n, -1/sqrt(2))
[A009116(n) for n in range(41)] # G. C. Greubel, Apr 17 2023

Real part of (-1-i)^n. See A009545 for imaginary part. - Marc LeBrun
a(n) = -2 * (a(n-1) + a(n-2)); a(0)=1, a(1)=-1. - Michael Somos, Nov 17 2002
G.f.: (1 + x) / (1 + 2*x + 2*x^2).
E.g.f.: cos(x) / exp(x).
a(n) = Sum_{k=0..n} (-1)^k*A098158(n,k). - Philippe Deléham, Dec 04 2006
a(n)*(-1)^n = A099087(n) - A099087(n-1). - R. J. Mathar, Nov 18 2007
a(n) = (-1)^n*A146559(n). - Philippe Deléham, Dec 01 2008
From Paul Curtz, Jul 22 2011: (Start)
a(n) = -4*a(n-4).
a(n) = A016116(n) * A075553(n+6). (End)
E.g.f.: cos(x)/exp(x) = 1 - x/(G(0)+1), where G(k) = 4k+1-x+(x^2)*(4k+1)/((2k+1)*(4k+3)-(x^2)+x*(2k+1)*(4k+3)/( 2k+2-x+x*(2k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+2) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 20 2013
a(n) = (-1)^n*2^(n/2)*cos(n*Pi/4). - Nordine Fahssi, Dec 18 2013
a(n) = (-1)^floor((n+1)/2)*2^(n-1)*H(n, n mod 2, 1/2) for n >= 3 where H(n, a, b) = hypergeom([a - n/2, b - n/2], [1 - n], 2). - Peter Luschny, Sep 03 2019
a(n) = 2^(n/2)*ChebyshevT(n, -1/sqrt(2)). - G. C. Greubel, Apr 17 2023
a(n) = A108520(n-1)+A108520(n). - R. J. Mathar, May 09 2023

Extended with signs by Olivier Gérard, Mar 15 1997

Definition corrected by Joerg Arndt, Apr 29 2011

A342726 Niven numbers in base i-1: numbers that are divisible by the sum of their digits in base i-1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 12, 15, 16, 18, 20, 24, 25, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 50, 54, 60, 64, 65, 66, 70, 77, 80, 88, 90, 96, 99, 100, 110, 112, 120, 124, 125, 126, 130, 140, 144, 145, 147, 150, 156, 160, 168, 170, 180, 182, 184, 185, 186, 190, 192

Offset: 1

Amiram Eldar, Mar 19 2021

Numbers k that are divisible by A066323(k).

Equivalently, Niven numbers in base -4, since A066323(k) is also the sum of the digits of k in base -4.

			2 is a term since its representation in base i-1 is 1100 and 1+1+0+0 = 2 is a divisor of 2.
10 is a term since its representation in base i-1 is 111001100 and 1+1+1+0+0+1+1+0+0 = 5 is a divisor of 10.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Walter Penney, A "binary" system for complex numbers, Journal of the ACM, Vol. 12, No. 2 (1965), pp. 247-248.

Cf. A007608, A066321, A066323, A271472, A342725, A342727, A342728, A342729.

Similar sequences: A005349 (decimal), A049445 (binary), A064150 (ternary), A064438 (quaternary), A064481 (base 5), A118363 (factorial), A328208 (Zeckendorf), A328212 (lazy Fibonacci), A331085 (negaFibonacci), A333426 (primorial), A334308 (base phi), A331728 (negabinary), A342426 (base 3/2).

v = {{0, 0, 0, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}, {1, 1, 0, 1}}; q[n_] := Divisible[n, Total[Flatten @ v[[1 + Reverse @ Most[Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]]]]]]; Select[Range[200], q]

A271472 Binary representation of n in base i-1.

Original entry on oeis.org

0, 1, 1100, 1101, 111010000, 111010001, 111011100, 111011101, 111000000, 111000001, 111001100, 111001101, 100010000, 100010001, 100011100, 100011101, 100000000, 100000001, 100001100, 100001101, 110011010000, 110011010001, 110011011100, 110011011101, 110011000000, 110011000001

Offset: 0

N. J. A. Sloane, Apr 08 2016

This is A066321 converted from base 10 to base 2.
Every Gaussian integer r+s*i (r, s ordinary integers) has a unique representation as a sum of powers of t = i-1. For example 3 = 1+b^2+b^3, that is, "1101" in binary, which explains a(3) = 1101. See A066321 for further information.
From Jianing Song, Jan 22 2023: (Start)
Also binary representation of n in base -1-i.
Write out n in base -4 (A007608), then change each digit 0, 1, 2, 3 to 0000, 0001, 1100, 1101 respectively. (End)

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 172. (See also exercise 16, p. 177; answer, p. 494.)
W. J. Penney, A "binary" system for complex numbers, JACM 12 (1965), 247-248.

Chai Wah Wu, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Table of n, (I-1)^n for n=0..100
Wikipedia, Complex-base system

Cf. A066321.

a(n) = my(v = [n,0], x=0, digit=0, a, b); while(v!=[0,0], a=v[1]; b=v[2]; v[1]=-2*(a\2)+b; v[2]=-(a\2); x+=(a%2)*10^digit; digit++); x \\ Jianing Song, Jan 22 2023; [a,b] represents the number a + b*(-1+i)

from gmpy2 import c_divmod
u = ('0000','1000','0011','1011')
def A271472(n):
    if n == 0:
        return 0
    else:
        s, q = '', n
        while q:
            q, r = c_divmod(q, -4)
            s += u[r]
        return int(s[::-1]) # Chai Wah Wu, Apr 09 2016

A073791 Replace 4^k with (-4)^k in base 4 expansion of n.

Original entry on oeis.org

0, 1, 2, 3, -4, -3, -2, -1, -8, -7, -6, -5, -12, -11, -10, -9, 16, 17, 18, 19, 12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 32, 33, 34, 35, 28, 29, 30, 31, 24, 25, 26, 27, 20, 21, 22, 23, 48, 49, 50, 51, 44, 45, 46, 47, 40, 41, 42, 43, 36, 37, 38, 39, -64, -63, -62, -61, -68, -67, -66, -65, -72, -71, -70, -69

Offset: 0

Robert G. Wilson v, Aug 12 2002

Base 4 representation for n converted from base -4 to base 10.

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], 2020-2021.

Cf. A007090, A007608, A053985, A065369, A073792, A073793, A073794, A073795, A073796 & A073835.

Cf. A066321, A320283.

f[n_Integer, b_Integer] := Block[{l = IntegerDigits[n]}, Sum[l[[ -i]]*(-b)^(i - 1), {i, 1, Length[l]}]]; a = Table[ FromDigits[ IntegerDigits[n, 4]], {n, 0, 80}]; b = {}; Do[b = Append[b, f[a[[n]], 4]], {n, 1, 80}]; b

a(n) = subst(Pol(digits(n,4)), x, -4); \\ Michel Marcus, Jan 30 2019

a(4*k+m) = -4*a(k)+m for 0 <= m < 4. - Chai Wah Wu, Jan 16 2020

A066323 Number of one bits in binary representation of base i-1 expansion of n (where i = sqrt(-1)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 7, 8, 4, 5, 6, 7, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 8, 9, 5, 6, 7, 8, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 9, 10, 6, 7, 8, 9, 5, 6, 7, 8, 4, 5, 6, 7, 8, 9, 10, 11, 7, 8, 9, 10, 6, 7, 8, 9, 5, 6, 7, 8, 9, 10, 11, 12, 8, 9, 10, 11, 7, 8, 9, 10, 6, 7, 8, 9

Offset: 0

Marc LeBrun, Dec 14 2001

First differences are usually +1, occasionally -4 (because in base i-1 [3]+[7]=(+i)+(-i)=0) hence often a(i+j)=a(i)+a(j). Differences terms given here are period-16, but for full sequence is actually period-256 at least.

a(n) is the sum of the digits of n when written in base -4 using digits 0 to 3 (A007608). This is since in Penney's digit substitution for A066321, the base -4 digits 0 to 3 become bit strings of exactly 0 to 3 many 1-bits each respectively. - Kevin Ryde, Sep 09 2019

			A066321(4) = 464 = 111010000 (binary) so a(4) = 4.  Or A007608(4) == 130 in base -4 and sum of digits is a(4) = 1+3+0 = 4.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 172. (Also exercise 16, p. 177, answer, p. 494.)

David A. Corneth, Table of n, a(n) for n = 0..9999
Walter Penney, A Binary System for Complex Numbers, NSA Technical Journal, 1965.
Walter Penney, A Binary System for Complex Numbers, Journal of the Association for Computing Machinery (JACM), volume 12, number 2, April 1965, pages 247-248.

Cf. A066321, A000120, A007608, A342728.

a[n_] := Plus @@ Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]; Array[a, 100, 0] (* Amiram Eldar, Mar 22 2021 *)

a(n) = my(ret=0); while(n, ret+=n%4; n\=-4); ret; \\ Kevin Ryde, Sep 09 2019

a(n) = A000120(A066321(n)) = A007953(A007608(n)). - Kevin Ryde, Sep 09 2019

A342729 Self numbers in base i-1: numbers not of the form k + A066323(k).

Original entry on oeis.org

1, 3, 5, 7, 9, 22, 24, 26, 39, 41, 43, 56, 58, 60, 73, 75, 77, 90, 92, 94, 107, 109, 111, 136, 138, 140, 153, 155, 157, 170, 172, 174, 199, 201, 203, 216, 218, 220, 233, 235, 237, 262, 264, 266, 279, 281, 283, 296, 298, 300, 313, 315, 317, 330, 332, 334, 347, 349

Offset: 1

Amiram Eldar, Mar 19 2021

Equivalently, self numbers in base -4, since A066323(k) is also the sum of the digits of k in base -4.
Analogous to self numbers (A003052) using base i-1 representation (A271472) instead of decimal expansion.
The number of terms not exceeding 10^k, for k=1,2,..., is 5, 20, 155, 1507, 15008, 150007, 1500014, 15000011. Is the asymptotic density of this sequence exactly 3/20?

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Walter Penney, A "binary" system for complex numbers, Journal of the ACM, Vol. 12, No. 2 (1965), pp. 247-248.
Eric Weisstein's World of Mathematics, Self Number.
Wikipedia, Self number.

Cf. A007608, A066323, A066321, A271472, A342725, A342726, A342727, A342728.

Similar sequences: A003052 (decimal), A010061 (binary), A010064 (base 4), A010067 (base 6), A010070 (base 8), A339211 (Zeckendorf), A339212 (dual Zeckendorf), A339213 (base phi), A339214 (factorial base), A339215 (primorial base).

s[n_] := Module[{v = {{0, 0, 0, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}, {1, 1, 0, 1}}}, Plus @@ Flatten @ v[[1 + Reverse @ Most[Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]]]]]; f[n_] := n + s[n]; m = 1000; Complement[Range[m], Select[Union@Array[f, m], # <= m &]]

A342727 Digitally balanced numbers in base i-1: numbers that in base i-1 have the same number of 0's as 1's.

Original entry on oeis.org

2, 21, 26, 31, 36, 41, 46, 51, 310, 315, 325, 330, 335, 340, 345, 350, 355, 360, 365, 370, 375, 390, 395, 405, 410, 415, 420, 425, 430, 435, 455, 470, 475, 485, 490, 495, 535, 550, 555, 565, 570, 575, 580, 585, 590, 595, 600, 605, 610, 620, 625, 630, 635, 645

Offset: 1

Amiram Eldar, Mar 19 2021

			2 is a term since its representation in base i-1, 1100, has 2 0's and 2 1's.
21 is a term since its representation in base i-1, 110011010001, has 6 0's and 6 1's.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Walter Penney, A "binary" system for complex numbers, Journal of the ACM, Vol. 12, No. 2 (1965), pp. 247-248.

Cf. A066321, A066323, A271472, A342725, A342726, A342728, A342729.

Similar sequences: A031443 (binary), A210619 (Zeckendorf).

v = {{0, 0, 0, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}, {1, 1, 0, 1}}; balQ[n_] := Plus @@ (d = IntegerDigits[n]) == Length[d]/2; q[n_] := balQ @ FromDigits[Flatten@v[[1 + Reverse @ Most[Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]]]]]; Select[Range[1000], q]

A342728 a(n) is the least number k such that A066323(k) = n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 23, 39, 55, 71, 87, 103, 359, 615, 871, 1127, 1383, 1639, 5735, 9831, 13927, 18023, 22119, 26215, 91751, 157287, 222823, 288359, 353895, 419431, 1468007, 2516583, 3565159, 4613735, 5662311, 6710887, 23488103, 40265319, 57042535, 73819751

Offset: 0

Amiram Eldar, Mar 19 2021

a(n) is the least number k whose sum of digits in base i-1 (or in base -4) is n.

Amiram Eldar, Table of n, a(n) for n = 0..1000
Walter Penney, A "binary" system for complex numbers, Journal of the ACM, Vol. 12, No. 2 (1965), pp. 247-248.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,16,-16).

Cf. A007608, A066321, A066323, A271472, A342725, A342726, A342727, A342729.

Join[{0}, LinearRecurrence[{1,0,0,0,0,16,-16}, Range[7], 50]]

a(n) = n for n <= 7, and a(n) = a(n-1) + 16*a(n-6) - 16*a(n-7) for n > 7.
G.f.: x*(1 + x + x^2 + x^3 + x^4 + x^5 - 15*x^6)/(1 - x - 16*x^6 + 16*x^7). - Stefano Spezia, Mar 20 2021
From Greg Dresden, Jun 21 2021: (Start)
a(3*n+1) = (24 + (4^n)*(25 -  9*(-1)^n))/40.
a(3*n+2) = (24 + (4^n)*(50 +  6*(-1)^n))/40.
a(3*n+3) = (24 + (4^n)*(75 + 21*(-1)^n))/40. (End)

A342725 Numbers that are palindromic in base i-1.

Original entry on oeis.org

0, 1, 13, 17, 189, 205, 257, 273, 3005, 3069, 3277, 3341, 4033, 4097, 4305, 4369, 48061, 48317, 49149, 49405, 52173, 52429, 53261, 53517, 64449, 64705, 65537, 65793, 68561, 68817, 69649, 69905, 768957, 769981, 773309, 774333, 785405, 786429, 789757, 790781, 834509

Offset: 1

Amiram Eldar, Mar 19 2021

Amiram Eldar, Table of n, a(n) for n = 1..512
Walter Penney, A "binary" system for complex numbers, Journal of the ACM, Vol. 12, No. 2 (1965), pp. 247-248.

Cf. A066321, A066323, A271472, A342726, A342727, A342728, A342729.

Similar sequences: A002113 (decimal), A006995 (binary), A014190 (base 3), A014192 (base 4), A029952 (base 5), A029953 (base 6), A029954 (base 7), A029803 (base 8), A029955 (base 9), A046807 (factorial base), A094202 (Zeckendorf), A331191 (dual Zeckendorf), A331891 (negabinary), A333423 (primorial base).

v = {{0, 0, 0, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}, {1, 1, 0, 1}}; q[n_] := PalindromeQ @ FromDigits[Flatten @ v[[1 + Reverse @ Most[Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]]]]]; Select[Range[0, 10^4], q]

13 is a term since its base-(i-1) presentation is 100010001 which is palindromic.

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A009545 Expansion of e.g.f. sin(x)*exp(x).

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A009116 Expansion of e.g.f. cos(x) / exp(x).

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A342726 Niven numbers in base i-1: numbers that are divisible by the sum of their digits in base i-1.

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A271472 Binary representation of n in base i-1.

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A073791 Replace 4^k with (-4)^k in base 4 expansion of n.

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A066323 Number of one bits in binary representation of base i-1 expansion of n (where i = sqrt(-1)).

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