A231664 -id:A231664 - cp's OEIS frontend

A053737 Sum of digits of (n written in base 4).

0, 1, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 7, 2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 7, 5, 6, 7, 8, 3, 4, 5, 6, 4, 5, 6, 7, 5, 6, 7, 8, 6, 7, 8, 9, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 7, 2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 7, 5, 6, 7, 8, 3, 4, 5, 6, 4, 5, 6, 7, 5

Offset: 0

Henry Bottomley, Mar 28 2000

Also the fixed point of the morphism 0->{0,1,2,3}, 1->{1,2,3,4}, 2->{2,3,4,5}, etc. - Robert G. Wilson v, Jul 27 2006

			a(20) = 1+1+0 = 2 because 20 is written as 110 base 4.
From _Omar E. Pol_, Feb 21 2010: (Start)
This can be written as a triangle (cf. A000120):
  0,
  1,2,3,
  1,2,3,4,2,3,4,5,3,4,5,6,
  1,2,3,4,2,3,4,5,3,4,5,6,4,5,6,7,2,3,4,5,3,4,5,6,4,5,6,7,5,6,7,8,3,4,5,6,4,5,6,7,5,6,7,8,6,7,8,9,
  1,2,3,4,2,3,4,5,3,4,5,6,4,5,6,7,2,3,4,5,3,4,5,6,4,5,6,7,5,6,7,8,3,4,5,6,4,...
where the rows converge to A173524.
(End)

Donovan Johnson, Table of n, a(n) for n = 0..10000
F. T. Adams-Watters and F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS Vol. 12 (2009), Article 09.5.6.
Steve Butler and Ron L. Graham, Shuffling with ordered cards, arXiv 1003:4422 [math.CO], 2010.
Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Digit Sum.

Cf. A231664-A231667, A138530.
Cf. A173524. - Omar E. Pol, Feb 21 2010
Sum of digits of n written in bases 2-16: A000120, A053735, this sequence, A053824, A053827, A053828, A053829, A053830, A007953, A053831, A053832, A053833, A053834, A053835, A053836.
Related base-4 sequences: A053737, A230631, A230632, A010064, A230633, A230634, A230635, A230636, A230637, A230638, A010065 (trajectory of 1).
Cf. A007090, A239690.

a053737 n = if n == 0 then 0 else a053737 m + r where (m, r) = divMod n 4
-- Reinhard Zumkeller, Mar 19 2015

for u=0:104; sol(u+1)=sum(dec2base(u,4)-'0');end
sol % Marius A. Burtea, Jan 17 2019

[&+Intseq(n,4):n in [0..104]]; // Marius A. Burtea, Jan 17 2019

A053737 := proc(n)
    add(d,d=convert(n,base,4)) ;
end proc: # R. J. Mathar, Oct 31 2012

Table[Plus @@ IntegerDigits[n, 4], {n, 0, 100}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> {a, a+1, a+2, a+3}] &, {0}, 4] (* Robert G. Wilson v, Jul 27 2006 *)
DigitSum[Range[0, 100], 4] (* Paolo Xausa, Aug 01 2024 *)

PARI
```
a(n)=if(n<1,0,if(n%4,a(n-1)+1,a(n/4)))
```

a(n) = sumdigits(n, 4); \\ Michel Marcus, Aug 24 2019

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(4n+i) = a(n)+i for 0 <= i <= 3.
a(n) = n - 3*Sum_{k>0} floor(n/4^k) = n - 3*A054893(n). (End)
G.f.: (Sum_{k>=0} (x^(4^k) + 2*x^(2*4^k) + 3*x^(3*4^k))/(1 + x^(4^k) + x^(2*4^k) + x^(3*4^k)))/(1-x). - Franklin T. Adams-Watters, Nov 03 2005
a(n) = A138530(n,4) for n > 3. - Reinhard Zumkeller, Mar 26 2008
a(n) = Sum_{k>=0} A030386(n,k). - Philippe Deléham, Oct 21 2011
a(n) = A007953(A007090(n)). - Reinhard Zumkeller, Mar 19 2015
a(0) = 0; a(n) = a(n - 4^floor(log_4(n))) + 1. - Ilya Gutkovskiy, Aug 23 2019
Sum_{n>=1} a(n)/(n*(n+1)) = 4*log(4)/3 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

A231667 a(n) = Sum_{i=0..n} digsum_4(i)^4, where digsum_4(i) = A053737(i).

Original entry on oeis.org

0, 1, 17, 98, 99, 115, 196, 452, 468, 549, 805, 1430, 1511, 1767, 2392, 3688, 3689, 3705, 3786, 4042, 4058, 4139, 4395, 5020, 5101, 5357, 5982, 7278, 7534, 8159, 9455, 11856, 11872, 11953, 12209, 12834, 12915, 13171, 13796, 15092, 15348, 15973, 17269, 19670, 20295, 21591, 23992, 28088, 28169, 28425, 29050, 30346, 30602, 31227, 32523, 34924, 35549, 36845, 39246, 43342

Offset: 0

N. J. A. Sloane, Nov 13 2013

Marius A. Burtea, Table of n, a(n) for n = 0..10000
Jean Coquet, Power sums of digital sums, J. Number Theory 22 (1986), no. 2, 161-176.
P. J. Grabner, P. Kirschenhofer, H. Prodinger, R. F. Tichy, On the moments of the sum-of-digits function, PDF, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), 263-271, Kluwer Acad. Publ., Dordrecht, 1993.
J.-L. Mauclaire, Leo Murata, On q-additive functions. I, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.
J.-L. Mauclaire, Leo Murata, On q-additive functions. II, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 9, 441-444.
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag. 41 1968 21-25.

Cf. A053737, A231664, A231665, A231666.

a(n) = sum(i=0, n, sumdigits(i, 4)^4); \\ Michel Marcus, Sep 20 2017

A231665 a(n) = Sum_{i=0..n} digsum_4(i)^2, where digsum_4(i) = A053737(i).

Original entry on oeis.org

0, 1, 5, 14, 15, 19, 28, 44, 48, 57, 73, 98, 107, 123, 148, 184, 185, 189, 198, 214, 218, 227, 243, 268, 277, 293, 318, 354, 370, 395, 431, 480, 484, 493, 509, 534, 543, 559, 584, 620, 636, 661, 697, 746, 771, 807, 856, 920, 929, 945, 970, 1006, 1022, 1047, 1083, 1132, 1157, 1193, 1242, 1306, 1342, 1391, 1455, 1536, 1537, 1541, 1550, 1566, 1570, 1579, 1595, 1620, 1629

Offset: 0

N. J. A. Sloane, Nov 13 2013

Marius A. Burtea, Table of n, a(n) for n = 0..10000
Jean Coquet, Power sums of digital sums, J. Number Theory 22 (1986), no. 2, 161-176.
P. J. Grabner, P. Kirschenhofer, H. Prodinger, R. F. Tichy, On the moments of the sum-of-digits function, PDF, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), 263-271, Kluwer Acad. Publ., Dordrecht, 1993.
J.-L. Mauclaire, Leo Murata, On q-additive functions. I, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.
J.-L. Mauclaire, Leo Murata, On q-additive functions. II, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 9, 441-444.
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag. 41 1968 21-25.

Cf. A053737, A231664, A231666, A231667.

for u=0:2000; v(u+1)=sum(dec2base(u,4)-'0');end
sol=cumsum(v.^2); % Marius A. Burtea, Jan 18 2019

a(n) = sum(i=0, n, sumdigits(i, 4)^2); \\ Michel Marcus, Sep 20 2017

A231666 a(n) = Sum_{i=0..n} digsum_4(i)^3, where digsum_4(i) = A053737(i).

Original entry on oeis.org

0, 1, 9, 36, 37, 45, 72, 136, 144, 171, 235, 360, 387, 451, 576, 792, 793, 801, 828, 892, 900, 927, 991, 1116, 1143, 1207, 1332, 1548, 1612, 1737, 1953, 2296, 2304, 2331, 2395, 2520, 2547, 2611, 2736, 2952, 3016, 3141, 3357, 3700, 3825, 4041, 4384, 4896, 4923, 4987, 5112, 5328, 5392, 5517, 5733, 6076, 6201, 6417, 6760, 7272, 7488, 7831, 8343, 9072, 9073, 9081, 9108, 9172

Offset: 0

N. J. A. Sloane, Nov 13 2013

Marius A. Burtea, Table of n, a(n) for n = 0..10000
Jean Coquet, Power sums of digital sums, J. Number Theory 22 (1986), no. 2, 161-176.
P. J. Grabner, P. Kirschenhofer, H. Prodinger, R. F. Tichy, On the moments of the sum-of-digits function, PDF, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), 263-271, Kluwer Acad. Publ., Dordrecht, 1993.
J.-L. Mauclaire, Leo Murata, On q-additive functions. I, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.
J.-L. Mauclaire, Leo Murata, On q-additive functions. II, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 9, 441-444.
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag. 41 1968 21-25.

Cf. A053737, A231664, A231665, A231667.

for u=0:2000; v(u+1)=sum(dec2base(u,4)-'0'); end
sol=cumsum(v.^3); % Marius A. Burtea, Jan 18 2019

a(n) = sum(i=0, n, sumdigits(i, 4)^3); \\ Michel Marcus, Sep 20 2017

A333596 a(0) = 0; for n > 0, a(n) = a(n-1) + (number of 1's and 3's in base-4 representation of n) - (number of 0's and 2's in base-4 representation of n).

Original entry on oeis.org

0, 1, 0, 1, 1, 3, 3, 5, 3, 3, 1, 1, 1, 3, 3, 5, 4, 5, 4, 5, 6, 9, 10, 13, 12, 13, 12, 13, 14, 17, 18, 21, 18, 17, 14, 13, 12, 13, 12, 13, 10, 9, 6, 5, 4, 5, 4, 5, 4, 5, 4, 5, 6, 9, 10, 13, 12, 13, 12, 13, 14, 17, 18, 21, 19, 19, 17, 17, 17, 19, 19, 21, 19, 19

Offset: 0

Alexander Van Plantinga, Mar 27 2020

Local maxima values minus 1 are divisible by 4.

For a digit-wise recurrence, it's convenient to sum n terms so b(n) = a(n-1) = Sum_{i=0..n-1} A334841(i). Then b(4n+r) = 4*b(n) + r*A334841(n) + (1 if r even), for 0 <= r <= 3 and 4n+r >= 1. This is 4 copies of terms 0..n-1 and r copies of the following n. The new lowest digits cancel when r is odd, or net +1 when r is even. Repeatedly expanding gives the PARI code below. - Kevin Ryde, Jun 02 2020

			      n in    #odd    #even
  n  base 4  digits - digits + a(n-1) = a(n)
  =  ======  ===============================
  0    0        0   -                     0
  1    1        1   -    0   +    0   =   1
  2    2        0   -    1   +    1   =   0
  3    3        1   -    0   +    0   =   1
  4   10        1   -    1   +    1   =   1
  5   11        2   -    0   +    1   =   3
  6   12        1   -    1   +    3   =   3
  7   13        2   -    0   +    3   =   5

Cf. A053737, A010065, A037863, A268289, A007090, A231664, A301336, A334841.

a:= proc(n) option remember; `if`(n=0, 0, a(n-1) +add(
     `if`(i in [1, 3], 1, -1), i=convert(n, base, 4)))
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, May 30 2020

f[n_] := Total[(-1)^(r = Range[0, 3]) * DigitCount[n, 4, r]]; a[0] = 0; a[n_] := a[n] = a[n - 1] - f[n]; Array[a, 100, 0] (* Amiram Eldar, Apr 24 2020 *)

a(n) = my(v=digits(n+1,4),s=0); for(i=1,#v, my(t=v[i]); v[i]=t*s+!(t%2); s-=(-1)^t); fromdigits(v,4); \\ Kevin Ryde, May 30 2020

b(n)=my(d=digits(n,4)); -sum(i=1,#d,(-1)^d[i])
first(n)=my(s); concat(0,vector(n,k,s+=b(k))) \\ Charles R Greathouse IV, Jul 04 2020

import numpy as np
def qnary(n):
    e = n//4
    q = n%4
    if n == 0 : return 0
    if e == 0 : return q
    if e != 0 : return np.append(qnary(e), q)
m = 400
v = [0]
for i in range(1,m+1) :
    t = np.array(qnary(i))
    t[t%2 != 0] = 1
    t[t%2 == 0] = -1
    v = np.append(v, np.sum([np.sum(t), v[i-1]]))

from itertools import accumulate
def A334841(n):
    return 2*bin(n)[-1:1:-2].count('1')-(len(bin(n))-1)//2 if n > 0 else 0
A333596_list = list(accumulate(A334841(n) for n in range(10000))) # Chai Wah Wu, Sep 03 2020

qnary = function(n, e, q){
  e = floor(n/4)
  q = n%%4
  if(n == 0 ){return(0)}
  if(e == 0){return(q)}
  else{return(c(qnary(e), (q)))}
}
m = 400
s = seq(2,m)
v = c(0)
for(i in s){
  x = qnary(i-1)
  x[which(x%%2!=0)] = 1
  x[which(x%%2==0)] = -1
  v[i] = sum(x,v[i-1])
}

cp's OEIS Frontend

A053737 Sum of digits of (n written in base 4).

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A231667 a(n) = Sum_{i=0..n} digsum_4(i)^4, where digsum_4(i) = A053737(i).

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A231665 a(n) = Sum_{i=0..n} digsum_4(i)^2, where digsum_4(i) = A053737(i).

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A231666 a(n) = Sum_{i=0..n} digsum_4(i)^3, where digsum_4(i) = A053737(i).

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A333596 a(0) = 0; for n > 0, a(n) = a(n-1) + (number of 1's and 3's in base-4 representation of n) - (number of 0's and 2's in base-4 representation of n).

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