A033264 -id:A033264 - cp's OEIS frontend

A213629 In binary representation: T(n,k) = number of (possibly overlapping) occurrences of k in n, triangle read by rows, 1<=k<=n.

1, 1, 1, 2, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 0, 1, 3, 0, 2, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 0, 1, 2, 2, 0, 0, 1, 0, 0, 0, 0, 1, 3, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 2, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 3, 1, 1, 0, 1, 1, 0, 0

Offset: 1

Reinhard Zumkeller, Jun 17 2012

The definition is based on the definition of pattern functions in the paper of Allouche and Shallit;
sum of n-th row = A029931(n);
T(n,1) = A000120(n);
T(n,2) = A033264(n) for n > 1;
T(n,3) = A014081(n) for n > 2;
T(n,4) = A056978(n) for n > 3;
T(n,5) = A056979(n) for n > 4;
T(n,6) = A056980(n) for n > 5;
T(n,7) = A014082(n) for n > 6;
T(n,k) = 0 for k with floor(n/2) < k < n;
T(n,n) = 1;
A122953(n) = Sum_{k=1..n} A057427(T(n,k));
A005811(n) = T(n,1) + T(n,2) - T(n,3);
A007302(n) = A000120(n) - sum (A213629(n,A136412(k))).

			The triangle begins:
.   1:                        1
.   2:                      1   1
.   3:                    2   0   1
.   4:                  1   1   0   1
.   5:                2   1   0   0   1
.   6:              2   1   1   0   0   1
.   7:            3   0   2   0   0   0   1
.   8:          1   1   0   1   0   0   0   1
.   9:        2   1   0   1   0   0   0   0   1
.  10:      2   2   0   0   1   0   0   0   0   1
.  11:    3   1   1   0   1   0   0   0   0   0   1
.  12:  2   1   1   1   0   1   0   0   0   0   0   1.

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
J.-P. Allouche, J. Shallit, The Ring of k-regular Sequences II, Example 4, p. 12
Index entries for sequences related to binary expansion of n

Cf. A030308, A007088.

import Data.List (inits, tails, isPrefixOf)
a213629 n k = a213629_tabl !! (n-1) !! (k-1)
a213629_row n = a213629_tabl !! (n-1)
a213629_tabl = map f $ tail $ inits $ tail $ map reverse a030308_tabf where
   f xss = map (\xs ->
           sum $ map (fromEnum . (xs `isPrefixOf`)) $ tails $ last xss) xss

t[n_, k_] := (idn = IntegerDigits[n, 2]; idk = IntegerDigits[k, 2]; ln = Length[idn]; lk = Length[idk]; For[cnt = 0; i = 1, i <= ln - lk + 1, i++, If[idn[[i ;; i + lk - 1]] == idk, cnt++]]; cnt); Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 22 2012 *)

A072219 Any number n can be written uniquely in the form n = 2^k_1 - 2^k_2 + 2^k_3 - ... + 2^k_{2r+1} where the signs alternate, there are an odd number of terms, and k_1 > k_2 > k_3 > ... > k_{2r+1} >= 0; sequence gives number of terms 2r+1.

Original entry on oeis.org

1, 1, 3, 1, 3, 3, 3, 1, 3, 3, 5, 3, 3, 3, 3, 1, 3, 3, 5, 3, 5, 5, 5, 3, 3, 3, 5, 3, 3, 3, 3, 1, 3, 3, 5, 3, 5, 5, 5, 3, 5, 5, 7, 5, 5, 5, 5, 3, 3, 3, 5, 3, 5, 5, 5, 3, 3, 3, 5, 3, 3, 3, 3, 1, 3, 3, 5, 3, 5, 5, 5, 3, 5, 5, 7, 5, 5, 5, 5, 3, 5, 5, 7, 5, 7, 7, 7, 5, 5, 5, 7, 5, 5, 5, 5, 3, 3, 3, 5, 3, 5, 5, 5, 3, 5

Offset: 1

N. J. A. Sloane, Jul 05 2002

2^k_1 is smallest power of 2 that is >= n.
The first Mathematica program computes the sequence for numbers 1 to 2^m. - T. D. Noe, Jul 15 2002
a(A000079(n)) = 1; a(A238246(n)) = 3; a(A238247(n)) = 5; a(A238248(n)) = 7. - Reinhard Zumkeller, Feb 20 2014
Add 1 to every other terms of A005811. - N. J. A. Sloane, Jan 14 2017

			1=1, 2=2, 3=4-2+1, 4=4, 5=8-4+1, 6=8-4+2, ...

P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, pp. 61-62.

Reinhard Zumkeller, Table of n, a(n) for n = 1..16384
S. Kropf and S. Wagner, q-Quasiadditive functions, arXiv:1605.03654 [math.CO], 2016. See section "The number of runs and the Gray code".
Index entries for sequences related to binary expansion of n

Cf. A000079, A005811, A030308, A072339, A065621, A238246, A238247, A238248.

Equals 2*A033264(n-1) + 1.

a072219 = (+ 1) . (* 2) . a033264 . subtract 1
-- Reinhard Zumkeller, Feb 20 2014

Needs["DiscreteMath`Combinatorica`"]; sumit[s_List] := Module[{i, ss=0}, Do[If[OddQ[i], ss+=s[[i]], ss-=s[[i]]], {i, Length[s]}]; ss]; m=8; powers=Table[2^i, {i, 0, m}]; lst=Table[0, {2^m}]; sets={}; Do[sets=Union[sets, KSubsets[powers, i]], {i, 1, m+1, 2}]; Do[t=sets[[i]]; lst[[sumit[t]]]=Length[t], {i, Length[sets]}]; lst
(* second program *)
a[n_] := 2 Count[Split[IntegerDigits[n-1, 2], #1 == 1 && #2 == 0 &], {1, 0} ] + 1; Array[a, 105] (* Jean-François Alcover, Apr 01 2016 *)

G.f.: 1/(1+x) + (1/(1-x)) * Sum_{r>=0} x^(2^r) / (1+x^(2^(r+1))). - Ramasamy Chandramouli, Dec 22 2012

More terms from T. D. Noe, Jul 15 2002

A175047 Write n in binary, then increase each run of 0's by one 0. a(n) is the decimal equivalent of the result.

Original entry on oeis.org

1, 4, 3, 8, 9, 12, 7, 16, 17, 36, 19, 24, 25, 28, 15, 32, 33, 68, 35, 72, 73, 76, 39, 48, 49, 100, 51, 56, 57, 60, 31, 64, 65, 132, 67, 136, 137, 140, 71, 144, 145, 292, 147, 152, 153, 156, 79, 96, 97, 196, 99, 200, 201, 204, 103, 112, 113, 228, 115, 120, 121, 124, 63, 128

Offset: 1

Leroy Quet, Dec 02 2009

From Reinhard Zumkeller, Dec 12 2009: (Start)
A070939(a(n)) = A070939(n) + A033264(n);
A171598 and A171599 give record values and where they occur. (End)

			12 in binary is 1100. Increase each run of 0 by one digit to get 11000, which is 24 in decimal. So a(12) = 24.

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A175046, A175048.
Cf. A084471. - Robert G. Wilson v, Dec 11 2009
Cf. A030308, A106151, A007088.

import Data.List (group)
a175047 = foldr (\b v -> 2 * v + b) 0 . concatMap
   (\bs@(b:_) -> if b == 0 then 0 : bs else bs) . group . a030308_row
-- Reinhard Zumkeller, Jul 05 2013

f[n_] := Block[{s = Split@ IntegerDigits[n, 2]}, FromDigits[ Flatten@ Insert[s, {0}, Table[{2 i}, {i, Floor[ Length@s/2]} ]], 2]]; Array[ f, 64] (* Robert G. Wilson v, Dec 11 2009 *)

from re import split
def A175047(n):
  return int(''.join(d+'0' if '0' in d else d for d in split('(0+)|(1+)',bin(n)[2:]) if d != '' and d != None),2) # Chai Wah Wu, Nov 21 2018

a(n) = if n<2 then n else 2*(1 + 0^((n+2) mod 4))*a([n/2]) + n mod 2. - Reinhard Zumkeller, Jan 20 2010

a(2^n) = 2^(n+1). - Chai Wah Wu, Nov 21 2018

Extended by Ray Chandler, Dec 18 2009

a(11) onwards from Robert G. Wilson v and Reinhard Zumkeller, Dec 11 2009

A196521 Decimal expansion of Pi/4-log(2)/2.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Oct 03 2011

			0.438824573117475654907044785090787437011542282663648828183396143330257...

L. B. W. Jolley, Summation of series, Dover Publications Inc., New York, 1961, p. 14 (eq. 72).

Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.

Cf. A003881, A016655 (10*log(2)/2), A033264.

Cf. A231902 (Pi/4+log(2)/2), A342316.

Digits:=100; evalf(Pi/4-log(2)/2); # Wesley Ivan Hurt, Dec 06 2013

RealDigits[Pi/4 - Log[2]/2, 10, 100] (* Wesley Ivan Hurt, Dec 06 2013 *)

Pi/4-log(2)/2 \\ Altug Alkan, Dec 14 2015

Equals 1 - 1/2 - 1/3 + 1/4 + 1/5 - ....
Equals Sum_{n>=0} 2/((4*n+2)*(4*n+3)). - Peter Luschny, Dec 06 2013
Equals Sum_{n>=1} (-1)^(n+1)/((2*n-1)*(2*n)). - Robert FERREOL, Dec 14 2015
Equals Integral_{x=0..1} (arctan(x)) dx = Integral_{x=0..Pi/4} (x / cos(x)^2) dx = Integral_{x=0..1/sqrt(2)} (arcsin(x)/(1-x^2)^(3/2)) dx. - Robert FERREOL, Dec 14 2015
Equals Integral_{x>=0} (exp(x) - 1)/(exp(2*x) + 1) dx. - Peter Bala, Nov 01 2019
From Bernard Schott, Sep 07 2020: (Start)
Equals Sum_{n>=1} (-1)^(n*(n-1)/2) / n [compare with A231902 formula].
Equals Sum_{n>=0} (8*n+5) / (4*(n+1)*(2*n+1)*(4*n+1)*(4*n+3)). (End)
Equals Sum_{k>=1} A033264(k)/(k*(k+1)) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021
From Peter Bala, Mar 04 2025: (Start)
Equals (1/2) * A342316.
Equals Integral_{x = 0..1} x/(x^2 - 2*x + 2) = Integral_{x = 0..1} x*(1 + x)/(2 - x^2*(1 - x)) dx.
Equals (5/2)*Sum_{n >= 1} 1/(n*binomial(3*n, n)*2^n). The first 10 terms of the series gives the approximate value 0.43882457311(68...), correct to 11 decimal places. (End)

A384715 a(n) = Sum_{k=0..n} (binomial(n, k) mod 4).

Original entry on oeis.org

1, 2, 4, 8, 4, 8, 12, 16, 4, 8, 12, 24, 12, 24, 24, 32, 4, 8, 12, 24, 12, 24, 32, 48, 12, 24, 32, 48, 24, 48, 48, 64, 4, 8, 12, 24, 12, 24, 32, 48, 12, 24, 32, 64, 32, 64, 64, 96, 12, 24, 32, 48, 32, 64, 64, 96, 24, 48, 64, 96, 48, 96, 96, 128, 4, 8, 12, 24

Offset: 0

David Radcliffe, Jun 23 2025

This is a 2-automatic sequence.

			Let b(n) be the binary expansion of n. Then a(n) = (1 + p10 + p11) * 2^c, where c is the number of set bits in b(n), p10 is the number of '10' patterns in b(n), and p11 is 1 or 0 depending on whether the pattern '11' is occurring in b(n) or not. This formula is used by _Chai Wah Wu_ in the Python function below. For instance:
  n = 25 -> b(n) = 11001 -> a(n) = (1+1+1) * 2^3 = 24.
  n = 26 -> b(n) = 11010 -> a(n) = (1+2+1) * 2^3 = 32.
  n = 27 -> b(n) = 11011 -> a(n) = (1+1+1) * 2^4 = 48.
- _Peter Luschny_, Jun 25 2025

David Radcliffe, Table of n, a(n) for n = 0..10000
Kenneth S. Davis and William A. Webb, Pascal's triangle modulo 4, Fib. Quart., 29 (1991), 79-83.

Cf. A001316, A014081, A033264, A051638 (mod 3 analog), A085357. Row sums of triangle A034931.

A384715[n_] := 2^DigitSum[n, 2]*(StringCount[IntegerString[n, 2], "10"] - Boole[BitAnd[n,2*n] == 0] + 2);
Array[A384715, 100, 0] (* Paolo Xausa, Jun 26 2025 *)

a(n) = sum(k=0, n, binomial(n,k)%4); \\ Michel Marcus, Jun 25 2025

def A001316(n): return (1 + (n % 2)) * A001316(n // 2) if n else 1
def A033264(n): return (n % 4 == 2) + A033264(n // 2) if n else 0
def A085357(n): return int(n & (n<<1) == 0)
def A384715(n): return A001316(n) * (A033264(n) - A085357(n) + 2)

def A384715(n): return (((n>>1)&~n).bit_count()+bool(n&(n<<1))+1)<Chai Wah Wu, Jun 25 2025

def a(n: int) -> int:  # after Chai Wah Wu
    b = bin(n)[2:]; p = b.count("10"); q = b.count("11")
    return (p + (2 if q else 1)) * 2**n.bit_count()  # Peter Luschny, Jun 25 2025

a(n) = A001316(n) * (A033264(n) - A085357(n) + 2) for n > 0.
Recurrences:
a(4n) = a(2n),
a(4n+1) = 2a(2n),
a(8n+2) = a(4n+2) + 2a(2n) - a(2n+1),
a(8n+3) = a(4n+3) + 4a(2n) - 4a(n),
a(8n+6) = a(4n+3) + 2a(4n+2) - 2a(2n+1),
a(8n+7) = 2a(4n+3).

A163000 Count of integers x in [0,n] satisfying A000120(x) + A000120(n-x) = A000120(n) + 1.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 2, 0, 1, 2, 4, 4, 2, 4, 4, 0, 1, 2, 4, 4, 4, 8, 8, 8, 2, 4, 8, 8, 4, 8, 8, 0, 1, 2, 4, 4, 4, 8, 8, 8, 4, 8, 12, 16, 8, 16, 16, 16, 2, 4, 8, 8, 8, 16, 16, 16, 4, 8, 16, 16, 8, 16, 16, 0, 1, 2, 4, 4, 4, 8, 8, 8, 4, 8, 12, 16, 8, 16, 16, 16, 4, 8, 12, 16, 12, 24, 24, 32, 8, 16, 24, 32, 16

Offset: 0

Vladimir Shevelev, Jul 20 2009

For every solution x, binomial(n,x) is 2 times an odd integer.
A generalization: for every solution 0 <= x <= n of the equation A000120(x) + A000120(n-x) = A000120(n) + r, binomial(n,x) is 2^r times an odd integer.
Apparently this is also the number of 2's in the n-th row of A034931. - R. J. Mathar, Jul 28 2017

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Kenneth S. Davis and William A. Webb, Pascal's triangle modulo 4, Fib. Quart., 29 (1991), 79-83.
Vladimir Shevelev, Binomial predictors, arXiv:0907.3302 [math.NT], 2009.
L. Spiegelhofer and M. Wallner, Divisibility of binomial coefficients by powers of two, arXiv:1710.10884 [math.NT], 2017.

Cf. A000120, A007814.

A001316 and A163577 count binomial coefficients with 2-adic valuation 0 and 2. A275012 gives a measure of complexity of these sequences. - Eric Rowland, Mar 15 2017

A163000 := proc(n) local a,x; a := 0 ; for x from 0 to n do if A000120(x)+A000120(n-x) = A000120(n)+1 then a := a+1; fi; od: a; end:
seq(A163000(n),n=0..100) ; # R. J. Mathar, Jul 21 2009

okQ[x_, n_] := DigitCount[x, 2, 1] + DigitCount[n - x, 2, 1] == DigitCount[n, 2, 1] + 1; a[n_] := Count[Range[0, n], x_ /; okQ[x, n]]; Table[a[n], {n, 0, 92}] (* Jean-François Alcover, Jul 13 2017 *)

a(n) = my(z=hammingweight(n)+1); sum(x=0, n, hammingweight(x) + hammingweight(n-x) == z); \\ Michel Marcus, Jun 06 2021

a(n)=0 iff n=2^k-1, k>=0. a(n)=1 iff n=2^k, k>=1.

Conjecture: a(n) = A033264(n)* 2^(A000120(n)-1); from [Davis & Webb]. - R. J. Mathar, Jul 28 2017

Extended beyond a(22) by R. J. Mathar, Jul 21 2009

A056973 Number of blocks of {0,0} in the binary expansion of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, 2, 1, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 4, 3, 2, 2, 2, 1, 1, 1, 2, 1, 0, 0, 1, 0, 0, 0, 3, 2, 1, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 5, 4, 3, 3, 3, 2, 2, 2, 3, 2, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 4, 3, 2, 2, 2, 1, 1

Offset: 1

Eric W. Weisstein

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions.
Eric Weisstein's World of Mathematics, Digit Block.
Index entries for sequences related to binary expansion of n

Cf. A003754, A014081, A023416, A033264, A037800.

Cf. A107782.

a056973 = f 0 where
   f y x = if x == 0 then y else f (y + 0 ^ (mod x 4)) $ div x 2
-- Reinhard Zumkeller, Mar 31 2015

f:= proc(n) option remember;
     if n mod 4 = 0 then 1 + procname(n/2)
     else procname(floor(n/2))
     fi
end proc:
f(1):= 0:
map(f, [$1..200]); # Robert Israel, Sep 02 2015

f[n_] := Count[Partition[IntegerDigits[n, 2], 2, 1], {0, 0}]; Table[f@ n, {n, 0, 102}] (* Michael De Vlieger, Sep 01 2015, after Robert G. Wilson v at A014081 *)
SequenceCount[#,{0,0},Overlaps->True]&/@(IntegerDigits[#,2]&/@Range[0,120]) (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 24 2018 *)

a(n) = { my(x = bitor(n, n>>1));
         if (x == 0, 0, 1 + logint(x, 2) - hammingweight(x)) }
vector(102, i, a(i))  \\ Gheorghe Coserea, Sep 01 2015

a(2n) = a(n) + [n is even], a(2n+1) = a(n).
G.f.: 1/(1-x) * Sum_{k>=0} t^4/((1+t)*(1+t^2)) where t=x^(2^k). - Ralf Stephan, Sep 10 2003
a(n) = A023416(n) - A033264(n). - Ralf Stephan, Sep 10 2003
Sum_{n>=1} a(n)/(n*(n+1)) = 2 - 3*log(2)/2 - Pi/4 (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021

A324345 Lexicographically earliest positive sequence such that a(i) = a(j) => A005811(i) = A005811(j) and A278222(i) = A278222(j), for all i, j >= 0.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 6, 7, 3, 5, 8, 9, 6, 9, 10, 11, 3, 5, 8, 9, 8, 12, 13, 14, 6, 9, 13, 15, 10, 14, 16, 17, 3, 5, 8, 9, 8, 12, 13, 14, 8, 12, 18, 19, 13, 19, 20, 21, 6, 9, 13, 15, 13, 19, 22, 23, 10, 14, 20, 23, 16, 21, 24, 25, 3, 5, 8, 9, 8, 12, 13, 14, 8, 12, 18, 19, 13, 19, 20, 21, 8, 12, 18, 19, 18, 26, 27, 28, 13, 19, 27, 29, 20, 28, 30, 31, 6, 9, 13, 15, 13, 19

Offset: 0

Antti Karttunen, Feb 24 2019

nonn

Restricted growth sequence transform of the ordered pair [A005811(n), A278222(n)], or equally, of [A005811(n), A286622(n)].
For all i, j >= 1:
  a(i) = a(j) => A033264(i) = A033264(j).

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A005811, A278222, A286622.

Cf. also A033264, A323889, A324343, A324346.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A005811(n) = hammingweight(bitxor(n, n>>1)); \\ From A005811
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
Aux324345(n) = [A005811(n), A278222(n)];
v324345 = rgs_transform(vector(1+up_to,n,Aux324345(n-1)));
A324345(n) = v324345[1+n];

a(2^n) = 3 for all n >= 1.

A382723 Number of entries in the n-th row of Pascal's triangle not divisible by 4.

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 6, 8, 3, 6, 8, 12, 6, 12, 12, 16, 3, 6, 8, 12, 8, 16, 16, 24, 6, 12, 16, 24, 12, 24, 24, 32, 3, 6, 8, 12, 8, 16, 16, 24, 8, 16, 20, 32, 16, 32, 32, 48, 6, 12, 16, 24, 16, 32, 32, 48, 12, 24, 32, 48, 24, 48, 48, 64, 3, 6, 8, 12, 8, 16, 16, 24, 8, 16, 20, 32, 16, 32, 32, 48, 8

Offset: 0

N. J. A. Sloane, Apr 23 2025

nonn

Cf. A000120, A001316, A006047, A033264, A194459, A382720-A382725.

a(n) = sum(k=0, n, (binomial(n, k) % 4) != 0); \\ Michel Marcus, Apr 23 2025

def A382723(n): return bin(n)[2:].count('10')+2<Chai Wah Wu, Aug 10 2025

a(n) = (A033264(n)+2)*2^(A000120(n)-1). - Chai Wah Wu, Aug 10 2025

cp's OEIS Frontend

A213629 In binary representation: T(n,k) = number of (possibly overlapping) occurrences of k in n, triangle read by rows, 1<=k<=n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

A072219 Any number n can be written uniquely in the form n = 2^k_1 - 2^k_2 + 2^k_3 - ... + 2^k_{2r+1} where the signs alternate, there are an odd number of terms, and k_1 > k_2 > k_3 > ... > k_{2r+1} >= 0; sequence gives number of terms 2r+1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

Extensions

A175047 Write n in binary, then increase each run of 0's by one 0. a(n) is the decimal equivalent of the result.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Python

Formula

Extensions

A196521 Decimal expansion of Pi/4-log(2)/2.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A384715 a(n) = Sum_{k=0..n} (binomial(n, k) mod 4).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Python

Python

Formula

A163000 Count of integers x in [0,n] satisfying A000120(x) + A000120(n-x) = A000120(n) + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula