A001316 -id:A001316 - cp's OEIS frontend

A060632 a(n) = 2^wt(floor(n/2)) (i.e., 2^A000120(floor(n/2)), or A001316(floor(n/2))).

1, 1, 2, 2, 2, 2, 4, 4, 2, 2, 4, 4, 4, 4, 8, 8, 2, 2, 4, 4, 4, 4, 8, 8, 4, 4, 8, 8, 8, 8, 16, 16, 2, 2, 4, 4, 4, 4, 8, 8, 4, 4, 8, 8, 8, 8, 16, 16, 4, 4, 8, 8, 8, 8, 16, 16, 8, 8, 16, 16, 16, 16, 32, 32, 2, 2, 4, 4, 4, 4, 8, 8, 4, 4, 8, 8, 8, 8, 16, 16, 4, 4, 8, 8, 8, 8, 16, 16, 8, 8, 16, 16, 16, 16, 32

Offset: 0

Avi Peretz (njk(AT)netvision.net.il), Apr 15 2001

nonn

Number of conjugacy classes in the symmetric group S_n that have odd number of elements.
Also sequence A001316 doubled.
Number of even numbers whose binary expansion is a child of the binary expansion of n. - Nadia Heninger and N. J. A. Sloane, Jun 06 2008
First differences of A151566. Sequence gives number of toothpicks added at the n-th generation of the leftist toothpick sequence A151566. - N. J. A. Sloane, Oct 20 2010
The Fi1 and Fi1 triangle sums, see A180662 for their definitions, of Sierpiński's triangle A047999 equal this sequence. - Johannes W. Meijer, Jun 05 2011
Also number of odd entries in n-th row of triangle of Stirling numbers of the first kind. - Istvan Mezo, Jul 21 2017

			a(3) = 2 because in S_3 there are two conjugacy classes with odd number of elements, the trivial conjugacy class and the conjugacy class of transpositions consisting of 3 elements: (12),(13),(23).
From _Omar E. Pol_, Oct 12 2011 (Start):
Written as a triangle:
1,
1,
2,2,
2,2,4,4,
2,2,4,4,4,4,8,8,
2,2,4,4,4,4,8,8,4,4,8,8,8,8,16,16,
2,2,4,4,4,4,8,8,4,4,8,8,8,8,16,16,4,4,8,8,8,8,16,16,8,...
(End)

I. G. MacDonald: Symmetric functions and Hall polynomials Oxford: Clarendon Press, 1979. Page 21.

Indranil Ghosh, Table of n, a(n) for n = 0..65536 (terms 0..1000 from Harry J. Smith)
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Christina Talar Bekaroğlu, Analyzing Dynamics of Larger than Life: Impacts of Rule Parameters on the Evolution of a Bug's Geometry, Master's thesis, Calif. State Univ. Northridge (2023). See p. 92.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000120, A001316, A139251, A151566, A160407.

a000120:=func< n | &+Intseq(n, 2) >; [ 2^a000120(Floor(n/2)): n in [0..100] ]; // Klaus Brockhaus, Oct 15 2010

A060632 := proc(n) local k; add(binomial(n,2*k) mod 2, k=0..floor(n/2)); end: seq(A060632(n),n=0..94); # edited by Johannes W. Meijer, May 28 2011
A060632 := n -> 2^add(i, i = convert(iquo(n,2), base, 2)); # Peter Luschny, Jun 30 2011
A060632 := n -> igcd(2^n, n! / iquo(n,2)!^2);  # Peter Luschny, Jun 30 2011

a[n_] := 2^DigitCount[Floor[n/2], 2, 1]; Table[a[n], {n, 0, 94}] (* Jean-François Alcover, Feb 25 2014 *)

for (n=0, 1000, write("b060632.txt", n, " ", sum(k=0, floor(n/2), binomial(n, 2*k) % 2)) ) \\ Harry J. Smith, Sep 14 2009

a(n)=2^hammingweight(n\2) \\ Charles R Greathouse IV, Feb 06 2017

def A060632(n):
    return 2**bin(n/2)[2:].count("1") # Indranil Ghosh, Feb 06 2017

a(n) = sum{k=0..floor(n/2), C(n, 2k) mod 2} - Paul Barry, Jan 03 2005, Edited by Harry J. Smith, Sep 15 2009
a(n) = gcd(A056040(n), 2^n). - Peter Luschny, Jun 30 2011
G.f.: (1 + x) * Product_{k>=0} (1 + 2*x^(2^(k+1))). - Ilya Gutkovskiy, Jul 19 2019

More terms from James Sellers, Apr 16 2001
Edited by N. J. A. Sloane, Jun 06 2008; Oct 11 2010
a(0) = 1 added by N. J. A. Sloane, Sep 14 2009
Formula corrected by Harry J. Smith, Sep 15 2009

A295989 Irregular triangle T(n, k), read by rows, n >= 0 and 0 <= k < A001316(n): T(n, k) is the (k+1)-th nonnegative number m such that n AND m = m (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 1, 2, 3, 0, 4, 0, 1, 4, 5, 0, 2, 4, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 8, 0, 1, 8, 9, 0, 2, 8, 10, 0, 1, 2, 3, 8, 9, 10, 11, 0, 4, 8, 12, 0, 1, 4, 5, 8, 9, 12, 13, 0, 2, 4, 6, 8, 10, 12, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0

Offset: 0

Rémy Sigrist, Dec 02 2017

The (n+1)-th row has A001316(n) terms and sums to n * A001316(n) / 2.
For any n >= 0 and k such that 0 <= k < A001316(n):
- if A000120(n) > 0 then T(n, 1) = A006519(n),
- if A000120(n) > 1 then T(n, 2) = 2^A285099(n),
- if A000120(n) > 0 then T(n, A001316(n)/2 - 1) = A053645(n),
- if A000120(n) > 0 then T(n, A001316(n)/2) = 2^A000523(n),
- if A000120(n) > 0 then T(n, A001316(n) - 2) = A129760(n),
- T(n, A001316(n) - 1) = n,
- the six previous relations correspond respectively (when applicable) to the second term, the third term, the pair of central terms, the penultimate term and the last term of a row,
- T(n, k) AND T(n, A001316(n) - k - 1) = 0,
- T(n, k) + T(n, A001316(n) - k - 1) = n,
- T(n, k) = k for any k < A006519(n+1),
- A000120(T(n, k)) = A000120(k).
If we plot (n, T(n,k)) then we obtain a skewed Sierpinski triangle (see Links section).
If interpreted as a flat sequence a(n) for n >= 0:
- a(n) = 0 iff n = A006046(k) for some k >= 0,
- a(n) = 1 iff n = A006046(2*k + 1) + 1 for some k >= 0,
- a(A006046(k) - 1) = k - 1 for any k > 0.

			Triangle begins:
  0:   [0]
  1:   [0, 1]
  2:   [0, 2]
  3:   [0, 1, 2, 3]
  4:   [0, 4]
  5:   [0, 1, 4, 5]
  6:   [0, 2, 4, 6]
  7:   [0, 1, 2, 3, 4, 5, 6, 7]
  8:   [0, 8]
  9:   [0, 1, 8, 9]
  10:  [0, 2, 8, 10]
  11:  [0, 1, 2, 3, 8, 9, 10, 11]
  12:  [0, 4, 8, 12]
  13:  [0, 1, 4, 5, 8, 9, 12, 13]
  14:  [0, 2, 4, 6, 8, 10, 12, 14]
  15:  [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]

Rémy Sigrist, Rows n = 0..256, flattened
Rémy Sigrist, Scatterplot of (n, T(n, k)) for n = 0..1023 and k = 0..A001316(n)-1

Cf. A000120, A000523, A001316 (row lengths), A006046, A006519, A053645, A129760, A285099.

First column of array in A352909.

A295989row[n_] := Select[Range[0, n], BitAnd[#, n-#] == 0 &];
Array[A295989row, 25, 0] (* Paolo Xausa, Feb 24 2024 *)

T(n,k) = if (k==0, 0, n%2==0, 2*T(n\2,k), k%2==0, 2*T(n\2, k\2), 2*T(n\2, k\2)+1)

For any n >= 0 and k such that 0 <= k < A001316(n):
- T(n, 0) = 0,
- T(2*n, k) = 2*T(n, k),
- T(2*n+1, 2*k) = 2*T(n, k),
- T(2*n+1, 2*k+1) = 2*T(n, k) + 1.

A105321 Convolution of binomial(1,n) and Gould's sequence A001316.

Original entry on oeis.org

1, 3, 4, 6, 6, 6, 8, 12, 10, 6, 8, 12, 12, 12, 16, 24, 18, 6, 8, 12, 12, 12, 16, 24, 20, 12, 16, 24, 24, 24, 32, 48, 34, 6, 8, 12, 12, 12, 16, 24, 20, 12, 16, 24, 24, 24, 32, 48, 36, 12, 16, 24, 24, 24, 32, 48, 40, 24, 32, 48, 48, 48, 64, 96, 66, 6, 8, 12, 12, 12, 16, 24, 20, 12, 16

Offset: 0

Paul Barry, Apr 01 2005

A universal function related to the spherical growth of repeated truncations of maps.

a(n) = (number of ones in row n of triangle A249133) = (number of odd terms in row n of triangle A249095) = A000120(A249184(n)). - Reinhard Zumkeller, Nov 14 2014

			From _Omar E. Pol_, May 29 2010: (Start)
If written as a triangle:
  1;
  3;
  4;
  6,6;
  6,8,12,10;
  6,8,12,12,12,16,24,18;
  6,8,12,12,12,16,24,20,12,16,24,24,24,32,48,34;
  6,8,12,12,12,16,24,20,12,16,24,24,24,32,48,36,12,16,24,24,24,32,48,40,24,32,48,48,48,64,96,66; (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
T. Pisanski and T. W. Tucker, Growth in Repeated Truncations of Maps, Preprint series, Univ. of Ljubljana, Vol. 38 (2000), 696.
T. Pisanski and T. W. Tucker, Growth in Repeated Truncations of Maps, Atti. Sem. Mat. Fis. Univ. Modena, Vol. 49 (2001), 167-176.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS - _Omar E. Pol_, May 29 2010

Cf. A139250, A139251, A173522, A048460.

Cf. A001316, A000120, A249184, A249095, A249133.

a105321 n = if n == 0 then 1 else a001316 n + a001316 (n - 1)
-- Reinhard Zumkeller, Nov 14 2014

nmax := 74: A001316 := n -> if n <= -1 then 0 else 2^add(i, i=convert(n, base, 2)) fi: for p from 0 to ceil(log[2](nmax)) do for n from 1 to nmax/(p+2)+1 do a((2*n-1)*2^p) := (2^p+2)  * A001316(n-1) od: od: a(0) :=1: seq(a(n), n=0..nmax); # Johannes W. Meijer, Jan 28 2013

f[n_] := Sum[Binomial[1, n - k]Mod[Binomial[k, j], 2], {k, 0, n}, {j, 0, k}]; Array[f, 75, 0] (* Robert G. Wilson v, Jun 28 2010 *)

a(n) = sum(k=0, n, binomial(1, n-k)*sum(j=0, k, binomial(k, j) % 2)); \\ Michel Marcus, Apr 29 2018

def A105321(n): return (1<Chai Wah Wu, Jul 30 2025

# (fast way for big vectors)
import numpy # (version >= 2.0.0)
n_up_to = 2**22
A000079 = 1 << numpy.arange(n_up_to.bit_length())
A001316 = A000079[(numpy.bitwise_count(numpy.arange(n_up_to)))]
A105321 = A001316
A105321[1:] += A001316[0:-1]
print(A105321[0:100]) # Karl-Heinz Hofmann, Aug 01 2025

G.f. (1+x)*Product{k>=0, 1+2x^(2^k)};
a(n) = Sum_{k=0..n, binomial(1, n-k)*Sum_{j=0..k, binomial(k, j) mod 2}}.
a(n) = 2*A048460(n) for n>=2. - Omar E. Pol, Jan 02 2011
a((2*n-1)*2^p) = (2^p+2)*A001316(n-1), p >= 0 and n >= 1, with a(0) = 1. - Johannes W. Meijer, Jan 28 2013
a(n) = A001316(n) + A001316(n-1) for n > 0. - Reinhard Zumkeller, Nov 14 2014

A064405 Number of even entries (A048967) minus the number of odd entries (A001316) in row n of Pascal's triangle (A007318).

Original entry on oeis.org

-1, -2, -1, -4, 1, -2, -1, -8, 5, 2, 3, -4, 5, -2, -1, -16, 13, 10, 11, 4, 13, 6, 7, -8, 17, 10, 11, -4, 13, -2, -1, -32, 29, 26, 27, 20, 29, 22, 23, 8, 33, 26, 27, 12, 29, 14, 15, -16, 41, 34, 35, 20, 37, 22, 23, -8, 41, 26, 27, -4, 29, -2, -1, -64, 61, 58, 59, 52, 61, 54, 55, 40, 65, 58, 59, 44, 61, 46, 47, 16, 73, 66, 67, 52, 69, 54

Offset: 0

Robert G. Wilson v, Sep 29 2001

Seiichi Manyama, Table of n, a(n) for n = 0..8191

Cf. A000079, A001316, A007318, A048967.

Table[ n + 1 - 2Sum[ Mod[ Binomial[ n, k ], 2 ], {k, 0, n} ], {n, 0, 100} ]

PARI
```
a(n)=sum(i=0,n,(-1)^binomial(n,i))
```

a(n)=if(n<1,-1,if(n%2==0,a(n/2)+n/2,2*a((n-1)/2)))

a(n) = Sum_{k=0..n} (-1)^binomial(n, k); a(2^n) = 2^n-3; a(2^n+1)=2^n-6; more generally there's a sequence z(k) such that for any k>=0 and for 2^n >k, a(2^n+k) = 2^n+z(k); for k=0, 1, 2, 3, 4, 5, 6, 7, 8... z(k) = -3, -6, -5, -12, -3, -10, -9, -24, 1, ... - Benoit Cloitre, Oct 18 2002
a(2n) = a(n) + n, a(2n+1) = 2a(n). - Ralf Stephan, Mar 05 2004
a(n) = -Sum_{k=0..n} moebius(binomial(n, k) mod 2). - Paul Barry, Apr 29 2005
a(2^n-1) = -2^n. - Seiichi Manyama, Aug 24 2022

A286575 Run-length transform of A001316.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 4, 4, 2, 4, 4, 2, 2, 4, 4, 4, 4, 8, 4, 8, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 4, 4, 4, 8, 4, 8, 4, 8, 8, 8, 4, 8, 8, 4, 2, 4, 4, 4, 4, 8, 4, 8, 4, 8, 8, 8, 2, 4, 4, 4, 2, 4, 4, 4, 4, 8, 4, 8, 4, 8, 8, 8, 4, 8, 8, 4, 4, 8, 8, 8, 8, 16, 8, 16, 4, 8, 8, 8, 8, 16, 4, 8, 2, 4, 4, 4, 4, 8, 4, 8, 4, 8, 8, 8, 4

Offset: 0

Antti Karttunen, May 28 2017

			For n = 0, there are no 1-runs, and thus a(0) = 1 as an empty product.
For n = 29, "11101" in binary, there are two 1-runs, of lengths 1 and 3, thus a(29) = A001316(1) * A001316(3) = 2*4 = 8.

Cf. A000079, A001316, A005940, A037445, A227349, A247282, A286574.

Table[Times @@ Map[Sum[Mod[#, 2] &@ Binomial[#, k], {k, 0, #}] &@ Length@ # &, DeleteCases[Split@ IntegerDigits[n, 2], ?(First@ # == 0 &)]], {n, 0, 108}] (* _Michael De Vlieger, May 29 2017 *)

from sympy import factorint, prime, log
import math
def wt(n): return bin(n).count("1")
def a037445(n):
    f=factorint(n)
    return 2**sum([wt(f[i]) for i in f])
def A(n): return n - 2**int(math.floor(log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a(n): return a037445(b(n)) # Indranil Ghosh, May 30 2017

# use RLT function from A278159
def A286575(n): return RLT(n,lambda m: 2**(bin(m).count('1'))) # Chai Wah Wu, Feb 04 2022

(define (A286575 n) (fold-left (lambda (a r) (* a (A001316 r))) 1 (bisect (reverse (binexp->runcount1list n)) (- 1 (modulo n 2)))))
(define (bisect lista parity) (let loop ((lista lista) (i 0) (z (list))) (cond ((null? lista) (reverse! z)) ((eq? i parity) (loop (cdr lista) (modulo (1+ i) 2) (cons (car lista) z))) (else (loop (cdr lista) (modulo (1+ i) 2) z)))))
(define (binexp->runcount1list n) (if (zero? n) (list) (let loop ((n n) (rc (list)) (count 0) (prev-bit (modulo n 2))) (if (zero? n) (cons count rc) (if (eq? (modulo n 2) prev-bit) (loop (floor->exact (/ n 2)) rc (1+ count) (modulo n 2)) (loop (floor->exact (/ n 2)) (cons count rc) 1 (modulo n 2)))))))
(define (A001316 n) (let loop ((n n) (z 1)) (cond ((zero? n) z) ((even? n) (loop (/ n 2) z)) (else (loop (/ (- n 1) 2) (* z 2))))))

a(n) = A037445(A005940(1+n)).

a(n) = A000079(A286574(n)).

A191488 A companion to Gould’s sequence A001316.

Original entry on oeis.org

4, 6, 8, 10, 8, 12, 16, 18, 8, 12, 16, 20, 16, 24, 32, 34, 8, 12, 16, 20, 16, 24, 32, 36, 16, 24, 32, 40, 32, 48, 64, 66, 8, 12, 16, 20, 16, 24, 32, 36, 16, 24, 32, 40, 32, 48, 64, 68, 16, 24, 32, 40, 32, 48, 64, 72, 32, 48, 64, 80, 64, 96, 128

Offset: 0

Johannes W. Meijer, Jun 05 2011

The row sums of the Sierpinski-Stern triangle A191372 are given by sequence A191487.
The differences diff1(n) = A191487(2*n+3) - A191487(2*n+1) lead to a peculiar number triangle, see the examples. The leading terms of the rows of the diff1(n) triangle clearly stand out from the rest of the terms and are given by A001550(p+1), p>=1; for p=0 this term is 7.
If we ignore the first term of the diff1(n) rows and reverse the order of the remaining terms we get sequence A191488, see the examples; more terms require a higher row number.
Both the diff1(n) and the diff2(n) sequences are related to Gould’s sequence A001316. We ignore the first term and reverse the order of the rest of the terms. The diff2(n) sequence leads directly to A001316, see A191487, while the diff1(n) sequence leads to A001316 in a slightly more complex way. We observe that for Gould’s sequence equation A001316((2*n+1)*2^p-1) = C(p)*A001316(n) with C(p) = 2^p holds, while for its companion A191488 equation A191488((2*n+1)*2^p-1) = C(p)*A001316(n) with C(p) = 2^(p+1)+2 holds; see the Maple program. Furthermore for both sequences a(2^p - 1) = C(p).

			The first few rows of diff1(n) as a triangle, row lengths A000079(p) with p>=0, are:
[7]
[14, 4]
[36, 8, 6, 4]
[98, 16, 12, 8, 10, 8, 6, 4]
[276, 32, 24, 16, 20, 16, 12, 8, 18, 16, 12, 8, 10, 8, 6, 4]
[794, 64, 48, 32, 40, 32, 24, 16, 36, 32, 24, 16, 20, 16, 12, 8, 34, 32, 24, 16, 20, 16, 12, 8, 18, 16, 12, 8, 10, 8, 6, 4]
The first few rows of diff1(n) reversed minus the first term are:
[4]
[4, 6, 8]
[4, 6, 8, 10, 8, 12, 16]
[4, 6, 8, 10, 8, 12, 16, 18, 8, 12, 16, 20, 16, 24, 32]
[4, 6, 8, 10, 8, 12, 16, 18, 8, 12, 16, 20, 16, 24, 32, 34, 8, 12, 16, 20, 16, 24, 32, 36, 16, 24, 32, 40, 32, 48, 64]

Cf. A001316, A191372, A191487.

nmax:=2^6; pmax:=ceil(log(nmax)/log(2)); A001316 := n -> if n<=-1 then 0 else 2^add(i, i=convert(n, base, 2)) fi: C := proc(p): C(p) := 2^(p+1)+2 end: for p from 0 to pmax do for n from 0 to nmax do a((2*n+1)*2^p-1):= C(p)*A001316(n) od: od: seq(a(n), n=0..nmax-2);

a((2*n+1)*2^p - 1) = C(p) * A001316(n) with C(p) = (2^(p+1)+2), p>=0.

a(2^p - 1) = 2^(p+1)+2 = A052548(p+1), p>=0.

A368229 Irregular table of nonnegative integers T(n, k), n >= 0, k = 1..A001316(n), read by rows: the 1's in the binary expansion of n exactly match the nonzero digits in the ternary expansions of the terms in the n-th row.

Original entry on oeis.org

0, 1, 2, 3, 6, 4, 5, 7, 8, 9, 18, 10, 11, 19, 20, 12, 15, 21, 24, 13, 14, 16, 17, 22, 23, 25, 26, 27, 54, 28, 29, 55, 56, 30, 33, 57, 60, 31, 32, 34, 35, 58, 59, 61, 62, 36, 45, 63, 72, 37, 38, 46, 47, 64, 65, 73, 74, 39, 42, 48, 51, 66, 69, 75, 78

Offset: 0

Rémy Sigrist, Dec 18 2023

As a flat sequence, this is a permutation of the nonnegative integers (with inverse A368230).

			Table T(n, k) begins:
    0;
    1, 2;
    3, 6;
    4, 5, 7, 8;
    9, 18;
    10, 11, 19, 20;
    12, 15, 21, 24;
    13, 14, 16, 17, 22, 23, 25, 26;
    27, 54;
    28, 29, 55, 56;
    30, 33, 57, 60;
    31, 32, 34, 35, 58, 59, 61, 62;
    36, 45, 63, 72;
    37, 38, 46, 47, 64, 65, 73, 74;
    39, 42, 48, 51, 66, 69, 75, 78;
    40, 41, 43, 44, 49, 50, 52, 53, 67, 68, 70, 71, 76, 77, 79, 80;
    81, 162;
    ...

Rémy Sigrist, Table of n, a(n) for n = 0..6560 (rows for n = 0..2^8-1 flattened)
Index entries for sequences that are permutations of the natural numbers

See A368225 for a similar sequence.

Cf. A001316, A005823, A005836, A289831, A368230 (inverse).

row(n) = { my (r = [0], b = binary(n)); for (k = 1, #b, r = [3*v+b[k]|v<-r]; if (b[k], r = concat(r, [v+1|v<-r]););); Set(r); }

T(n, 1) = A005836(n + 1).
T(n, A001316(n)) = A005823(n + 1).
A289831(T(n, k)) = n.

A268128 a(n) = (A000123(n) - A001316)/2.

Original entry on oeis.org

0, 0, 1, 1, 4, 5, 8, 9, 17, 21, 28, 33, 45, 53, 66, 75, 100, 117, 140, 161, 193, 221, 258, 291, 344, 389, 446, 499, 573, 639, 722, 797, 913, 1013, 1132, 1249, 1393, 1533, 1698, 1859, 2060, 2253, 2478, 2699, 2965, 3223, 3522, 3813, 4173, 4517, 4910, 5299, 5753

Offset: 0

Tom Edgar, Jan 26 2016

nonn

G. E. Andrews, A. S. Fraenkel, and J. A. Sellers, Characterizing the number of m-ary partitions modulo m, The American Mathematical Monthly, Vol. 122, No. 9 (November 2015), pp. 880-885.
G. E. Andrews, A. S. Fraenkel, and J. A. Sellers, Characterizing the number of m-ary partitions modulo m.
Tom Edgar, The distribution of the number of parts of m-ary partitions modulo m., arXiv:1603.00085 [math.CO], 2016.

Cf. A005704, A006047, A268127.

b[0] = 1; b[n_] := b[n] = b[Floor[n/2]] + b[n - 1];
c[n_] := Sum[Mod[Binomial[n, k], 2], {k, 0, n}];
a[n_] := (b[n] - c[n])/2;
Table[a[n], {n, 0, 52}] (* Jean-François Alcover, Dec 12 2018 *)

def b(n):
    A=[1]
    for i in [1..n]:
        A.append(A[i-1] + A[floor(i/2)])
    return A[n]
[(b(n)-prod(x+1 for x in n.digits(2)))/2 for n in [0..60]]

Let b(0) = 1 and b(n) = b(n-1) + b(floor(n/2)) and let c(n) = Product_{i=0..k}(n_i+1) where n = Sum_{i=0..k}n_i*2^i is the binary representation of n. Then a(n) = (1/2)*(b(n) - c(n)).

A085814 Even entries (A048967) minus the odd entries (A001316) in row n of Pascal's triangle (A007318).

Original entry on oeis.org

-1, -2, 0, -8, 12, 8, 0, -128, 252, 472, 840, 1120, 2112, 128, 0, -32768, 65532, 131000, 261528, 519648, 1029192, 1991728, 3865664, 6825984, 13835328, 21056128, 39611520, 20680192, 134234112, 32768, 0, -2147483648, 4294967292, 8589934456, 17179866936, 34359709664, 68719241112, 137436945552

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jul 25 2003

sign

Cf. A064405.

a(n) = sum((-1)^binomial(n, i)*binomial(n, i), i=0..n)

For n>1, a(2^n-2)=0. - Benoit Cloitre, Jun 13 2004

A128806 a(n) = A001316(n) + A046092(n).

Original entry on oeis.org

1, 6, 14, 28, 42, 64, 88, 120, 146, 184, 224, 272, 316, 372, 428, 496, 546, 616, 688, 768, 844, 932, 1020, 1120, 1204, 1308, 1412, 1528, 1632, 1756, 1876, 2016, 2114, 2248, 2384, 2528, 2668, 2820, 2972, 3136, 3284, 3452, 3620, 3800, 3968, 4156, 4340

Offset: 0

Philippe Deléham, Apr 08 2007

nonn

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A001316, A046092.

Table[Sum[Mod[Binomial[n,k],2],{k,0,n}]+2n(n+1),{n,0,46}] (* James C. McMahon, Jan 11 2025 *)
A128806[n_] := 2^DigitSum[n, 2] + 2*n*(n+1); Array[A128806, 100, 0] (* Paolo Xausa, Jul 31 2025 *)

def A128806(n): return (1<Chai Wah Wu, Jul 29 2025

a(32)-a(34) corrected by Georg Fischer, Jul 01 2020

cp's OEIS Frontend

A060632 a(n) = 2^wt(floor(n/2)) (i.e., 2^A000120(floor(n/2)), or A001316(floor(n/2))).

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A295989 Irregular triangle T(n, k), read by rows, n >= 0 and 0 <= k < A001316(n): T(n, k) is the (k+1)-th nonnegative number m such that n AND m = m (where AND denotes the bitwise AND operator).

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A105321 Convolution of binomial(1,n) and Gould's sequence A001316.

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A064405 Number of even entries (A048967) minus the number of odd entries (A001316) in row n of Pascal's triangle (A007318).

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A286575 Run-length transform of A001316.

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A191488 A companion to Gould’s sequence A001316.

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