A006046 -id:A006046 - cp's OEIS frontend

A130665 a(n) = Sum_{k=0..n} 3^wt(k), where wt() = A000120().

1, 4, 7, 16, 19, 28, 37, 64, 67, 76, 85, 112, 121, 148, 175, 256, 259, 268, 277, 304, 313, 340, 367, 448, 457, 484, 511, 592, 619, 700, 781, 1024, 1027, 1036, 1045, 1072, 1081, 1108, 1135, 1216, 1225, 1252, 1279, 1360, 1387, 1468, 1549, 1792, 1801, 1828, 1855

Offset: 0

N. J. A. Sloane, based on a message from Don Knuth, Jun 23 2007

Partial sums of A048883. - David Applegate, Jun 11 2009
From Gary W. Adamson, Aug 26 2016: (Start)
The formula of Mar 26 2010 is equivalent to the left-shifted vector of matrix powers (lim_{k->infinity} M^k), of the production matrix M:
  1, 0, 0, 0, 0, 0, ...
  4, 0, 0, 0, 0, 0, ...
  3, 1, 0, 0, 0, 0, ...
  0, 4, 0, 0, 0, 0, ...
  0, 3, 1, 0, 0, 0, ...
  0, 0, 4, 0, 0, 0, ...
  0, 0, 3, 1, 0, 0, ...
  ...
The sequence divided by its aerated variant is (1, 4, 3, 0, 0, 0, ...). (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Applegate, The movie version
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 27, 31-32.
Tanya Khovanova and Joshua Xiong, Nim Fractals, arXiv:1405.594291 [math.CO], 2014 and J. Int. Seq. 17 (2014) # 14.7.8.
D. E. Knuth, Problem 11320 Amer. Math. Monthly, Vol. 114 No. 9 (Nov 2007) p. 835.
Omar E. Pol, Illustration of initial terms: Fig. 1. Neighbors of the vertices, Fig. 2. Overlapping squares, Fig. 3. One-step bishop, (Nov 08 2009)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Mike Warburton, Ulam-Warburton Automaton - Counting Cells with Quadratics, arXiv:1901.10565 [math.CO], 2019.

Cf. A000120, A006046 (Sum 2^wt(n)), A116520, A130667, A147562, A151920, A151922, A160410, A160412, A360189.

a130665 = sum . map (3 ^) . (`take` a000120_list) . (+ 1)
-- Reinhard Zumkeller, Apr 18 2012

u:=3; a[1]:=1; M:=30; for n from 1 to M do a[2*n] := (u+1)*a[n]; a[2*n+1] := u*a[n] + a[n+1]; od; t1:=[seq( a[n], n=1..2*M )]; # Gives sequence with a different offset

f[n_] := Sum[3^Count[ IntegerDigits[k, 2], 1], {k, 0, n}]; Array[f, 51, 0] (* Robert G. Wilson v, Jun 28 2010 *)

def a(n):  # formula version, n=10^10000 takes ~1 second
    if n == 0:
        return 1
    msb = 1 << (n.bit_length() - 1)
    return msb**2 + 3 * a(n-msb) # Stefan Pochmann, Mar 15 2023

def a(n):  # optimized, n=10^50000 takes ~1 second
    n += 1
    total = 0
    power3 = 1
    while n:
        log = n.bit_length() - 1
        total += power3 << (2*log)
        n -= 1 << log
        power3 *= 3
    return total # Stefan Pochmann, Mar 15 2023

With a different offset: a(1) = 1; a(n) = max { 3*a(k)+a(n-k) | 1 <= k <= n/2 }, for n>1.
a(2n+1) = 4*a(n) and a(2n) = 3*a(n-1) + a(n).
a(n) = (A147562(n+1) - 1)*3/4 + 1. - Omar E. Pol, Nov 08 2009
a(n) = A160410(n+1)/4. - Omar E. Pol, Nov 12 2009
Let r(x) = (1 + 4x + 3x^2), then (1 + 4x + 7x^2 + 16x^3 + ...) =
r(x)* r(x^2) * r(x^4) * r(x^8) * ... - Gary W. Adamson, Mar 26 2010
For asymptotics see the discussion in the comments in A006046. - N. J. A. Sloane, Mar 11 2021
a(n) = Sum_{k=0..floor(log_2(n+1))} 3^k * A360189(n,k). - Alois P. Heinz, Mar 06 2023
a(n) = msb^2 + 3*a(n-msb), where msb = A053644(n). - Stefan Pochmann, Mar 15 2023

Simpler definition (and new offset) from David Applegate, Jun 11 2009

Lower limit of sum in definition changed from 1 to 0 by Robert G. Wilson v, Jun 28 2010

A194459 Number of entries in the n-th row of Pascal's triangle not divisible by 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 4, 6, 8, 10, 3, 6, 9, 12, 15, 4, 8, 12, 16, 20, 5, 10, 15, 20, 25, 2, 4, 6, 8, 10, 4, 8, 12, 16, 20, 6, 12, 18, 24, 30, 8, 16, 24, 32, 40, 10, 20, 30, 40, 50, 3, 6, 9, 12, 15, 6, 12, 18, 24, 30, 9, 18, 27, 36, 45, 12, 24, 36, 48, 60, 15, 30

Offset: 0

Paul Weisenhorn, Aug 24 2011

Pascal triangles modulo p with p prime have the dimension D = log(p*(p+1)/2)/log(p). [Corrected by Connor Lane, Nov 28 2022]

Also number of ones in row n of triangle A254609. - Reinhard Zumkeller, Feb 04 2015

			n = 32 = 112|_5: b(32,1) = 2, b(32,2) = 1, thus a(32) = 2^2 * 3^1 = 12.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Periodic minimum in the count of binomial coefficients not divisible by a prime, arXiv:2408.06817 [math.NT], 2024.

Cf. A006046, A001316 (for p=2).
Cf. A006048, A006047 (for p=3).
Cf. A194458 (for p=5).
Cf. A007318, A254609.

a194459 = sum . map (signum . flip mod 5) . a007318_row
-- Reinhard Zumkeller, Feb 04 2015

a:= proc(n) local l, m, t;
      m:= n;
      l:= [0$5];
      while m>0 do t:= irem(m, 5, 'm')+1; l[t]:=l[t]+1 od;
      mul(r^l[r], r=2..5)
    end:
seq(a(n), n=0..100);

Nest[Join[#, 2#, 3#, 4#, 5#]&, {1}, 4] (* Jean-François Alcover, Apr 12 2017, after code by Robert G. Wilson v in A006047 *)

from math import prod
from sympy.ntheory import digits
def A194459(n):
    s = digits(n,5)[1:]
    return prod((d+1)**s.count(d) for d in range(1,5)) # Chai Wah Wu, Jul 23 2025

a(n) = Product_{d=1..4} (d+1)^b(n,d) with b(n,d) = number of digits d in base 5 expansion of n. The formula generalizes to other prime bases p.

a(n) = A194458(n) - A194458(n-1).

Edited by Alois P. Heinz, Sep 06 2011

A116520 a(0) = 0, a(1) = 1; a(n) = max { 4*a(k) + a(n-k) | 1 <= k <= n/2 }, for n > 1.

Original entry on oeis.org

0, 1, 5, 9, 25, 29, 45, 61, 125, 129, 145, 161, 225, 241, 305, 369, 625, 629, 645, 661, 725, 741, 805, 869, 1125, 1141, 1205, 1269, 1525, 1589, 1845, 2101, 3125, 3129, 3145, 3161, 3225, 3241, 3305, 3369, 3625, 3641, 3705, 3769, 4025, 4089, 4345, 4601, 5625

Offset: 0

Roger L. Bagula, Mar 15 2006

Equivalently, a(n) = r*a(ceiling(n/2)) + s*a(floor(n/2)), a(0)=0, a(1)=1, for (r,s) = (1,4). - N. J. A. Sloane, Feb 16 2016
A 5-divide version of A084230.
Zero together with the partial sums of A102376. - Omar E. Pol, May 05 2010
Also, total number of cubic ON cells after n generations in a three-dimensional cellular automaton in which A102376(n-1) gives the number of cubic ON cells in the n-th level of the structure starting from the top. An ON cell remains ON forever. The structure looks like an irregular stepped pyramid, with n >= 1. - Omar E. Pol, Feb 13 2015
From Gary W. Adamson, Aug 27 2016: (Start)
The formula of Mar 26 2010 is equivalent to lim_{k->infinity} M^k of the following production matrix M:
  1, 0, 0, 0, 0, 0, ...
  5, 0, 0, 0, 0, 0, ...
  4, 1, 0, 0, 0, 0, ...
  0, 5, 0, 0, 0, 0, ...
  0, 4, 1, 0, 0, 0, ...
  0, 0, 5, 0, 0, 0, ...
  0, 0, 4, 1, 0, 0, ...
  0, 0, 0, 5, 0, 0, ...
  ...
The sequence with offset 1 divided by its aerated variant is (1, 5, 4, 0, 0, 0, ...). (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
K.-N. Chang and S.-C. Tsai, Exact solution of a minimal recurrence, Inform. Process. Lett. 75 (2000), 61-64.
H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 27, 32.
D. E. Knuth, Problem 11320, The American Mathematical Monthly, Vol. 114, No. 9 (Nov., 2007), p. 835.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant
Index entries for sequences related to cellular automata

Cf. A000120, A006046, A077465, A130665, A130667, A102376, A360189.

Sequences of the form a(n) = r*a(ceiling(n/2)) + s*a(floor(n/2)), a(1)=1, for (r,s) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), (1,4), (2,3), (3,2), (4,1): A000027, A006046, A064194, A130665, A073121, A268524, A116520, A268525, A268526, A268527.

import Data.List (transpose)
a116520 n = a116520_list !! n
a116520_list = 0 : zs where
   zs = 1 : (concat $ transpose
                      [zipWith (+) vs zs, zipWith (+) vs $ tail zs])
      where vs = map (* 4) zs
-- Reinhard Zumkeller, Apr 18 2012

a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 5*a(n/2) else 4*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n),n=0..52);

b[0] := 0 b[1] := 1 b[n_?EvenQ] := b[n] = 5*b[n/2] b[n_?OddQ] := b[n] = 4*b[(n - 1)/2] + b[(n + 1)/2] a = Table[b[n], {n, 1, 25}]

a(0) = 1, a(1) = 1; thereafter a(2n) = 5a(n) and a(2n+1) = 4a(n) + a(n+1).
Let r(x) = (1 + 5x + 4x^2). Then (1 + 5x + 9x^2 + 25x^3 + ...) = r(x) * r(x^2) * r(x^4) * r(x^8) * ... . - Gary W. Adamson, Mar 26 2010
a(n) = Sum_{k=0..n-1} 4^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019
a(n) = Sum_{k=0..floor(log_2(n))} 4^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

Edited by N. J. A. Sloane, Apr 16 2006, Jul 02 2008

A267610 Total number of OFF (white) cells after n iterations of the "Rule 182" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

0, 0, 2, 2, 4, 6, 12, 12, 14, 16, 22, 24, 30, 36, 50, 50, 52, 54, 60, 62, 68, 74, 88, 90, 96, 102, 116, 122, 136, 150, 180, 180, 182, 184, 190, 192, 198, 204, 218, 220, 226, 232, 246, 252, 266, 280, 310, 312, 318, 324, 338, 344, 358, 372, 402, 408, 422, 436

Offset: 0

Robert Price, Jan 18 2016

From Gus Wiseman, Mar 30 2019: (Start)
It appears that a(n) is also the number of increasing binary-containment pairs of distinct positive integers up to n + 1. A pair of positive integers is a binary containment if the positions of 1's in the reversed binary expansion of the first are a subset of the positions of 1's in the reversed binary expansion of the second. For example, the a(2) = 2 through a(8) = 14 pairs are:
  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}
  {2,3}  {2,3}  {1,5}  {1,5}  {1,5}  {1,5}  {1,5}
                {2,3}  {2,3}  {1,7}  {1,7}  {1,7}
                {4,5}  {2,6}  {2,3}  {2,3}  {1,9}
                       {4,5}  {2,6}  {2,6}  {2,3}
                       {4,6}  {2,7}  {2,7}  {2,6}
                              {3,7}  {3,7}  {2,7}
                              {4,5}  {4,5}  {3,7}
                              {4,6}  {4,6}  {4,5}
                              {4,7}  {4,7}  {4,6}
                              {5,7}  {5,7}  {4,7}
                              {6,7}  {6,7}  {5,7}
                                            {6,7}
                                            {8,9}
(End)

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A000120, A006046, A006218, A070939, A071038, A080572, A267700.

Cf. A325101, A325103, A325104, A325106, A325109, A325110, A325123, A325124.

rule=182; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) nbc=Table[Total[catri[[k]]],{k,1,rows}]; (* Number of Black cells in stage n *) nwc=Table[Length[catri[[k]]]-nbc[[k]],{k,1,rows}]; (* Number of White cells in stage n *) Table[Total[Take[nwc,k]],{k,1,rows}] (* Number of White cells through stage n *)

Conjecture: a(n) = A267700(n) - n. - Gus Wiseman, Mar 30 2019

G.f.: (1/x)*(A(x)/x - (x+1)/(1-x)^2) where A(x) is the g.f. for A006046 (conjectured). - John Tyler Rascoe, Jul 08 2024

A116528 a(0)=0, a(1)=1, and for n>=2, a(2n) = a(n), a(2n+1) = 2*a(n) + a(n+1).

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 3, 7, 1, 7, 5, 13, 3, 13, 7, 15, 1, 9, 7, 19, 5, 23, 13, 29, 3, 19, 13, 33, 7, 29, 15, 31, 1, 11, 9, 25, 7, 33, 19, 43, 5, 33, 23, 59, 13, 55, 29, 61, 3, 25, 19, 51, 13, 59, 33, 73, 7, 43, 29, 73, 15, 61, 31, 63, 1, 13

Offset: 0

Roger L. Bagula, Mar 15 2006

Equals row 2 of the array in A178239, an infinite set of sequences of the form a(n) = a(2n), a(2n+1) = r*a(n) + a(n+1). - Gary W. Adamson, May 23 2010

Given an infinite lower triangular matrix M with (1, 1, 2, 0, 0, 0, ...) in every column, shifted down twice for columns k>1; lim_{n->infinity} M^n = A116528, the left-shifted vector considered as a sequence with offset 1. - Gary W. Adamson, May 05 2010

G. C. Greubel, Table of n, a(n) for n = 0..2500
H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant

Cf. A006046, A116529, A116552, A116553, A116554, A116555.

a:=func< n | n lt 2 select n else ((n mod 2) eq 0) select Self(Round((n+1)/2)) else (2*Self(Round(n/2)) + Self(Round((n+2)/2))) >;
[a(n): n in [0..70]]; // G. C. Greubel, Jul 07 2019

A116528 := proc(n)
   option remember;
   if n <= 1 then
      n;
   elif type(n,'even') then
      procname(n/2) ;
   else
      2* procname((n-1)/2)+procname((n+1)/2) ;
   end if;
end proc:
seq(A116528(n),n=0..70) ; # R. J. Mathar, Nov 16 2011

b[0]:= 0; b[1]:= 1; b[n_?EvenQ]:= b[n] = b[n/2]; b[n_?OddQ]:= b[n] = 2*b[(n-1)/2] + b[(n+1)/2]; a = Table[b[n], {n, 1, 70}]

a(n) = if(n<2, n, if(n%2==0, a(n/2), 2*a((n-1)/2) + a((n+1)/2))); \\ G. C. Greubel, Jul 07 2019

def a(n):
    if (n<2): return n
    elif (mod(n,2)==0): return a(n/2)
    else: return 2*a((n-1)/2) + a((n+1)/2)
[a(n) for n in (0..70)] # G. C. Greubel, Jul 07 2019

G.f.: x * Product_{k>=0} (1 + x^(2^k) + 2*x^(2^(k+1))). - Ilya Gutkovskiy, Jul 07 2019

Edited by G. C. Greubel, Oct 30 2016

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.

A360189 Triangle T(n,k), n>=0, 0<=k<=floor(log_2(n+1)), read by rows: T(n,k) = number of nonnegative integers <= n having binary weight k.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Mar 04 2023

T(n,k) is defined for all n >= 0 and k >= 0. Terms that are not in the triangle are zero.

			T(6,2) = 3: 3, 5, 6, or in binary: 11_2, 101_2, 110_2.
T(15,3) = 4: 7, 11, 13, 14, or in binary: 111_2, 1011_2, 1101_2, 1110_2.
Triangle T(n,k) begins:
  1;
  1, 1;
  1, 2;
  1, 2, 1;
  1, 3, 1;
  1, 3, 2;
  1, 3, 3;
  1, 3, 3, 1;
  1, 4, 3, 1;
  1, 4, 4, 1;
  1, 4, 5, 1;
  1, 4, 5, 2;
  1, 4, 6, 2;
  1, 4, 6, 3;
  1, 4, 6, 4;
  1, 4, 6, 4, 1;
  ...

Alois P. Heinz, Rows n = 0..2^12-1, flattened
Peter J. Taylor, Closed form for Sum_{k=0..n} [wt(k) = m] where wt(n) is the binary weight of n, answer to question on MathOverflow (2024).
Wikipedia, Iverson bracket

Columns k=0-2 give: A000012, A029837(n+1) = A113473(n) for n>0, A340068(n+1).
Last elements of rows give A090996(n+1).
Cf. A000027, A000120, A000225, A000788, A006046, A007318, A116520, A116522, A116525, A116526, A130665, A130667, A161342, A231500, A231501, A231502, A272020, A361257.

b:= proc(n) option remember; `if`(n<0, 0,
      b(n-1)+x^add(i, i=Bits[Split](n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..23);

T(n,k) = my(v1); v1 = Vecrev(binary(n+1)); v1 = Vecrev(select(x->(x>0),v1,1)); sum(j=0, min(k,#v1-1), binomial(v1[j+1]-1,k-j)) \\ Mikhail Kurkov, Nov 27 2024

T(n,k) = T(n-1,k) + [A000120(n) = k] where [] is the Iverson bracket and T(n,k) = 0 for n<0.
T(2^n-1,k) = A007318(n,k) = binomial(n,k).
T(n,floor(log_2(n+1))) = A090996(n+1).
Sum_{k>=0} T(n,k) = n+1.
Sum_{k>=0} k * T(n,k) = A000788(n).
Sum_{k>=0} k^2 * T(n,k) = A231500(n).
Sum_{k>=0} k^3 * T(n,k) = A231501(n).
Sum_{k>=0} k^4 * T(n,k) = A231502(n).
Sum_{k>=0} 2^k * T(n,k) = A006046(n+1).
Sum_{k>=0} 3^k * T(n,k) = A130665(n).
Sum_{k>=0} 4^k * T(n,k) = A116520(n+1).
Sum_{k>=0} 5^k * T(n,k) = A130667(n+1).
Sum_{k>=0} 6^k * T(n,k) = A116522(n+1).
Sum_{k>=0} 7^k * T(n,k) = A161342(n+1).
Sum_{k>=0} 8^k * T(n,k) = A116526(n+1).
Sum_{k>=0} 10^k * T(n,k) = A116525(n+1).
Sum_{k>=0} n^k * T(n,k) = A361257(n).
T(n,k) = Sum_{j=0..min(k, A000120(n+1)-1)} binomial(A272020(n+1,j+1)-1,k-j) for n >= 0, k >= 0 (see Peter J. Taylor link). - Mikhail Kurkov, Nov 27 2024

A064194 a(2n) = 3a(n), a(2n+1) = 2a(n+1)+a(n), with a(1) = 1.

Original entry on oeis.org

1, 3, 7, 9, 17, 21, 25, 27, 43, 51, 59, 63, 71, 75, 79, 81, 113, 129, 145, 153, 169, 177, 185, 189, 205, 213, 221, 225, 233, 237, 241, 243, 307, 339, 371, 387, 419, 435, 451, 459, 491, 507, 523, 531, 547, 555, 563, 567, 599, 615, 631, 639, 655, 663, 671, 675

Offset: 1

Guillaume Hanrot and Paul Zimmermann, Sep 21 2001

Number of ring multiplications needed to multiply two degree-n polynomials using Karatsuba's algorithm.
Number of gates in the AND/OR problem (see Chang/Tsai reference).
a(n) is also the number of odd elements in the n X n symmetric Pascal matrix. - Stefano Spezia, Nov 14 2022

A. A. Karatsuba and Y. P. Ofman, Multiplication of multiplace numbers by automata. Dokl. Akad. Nauk SSSR 145, 2, 293-294 (1962).

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
K.-N. Chang and S.-C. Tsai, Exact solution of a minimal recurrence, Inform. Process. Lett. 75 (2000), 61-64.
P. J. Grabner and H.-K. Hwang, Digital sums and divide-and-conquer recurrences: Fourier expansions and absolute convergence, Constructive Approximation, Jan. 2005, Volume 21, Issue 2, pp. 149-179.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 27-28.

Cf. A023416, A267584, A047999 (Sierpinski triangle).
Cf. also A268514.
Sequences of form a(n)=r*a(ceil(n/2))+s*a(floor(n/2)), a(1)=1, for (r,s) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), (1,4), (2,3), (3,2), (4,1): A000027, A006046, A064194, A130665, A073121, A268524, A116520, A268525, A268526, A268527.

[n le 1 select 1 else Self(Floor(n/2)) + 2*Self(Ceiling(n/2)): n in [1..60]]; // Vincenzo Librandi, Aug 30 2016

f:=proc(n) option remember; if n=1 then 1 elif n mod 2 = 0 then 3*f(n/2) else 2*f((n+1)/2)+f((n-1)/2); fi; end; [seq(f(n),n=1..60)]; # N. J. A. Sloane, Jan 17 2016

a[n_] := a[n] = If[EvenQ[n], 3 a[n/2], 2 a[# + 1] + a[#] &[(n - 1)/2]]; a[1] = 1; Array[a, 56] (* Michael De Vlieger, Oct 29 2022 *)

a(n) = sum(i=0, n-1, sum(j=0, n-1, binomial(i+j, i) % 2)); \\ Michel Marcus, Aug 25 2013

Partial sums of the sequence { b(1)=1, b(n)=2^(e0(n-1)+1) } (essentially A267584), where e0(n)=A023416(n) is the number of zeros in the binary expansion of n. [Chang/Tsai] - Ralf Stephan, Jul 29 2003
a(1) = 1, a(n) = a(floor(n/2)) + 2*a(ceiling(n/2)), n > 1.
a(n+1) = Sum_{0<=i, j<=n} (binomial(i+j, i) mod 2). - Benoit Cloitre, Mar 07 2005
In particular, a(2^k)=3^k, a(3*2^k)=7*3^k. - N. J. A. Sloane, Jan 18 2016
a(n) = 2*A268514(n-1) + 1. - N. J. A. Sloane, Feb 07 2016

Edited with clearer definition by N. J. A. Sloane, Jan 18 2016

A080263 A014486-encoding of the branch-reduced binomial-mod-2 binary trees.

Original entry on oeis.org

2, 50, 906, 247986, 4072138, 1059204274, 272900475786, 17953590946285746, 287705670922216138, 73724537815637830834, 18880972926031430339466, 1237678872789190922262530226, 316876593058175709191975346890

Offset: 0

Antti Karttunen, Mar 02 2003

nonn

These trees are obtained from the successive generations of Rule 90 cellular automaton (A070886) or Pascal's triangle computed modulo 2 (A047999), with alive cells of the automaton (respectively: the odd binomials) forming the vertices of the zigzag tree.

J. C. P. Miller, Periodic Forests of Stunted Trees, Phil. Tran. Roy. Soc. London A266 (1970) 63; A293 (1980) 48.

A. Karttunen, Initial terms illustrated

Same sequence in binary: A080264. Cf. A080265. Breadth-first-wise encodings of the same trees: A080268. Corresponding branch-reduced zigzag trees: A080293.

Number of edges in general trees/internal nodes in binary trees: A006046, number of zigzag-edges (those colored black in illustrations) is one less: A074330. Cf. A080978.

A151566 Leftist toothpicks (see Comments for definition).

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 10, 14, 18, 20, 22, 26, 30, 34, 38, 46, 54, 56, 58, 62, 66, 70, 74, 82, 90, 94, 98, 106, 114, 122, 130, 146, 162, 164, 166, 170, 174, 178, 182, 190, 198, 202, 206, 214, 222, 230, 238, 254, 270, 274, 278, 286, 294, 302, 310, 326, 342, 350, 358, 374, 390, 406

Offset: 0

N. J. A. Sloane, May 23 2009

Similar to A139250, except that when we add toothpicks to horizontal toothpicks, we only add them at the left-hand end.
Sequence gives total number of toothpicks in the n-th generation. First differences are in A060632.
This is equivalent to the Sierpinski triangle A047999. Each inverted T formed by two toothpicks is equivalent to a triangle in the Sierpinski sieve. See Gould's sequence A001316. [From Omar E. Pol, May 23 2009]

Seiichi Manyama, 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.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A001316, A006046.

a(2n) = 2*A006046(n), a(2n+1) = a(2n) + A001316(n) = 2*A006046(n) + A001316(n).

G.f.: (x*(1+x)/(1-x)) * Product_{k>0} (1 + 2 * x^(2^k)). - Seiichi Manyama, Oct 12 2019

cp's OEIS Frontend

A130665 a(n) = Sum_{k=0..n} 3^wt(k), where wt() = A000120().

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Python

Python

Formula

Extensions

A194459 Number of entries in the n-th row of Pascal's triangle not divisible by 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Python

Formula

Extensions

A116520 a(0) = 0, a(1) = 1; a(n) = max { 4*a(k) + a(n-k) | 1 <= k <= n/2 }, for n > 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

Extensions

A267610 Total number of OFF (white) cells after n iterations of the "Rule 182" elementary cellular automaton starting with a single ON (black) cell.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A116528 a(0)=0, a(1)=1, and for n>=2, a(2*n) = a(n), a(2*n+1) = 2*a(n) + a(n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A116528 a(0)=0, a(1)=1, and for n>=2, a(2n) = a(n), a(2n+1) = 2*a(n) + a(n+1).

A064194 a(2n) = 3a(n), a(2n+1) = 2a(n+1)+a(n), with a(1) = 1.