A116520 -id:A116520 - cp's OEIS frontend

A006046 Total number of odd entries in first n rows of Pascal's triangle: a(0) = 0, a(1) = 1, a(2k) = 3a(k), a(2k+1) = 2a(k) + a(k+1). a(n) = Sum_{i=0..n-1} 2^wt(i).

0, 1, 3, 5, 9, 11, 15, 19, 27, 29, 33, 37, 45, 49, 57, 65, 81, 83, 87, 91, 99, 103, 111, 119, 135, 139, 147, 155, 171, 179, 195, 211, 243, 245, 249, 253, 261, 265, 273, 281, 297, 301, 309, 317, 333, 341, 357, 373, 405, 409, 417, 425, 441, 449, 465, 481, 513, 521

Offset: 0

Jeffrey Shallit

The graph has a blancmange or Takagi appearance. For the asymptotics, see the references by Flajolet with "Mellin" in the title. - N. J. A. Sloane, Mar 11 2021
The following alternative construction of this sequence is due to Thomas Nordhaus, Oct 31 2000: For each n >= 0 let f_n be the piecewise linear function given by the points (k /(2^n), a(k) / 3^n), k = 0, 1, ..., 2^n. f_n is a monotonic map from the interval [0,1] into itself, f_n(0) = 0, f_n(1) = 1. This sequence of functions converges uniformly. But the limiting function is not differentiable on a dense subset of this interval.
I submitted a problem to the Amer. Math. Monthly about an infinite family of non-convex sequences that solve a recurrence that involves minimization: a(1) = 1; a(n) = max { ua(k) + a(n-k) | 1 <= k <= n/2 }, for n > 1; here u is any real-valued constant >= 1. The case u=2 gives the present sequence. Cf. A130665 - A130667. - Don Knuth, Jun 18 2007
a(n) = sum of (n-1)-th row terms of triangle A166556. - Gary W. Adamson, Oct 17 2009
From Gary W. Adamson, Dec 06 2009: (Start)
Let M = an infinite lower triangular matrix with (1, 3, 2, 0, 0, 0, ...) in every column shifted down twice:
  1;
  3;
  2; 1;
  0, 3;
  0, 2, 1;
  0, 0, 3;
  0, 0, 2, 1;
  0, 0, 0, 3;
  0, 0, 0, 2, 1;
  ...
This sequence starting with "1" = lim_{n->infinity} M^n, the left-shifted vector considered as a sequence. (End)
a(n) is also the sum of all entries in rows 0 to n of Sierpiński's triangle A047999. - Reinhard Zumkeller, Apr 09 2012
The production matrix of Dec 06 2009 is equivalent to the following: Let p(x) = (1 + 3x + 2x^2). The sequence = P(x) * p(x^2) * p(x^4) * p(x^8) * .... The sequence divided by its aerated variant = (1, 3, 2, 0, 0, 0, ...). - Gary W. Adamson, Aug 26 2016
Also the total number of ON cells, rows 1 through n, for cellular automaton Rule 90 (Cf. A001316, A038183, also Mathworld Link). - Bradley Klee, Dec 22 2018

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.16.
Flajolet, Philippe, and Mordecai Golin. "Mellin transforms and asymptotics." Acta Informatica 31.7 (1994): 673-696.
Flajolet, Philippe, Mireille Régnier, and Robert Sedgewick. "Some uses of the Mellin integral transform in the analysis of algorithms." in Combinatorial algorithms on words. Springer, 1985. 241-254.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..16383 (first 1000 terms from T. D. Noe)
L. Carlitz, The number of binomial coefficients divisible by a fixed power of a prime, Rend. Circ. Mat. Palermo (2) 16 (1967), pp. 299-320.
K.-N. Chang and S.-C. Tsai, Exact solution of a minimal recurrence, Inform. Process. Lett. 75 (2000), 61-64.
Prerona Chatterjee, Kshitij Gajjar, and Anamay Tengse, Transparency Beyond VNP in the Monotone Setting, arXiv:2202.13103 [cs.CC], 2022.
S. R. Finch, P. Sebah, and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008.
N. J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly 54:10 (1947), pp. 89-92.
Philippe Flajolet, Peter Grabner, Peter Kirschenhofer, Helmut Prodinger, and Robert F. Tichy, Mellin Transforms And Asymptotics: Digital Sums, Theoret. Computer Sci. 23 (1994), 291-314.
Philippe Flajolet, Xavier Gourdon, and Philippe Dumas, Mellin transforms and asymptotics: harmonic sums Special volume on mathematical analysis of algorithms. Theoret. Comput. Sci. 144 (1995), no. 1-2, 3-58.
Philippe Flajolet and Robert Sedgewick, Mellin transforms and asymptotics: Finite differences and Rice's integrals, Theoretical Computer Science 144.1-2 (1995): 101-124.
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.
H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62.1 (1977), 19-22. (Annotated scanned copy)
F. T. Howard, The number of binomial coefficients divisible by a fixed power of 2, Proceedings of the American Mathematical Society, Vol. 29:2 (Jul 1971), pp. 236-242.
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. 6, 27, 29-31.
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. See p. 1.
Akhlesh Lakhtakia and Russell Messier, Self-similar sequences and chaos from Gauss sums, Computers & graphics 13.1 (1989): 59-62.
Akhlesh Lakhtakia and Russell Messier, Self-similar sequences and chaos from Gauss sums, Computers & Graphics 13.1 (1989), 59-60. (Annotated scanned copy)
A. Lakhtakia et al., Fractal sequences derived from the self-similar extensions of the Sierpinski gasket, J. Phys. A 21 (1988), 1925-1928.
Giuseppe Lancia and Paolo Serafini, Computational Complexity and ILP Models for Pattern Problems in the Logical Analysis of Data, Algorithms (2021) Vol. 14, No. 8, 235.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
K. B. Stolarsky, Power and exponential sums of digital sums related to binomial coefficient parity, SIAM J. Appl. Math., 32 (1977), 717-730. See B(n). - _N. J. A. Sloane_, Apr 05 2014
Eric Weisstein's World of Mathematics, Pascal's Triangle
Eric Weisstein's World of Mathematics, Rule 90
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant
Index entries for triangles and arrays related to Pascal's triangle

Partial sums of A001316.
See A130665 for Sum 3^wt(n).
a(n) = A074330(n-1) + 1 for n >= 2. A080978(n) = 2*a(n) + 1. Cf. A080263.
Cf. also A159912, A166556, A038183, A048896, A360189.
Sequences of 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.

a006046 = sum . concat . (`take` a047999_tabl)
-- Reinhard Zumkeller, Apr 09 2012

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

f:=proc(n) option remember;
if n <= 1 then n 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=0..130)]; # N. J. A. Sloane, Jul 29 2014

f[n_] := Sum[ Mod[ Binomial[n, k], 2], {k, 0, n} ]; Table[ Sum[ f[k], {k, 0, n} ], {n, 0, 100} ]
Join[{0},Accumulate[Count[#,?OddQ]&/@Table[Binomial[n,k],{n,0,60},{k,0,n}]]] (* _Harvey P. Dale, Dec 10 2014 *)
FoldList[Plus, 0, Total /@ CellularAutomaton[90, Join[Table[0, {#}], {1}, Table[0, {#}]], #]][[2 ;; -1]] &@50 (* Bradley Klee, Dec 23 2018 *)
Join[{0}, Accumulate[2^DigitCount[Range[0, 127], 2, 1]]] (* Paolo Xausa, Oct 24 2024 *)
Join[{0}, Accumulate[2^Nest[Join[#, #+1]&, {0}, 7]]] (* Paolo Xausa, Oct 24 2024, after IWABUCHI Yu(u)ki in A000120 *)

A006046(n)={ n<2 & return(n); A006046(n\2)*3+if(n%2,1<M. F. Hasler, May 03 2009

a(n) = if(!n, 0, my(r=0, t=1); forstep(i=logint(n, 2), 0, -1, r*=3; if(bittest(n, i), r+=t; t*=2)); r); \\ Ruud H.G. van Tol, Jul 06 2024

from functools import lru_cache
@lru_cache(maxsize=None)
def A006046(n):return n if n<=1 else 2*A006046((n-1)//2)+A006046((n+1)//2)if n%2 else 3*A006046(n//2) # Guillermo Hernández, Dec 31 2023

from math import prod
def A006046(n):
    d = list(map(lambda x:int(x)+1,bin(n)[:1:-1]))
    return sum((b-1)*prod(d[a:])*3**a for a, b in enumerate(d))>>1 # Chai Wah Wu, Aug 13 2025

a(n) = Sum_{k=0..n-1} 2^A000120(k). - Paul Barry, Jan 05 2005; simplified by N. J. A. Sloane, Apr 05 2014
For asymptotics see Stolarsky (1977). - N. J. A. Sloane, Apr 05 2014
a(n) = a(n-1) + A001316(n-1). a(2^n) = 3^n. - Henry Bottomley, Apr 05 2001
a(n) = n^(log_2(3))*G(log_2(n)) where G(x) is a function of period 1 defined by its Fourier series. - Benoit Cloitre, Aug 16 2002; formula modified by S. R. Finch, Dec 31 2007
G.f.: (x/(1-x))*Product_{k>=0} (1 + 2*x^2^k). - Ralf Stephan, Jun 01 2003; corrected by Herbert S. Wilf, Jun 16 2005
a(1) = 1, a(n) = 2*a(floor(n/2)) + a(ceiling(n/2)).
a(n) = 3*a(floor(n/2)) + (n mod 2)*2^A000120(n-1). - M. F. Hasler, May 03 2009
a(n) = Sum_{k=0..floor(log_2(n))} 2^k * A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

More terms from James Sellers, Aug 21 2000

Definition expanded by N. J. A. Sloane, Feb 16 2016

A102376 a(n) = 4^A000120(n).

Original entry on oeis.org

1, 4, 4, 16, 4, 16, 16, 64, 4, 16, 16, 64, 16, 64, 64, 256, 4, 16, 16, 64, 16, 64, 64, 256, 16, 64, 64, 256, 64, 256, 256, 1024, 4, 16, 16, 64, 16, 64, 64, 256, 16, 64, 64, 256, 64, 256, 256, 1024, 16, 64, 64, 256, 64, 256, 256, 1024, 64, 256, 256, 1024, 256, 1024, 1024

Offset: 0

Paul Barry, Jan 05 2005

Consider a simple cellular automaton, a grid of binary cells c(i,j), where the next state of the grid is calculated by applying the following rule to each cell: c(i,j) = ( c(i+1,j-1) + c(i+1,j+1) + c(i-1,j-1) + c(i-1,j+1) ) mod 2 If we start with a single cell having the value 1 and all the others 0, then the aggregate values of the subsequent states of the grid will be the terms in this sequence. - Andras Erszegi (erszegi.andras(AT)chello.hu), Mar 31 2006. See link for initial states. - N. J. A. Sloane, Feb 12 2015
This is the odd-rule cellular automaton defined by OddRule 033 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). - N. J. A. Sloane, Feb 25 2015
First differences of A116520. - Omar E. Pol, May 05 2010

			1 + 4*x + 4*x^2 + 16*x^3 + 4*x^4 + 16*x^5 + 16*x^6 + 64*x^7 + 4*x^8 + ...
From _Omar E. Pol_, Jun 07 2009: (Start)
Triangle begins:
  1;
  4;
  4,16;
  4,16,16,64;
  4,16,16,64,16,64,64,256;
  4,16,16,64,16,64,64,256,16,64,64,256,64,256,256,1024;
  4,16,16,64,16,64,64,256,16,64,64,256,64,256,256,1024,16,64,64,256,64,256,...
(End)

N. J. A. Sloane, 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.]
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
Nathan Epstein, Animation of CA generating A102376
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
N. J. A. Sloane, Illustration of generations 0-15 of the cellular automaton
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Alexander Yu. Vlasov, Modelling reliability of reversible circuits with 2D second-order cellular automata, arXiv:2312.13034 [nlin.CG], 2023. See page 13.
Index entries for sequences related to cellular automata

For generating functions Prod_{k>=0} (1+a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674.
A151783 is a very similar sequence.
Cf. A001316, A048883, A000079, A116520, A000302.
See A160239 for the analogous CA defined by Rule 204 on an 8-celled neighborhood.

a102376 = (4 ^) . a000120  -- Reinhard Zumkeller, Feb 13 2015

seq(4^convert(convert(n,base,2),`+`),n=0..100); # Robert Israel, Apr 30 2017

Table[4^DigitCount[n, 2, 1], {n, 0, 100}] (* Indranil Ghosh, Apr 30 2017 *)

{a(n) = if( n<0, 0, 4^subst( Pol( binary(n)), x, 1))} /* Michael Somos, May 29 2008 */
a(n) = 4^hammingweight(n); \\ Michel Marcus, Apr 30 2017

def a(n): return 4**bin(n)[2:].count("1") # Indranil Ghosh, Apr 30 2017

def A102376(n): return 1<<(n.bit_count()<<1) # Chai Wah Wu, Nov 15 2022

Formulas due to Paul D. Hanna: (Start)
G.f.: Product_{k>=0} 1 + 4x^(2^k).
a(n) = Product_{k=0..log_2(n)} 4^b(n, k), b(n, k)=coefficient of 2^k in binary expansion of n.
a(n) = Sum_{k=0..n} (C(n, k) mod 2)*3^A000120(n-k). (End)
a(n) = Sum_{k=0..n} (C(n, k) mod 2) * Sum_{j=0..k} (C(k, j) mod 2) * Sum_{i=0..j} (C(j, i) mod 2). - Paul Barry, Apr 01 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w * (u^2 - 2*u*v + 5*v^2) - 4*v^3. - Michael Somos, May 29 2008
Run length transform of A000302. - N. J. A. Sloane, Feb 23 2015

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

Original entry on oeis.org

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

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

A130667 a(1) = 1; a(n) = max{ 5*a(k) + a(n-k) | 1 <= k <= n/2 } for n > 1.

Original entry on oeis.org

1, 6, 11, 36, 41, 66, 91, 216, 221, 246, 271, 396, 421, 546, 671, 1296, 1301, 1326, 1351, 1476, 1501, 1626, 1751, 2376, 2401, 2526, 2651, 3276, 3401, 4026, 4651, 7776, 7781, 7806, 7831, 7956, 7981, 8106, 8231, 8856, 8881, 9006, 9131, 9756, 9881, 10506, 11131

Offset: 1

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

nonn

From Gary W. Adamson, Aug 27 2016: (Start)
The formula of Mar 26 2010 is equivalent to the following: Given the production matrix M below, lim_{k->infinity} M^k as a left-shifted vector generates the sequence.
  1, 0, 0, 0, 0, ...
  6, 0, 0, 0, 0, ...
  5, 1, 0, 0, 0, ...
  0, 6, 0, 0, 0, ...
  0, 5, 1, 0, 0, ...
  0, 0, 6, 0, 0, ...
  0, 0, 5, 1, 0, ...
  ...
The sequence divided by its aerated variant is (1, 6, 5, 0, 0, 0, ...). (End)

Alois P. Heinz, Table of n, a(n) for n = 1..16383
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, 32-33.
D. E. Knuth, Problem 11320, The American Mathematical Monthly, Vol. 114, No. 9 (Nov., 2007), p. 835.

Cf. A000120, A006046, A116520, A130665, A360189.

import Data.List (transpose)
a130667 n = a130667_list !! (n-1)
a130667_list = 1 : (concat $ transpose
   [zipWith (+) vs a130667_list, zipWith (+) vs $ tail a130667_list])
   where vs = map (* 5) a130667_list
-- Reinhard Zumkeller, Apr 18 2012

[&+[5^(2*k - Valuation(Factorial(2*k), 2)): k in [0..n]]: n in [0..50]]; // Vincenzo Librandi, Mar 15 2019

a:= proc(n) option remember;
      `if`(n=1, 1, `if`(irem(n, 2, 'm')=0, 6*a(m), 5*a(m)+a(n-m)))
    end:
seq(a(n), n=1..70); # Alois P. Heinz, Apr 09 2012

a[1]=1; a[n_] := a[n] = If[EvenQ[n], 6a[n/2], 5a[(n-1)/2]+a[(n+1)/2]]; Array[a, 50] (* Jean-François Alcover, Feb 13 2015 *)

first(n)=my(v=vector(n),r,t); v[1]=1; for(i=2,n, r=0; for(k=1,i\2, t=5*v[k]+v[i-k]; if(t>r, r=t)); v[i]=r); v \\ Charles R Greathouse IV, Aug 29 2016

a(2*n) = 6*a(n) and a(2*n+1) = 5*a(n) + a(n+1).
Let r(x) = (1 + 6*x + 5*x^2). Then (1 + 6*x + 11*x^2 + 36*x^3 + ...) = r(x) * r(x^2) * r(x^4) * r(x^8) * ... - Gary W. Adamson, Mar 26 2010
a(n) = Sum_{k=0..n} 5^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019
a(n) = Sum_{k=0..floor(log_2(n))} 5^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

A073121 a(n) = ra(ceiling(n/2)) + sa(floor(n/2)) with a(1)=1 and (r,s)=(2,2).

Original entry on oeis.org

1, 4, 10, 16, 28, 40, 52, 64, 88, 112, 136, 160, 184, 208, 232, 256, 304, 352, 400, 448, 496, 544, 592, 640, 688, 736, 784, 832, 880, 928, 976, 1024, 1120, 1216, 1312, 1408, 1504, 1600, 1696, 1792, 1888, 1984, 2080, 2176, 2272, 2368, 2464, 2560, 2656, 2752

Offset: 1

Jeffrey Shallit, Aug 25 2002

nonn

A recurrence occurring in the analysis of a regular expression algorithm.

			a(1)=1, a(2) = 2*(a(1)+a(1)) = 4, a(3) = 2*(a(2)+a(1)) = 10.

K. Ellul, J. Shallit and M.-w. Wang, Regular expressions: new results and open problems, in Descriptional Complexity of Formal Systems (DCFS), Proceedings of workshop, London, Ontario, Canada, 21-24 August 2002, pp. 17-34.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jean-Paul Allouche and Jeffrey Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
Jean-Paul Allouche and Jeffrey Shallit, The Ring of k-regular Sequences, II
F. Barbero, G. Gutin, M. Jones, and B. Sheng, Parameterized and approximation algorithms for the load coloring problem, Algorithmica 79, No. 1, 211-229 (2017). Prop 3.
Keh-Ning Chang and Shi-Chun Tsai, Exact solution of a minimal recurrence, Inform. Process. Lett. 75 (2000), 61-64.
Erik D. Demaine, Martin L. Demaine, Yair N. Minsky, Joseph S. B. Mitchell, Ronald L. Rivest, and Mihai Patrascu, Picture-Hanging Puzzles, arXiv:1203.3602 [cs.DS], 2012-2014.
Keith Ellul, Bryan Krawetz, Jeffrey Shallit, and Ming-wei Wang, Regular expressions: new results and open problems, Journal of Automata, Languages and Combinatorics, preprint.
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, 37.
Jean-Marc Luck, Revisiting log-periodic oscillations, arXiv:2403.00432 [cond-mat.stat-mech], 2024. See p. 14.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Cf. A053644, A053645, A254575.

Sequences of 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.

a073121 n = a053644 n * (fromIntegral n + 2 * a053645 n)
-- Reinhard Zumkeller, Mar 23 2012

a:= proc(n) option remember; `if`(n=1, 1,
      2*((m-> a(m)+a(n-m))(iquo(n, 2))))
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Feb 01 2015

a[n_] := a[n] = If[n == 1, 1, 2*(a[Quotient[n, 2]] + a[n - Quotient[n, 2]])]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)
a[ n_] := If[ n < 1, 0, Module[{m = 1, A = 1}, While[m < n, m *= 2; A = (Normal[A] /. x -> x^2) 2 (1 + x)^2 - 1 + O[x]^m]; Coefficient[A, x, n - 1]]]; (* Michael Somos, Jul 04 2017 *)

{a(n) = n--; if( n<0, 0, my(m=1, A = 1 + O(x)); while(m<=n, m*=2; A = subst(A, x, x^2) * 2 * (1 + x)^2 - 1); polcoeff(A, n))}; /* Michael Somos, Jul 04 2017 */

a(n) = 2*(a(floor(n/2)) + a(ceiling(n/2))) for n >= 2; alternatively, a(n) = 2^c(n+2b) where n = 2^c + b, 0 <= b < 2^c.
a(n) == 1 (mod 3), a(n+1)-a(n) = 3*A053644(n). If k >= 1: a(2^k)=4^k, a(3*2^k)=(10/9)*4^k. More generally a(m*2^k) = a(m)*4^k. Hence for any n, n^2 <= a(n) <= C*n^2 where C is a constant 1.125 < C < 1.14 and it seems that C = lim_{k->infinity} a(A001045(k))/A001045(k)^2 where A001045(k) =(2^n - (-1)^n)/3 is the Jacobsthal sequence. In other words, in the range 2^k <= n <= 2^(k+1) the maximum of a(n)/n^2 is reached for the only possible n in the Jacobsthal sequence. - Benoit Cloitre, Aug 26 2002
For any n, n^2 <= a(n) <= 9/8 * n^2. - Arnoud van der Leer, Sep 01 2019
a(n) = 2*(a(floor(n/2)) + a(ceiling(n/2))) for n >= 2; alternatively, a(n) = 2^c(n+2b) where n = 2^c + b, 0 <= b < 2^c
G.f.: 3*x/(1-x)^2 * ((2*x+1)/3 + Sum_{k>=1} 2^(k-1)*x^2^k). - Ralf Stephan, Apr 18 2003
G.f.: A(x) = 2 * (1/x + 2 + x) * A(x^2) - x. - Michael Somos, Jul 04 2017

Edited by N. J. A. Sloane, Feb 16 2016

A161342 Number of "ON" cubic cells at n-th stage in simple 3-dimensional cellular automaton: a(n) = A160428(n)/8.

Original entry on oeis.org

0, 1, 8, 15, 64, 71, 120, 169, 512, 519, 568, 617, 960, 1009, 1352, 1695, 4096, 4103, 4152, 4201, 4544, 4593, 4936, 5279, 7680, 7729, 8072, 8415, 10816, 11159, 13560, 15961, 32768, 32775, 32824, 32873, 33216, 33265, 33608, 33951, 36352, 36401, 36744, 37087, 39488

Offset: 0

Omar E. Pol, Jun 14 2009

nonn

First differences are in A161343. - Omar E. Pol, May 03 2015
From Gary W. Adamson, Aug 30 2016: (Start)
Let M =
  1, 0, 0, 0, 0, ...
  8, 0, 0, 0, 0, ...
  7, 1, 0, 0, 0, ...
  0, 8, 0, 0, 0, ...
  0, 7, 1, 0, 0, ...
  0, 0, 8, 0, 0, ...
  0, 0, 7, 1, 0, ...
  ...
Then M^k converges to a single nonzero column giving the sequence.
The sequence with offset 1 divided by its aerated variant is (1, 8, 7, 0, 0, 0, ...). (End)

Alois P. Heinz, Table of n, a(n) for n = 0..16383
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A160410, A160428, A161343, A006046, A130665, A116520, A130667, A116522, A116526, A116525, A360189.

b:= proc(n) option remember; `if`(n<0, 0,
      b(n-1)+x^add(i, i=Bits[Split](n)))
    end:
a:= n-> subs(x=7, b(n-1)):
seq(a(n), n=0..44);  # Alois P. Heinz, Mar 06 2023

A161342list[nmax_]:=Join[{0},Accumulate[7^DigitCount[Range[0,nmax-1],2,1]]];A161342list[100] (* Paolo Xausa, Aug 05 2023 *)

From Nathaniel Johnston, Nov 13 2010: (Start)
a(n) = Sum_{k=0..n-1} 7^A000120(k).
a(n) = 1 + 7 * Sum_{k=1..n-1} A151785(k), for n >= 1.
a(2^n) = 2^(3n).
(End)
a(n) = Sum_{k=0..floor(log_2(n))} 7^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

More terms from Nathaniel Johnston, Nov 13 2010

cp's OEIS Frontend

A006046 Total number of odd entries in first n rows of Pascal's triangle: a(0) = 0, a(1) = 1, a(2k) = 3*a(k), a(2k+1) = 2*a(k) + a(k+1). a(n) = Sum_{i=0..n-1} 2^wt(i).

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A102376 a(n) = 4^A000120(n).

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

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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.

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

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A130667 a(1) = 1; a(n) = max{ 5*a(k) + a(n-k) | 1 <= k <= n/2 } for n > 1.

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A006046 Total number of odd entries in first n rows of Pascal's triangle: a(0) = 0, a(1) = 1, a(2k) = 3a(k), a(2k+1) = 2a(k) + a(k+1). a(n) = Sum_{i=0..n-1} 2^wt(i).

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

A073121 a(n) = ra(ceiling(n/2)) + sa(floor(n/2)) with a(1)=1 and (r,s)=(2,2).

A268524 a(n) = ra(ceiling(n/2))+sa(floor(n/2)) with a(1)=1 and (r,s)=(3,1).