cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A147562 Number of "ON" cells at n-th stage in the "Ulam-Warburton" two-dimensional cellular automaton.

Original entry on oeis.org

0, 1, 5, 9, 21, 25, 37, 49, 85, 89, 101, 113, 149, 161, 197, 233, 341, 345, 357, 369, 405, 417, 453, 489, 597, 609, 645, 681, 789, 825, 933, 1041, 1365, 1369, 1381, 1393, 1429, 1441, 1477, 1513, 1621, 1633, 1669, 1705, 1813, 1849, 1957, 2065, 2389, 2401, 2437, 2473
Offset: 0

Views

Author

N. J. A. Sloane, based on emails from Franklin T. Adams-Watters, R. J. Mathar and David W. Wilson, Apr 29 2009

Keywords

Comments

Studied by Holladay and Ulam circa 1960. See Fig. 1 and Example 1 of the Ulam reference. - N. J. A. Sloane, Aug 02 2009.
Singmaster calls this the Ulam-Warburton cellular automaton. - N. J. A. Sloane, Aug 05 2009
On the infinite square grid, start with all cells OFF.
Turn a single cell to the ON state.
At each subsequent step, each cell with exactly one neighbor ON is turned ON, and everything that is already ON remains ON.
Here "neighbor" refers to the four adjacent cells in the X and Y directions.
Note that "neighbor" could equally well refer to the four adjacent cells in the diagonal directions, since the graph formed by Z^2 with "one-step rook" adjacencies is isomorphic to Z^2 with "one-step bishop" adjacencies.
Also toothpick sequence starting with a central X-toothpick followed by T-toothpicks (see A160170 and A160172). The sequence gives the number of polytoothpicks in the structure after n-th stage. - Omar E. Pol, Mar 28 2011
It appears that this sequence shares infinitely many terms with both A162795 and A169707, see Formula section and Example section. - Omar E. Pol, Feb 20 2015
It appears that the positive terms are also the odd terms (a bisection) of A151920. - Omar E. Pol, Mar 06 2015
Also, the number of active (ON, black) cells in the n-th stage of growth of two-dimensional cellular automaton defined by Wolfram's "Rule 558" or "Rule 686" based on the 5-celled von Neumann neighborhood. - Robert Price, May 10 2016
From Omar E. Pol, Mar 05 2019: (Start)
a(n) is also the total number of "hidden crosses" after 4*n stages in the toothpick structure of A139250, including the central cross, beginning to count the crosses when their nuclei are totally formed with 4 quadrilaterals.
a(n) is also the total number of "flowers with six petals" after 4*n stages in the toothpick structure of A323650.
Note that the location of the "nuclei of the hidden crosses" and the "flowers with six petals" in both toothpick structures is essentially the same as the location of the "ON" cells in the version "one-step bishop" of this sequence (see the illustration of initial terms, figure 2). (End)
This sequence has almost exactly the same graph as A187220, A162795, A169707 and A160164 which is twice A139250. - Omar E. Pol, Jun 18 2022

Examples

			If we label the generations of cells turned ON by consecutive numbers we get a rosetta cell pattern:
. . . . . . . . . . . . . . . . .
. . . . . . . . 4 . . . . . . . .
. . . . . . . 4 3 4 . . . . . . .
. . . . . . 4 . 2 . 4 . . . . . .
. . . . . 4 3 2 1 2 3 4 . . . . .
. . . . . . 4 . 2 . 4 . . . . . .
. . . . . . . 4 3 4 . . . . . . .
. . . . . . . . 4 . . . . . . . .
. . . . . . . . . . . . . . . . .
In the first generation, only the central "1" is ON, a(1)=1. In the next generation, we turn ON four "2", leading to a(2)=a(1)+4=5. In the third generation, four "3" are turned ON, a(3)=a(2)+4=9. In the fourth generation, each of the four wings allows three 4's to be turned ON, a(4)=a(3)+4*3=21.
From _Omar E. Pol_, Feb 18 2015: (Start)
Also, written as an irregular triangle T(j,k), j>=0, k>=1, in which the row lengths are the terms of A011782:
1;
5;
9,   21;
25,  37, 49, 85;
89, 101,113,149,161,197,233,341;
345,357,369,405,417,453,489,597,609,645,681,789,825,933,1041,1365;
...
The right border gives the positive terms of A002450.
(End)
It appears that T(j,k) = A162795(j,k) = A169707(j,k), if k is a power of 2, for example: it appears that the three mentioned triangles only share the elements from the columns 1, 2, 4, 8, 16, ... - _Omar E. Pol_, Feb 20 2015

References

  • S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962.
  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 928.

Links

Crossrefs

Cf. A000120, A139250, A147582 (number turned ON at n-th step), A147610, A130665, A151920, A151917, A160120, A160164, A160410, A160414, A162795, A169707, A187220, A246331, A323650.
See also A006046, A335794, A335795.

Programs

  • Maple
    Since this is the partial sum sequence of A147582, it is most easily obtained using the Maple code given in A147582.
# [x,y] coordinates of cells on
Lse := [[0,0]] ;
# enclosing rectangle of the cells on (that is, minima and maxima in Lse)
xmin := 0 ;
xmax := 0 ;
ymin := 0 ;
ymax := 0 ;
# count neighbors of x,y which are on; return 0 if [x,y] is in L
cntnei := proc(x,y,L)
local a,p,xpt,ypt;
a := 0 ;
if not [x,y] in L then
for p in Lse do
xpt := op(1,p) ;
ypt := op(2,p) ;
if ( abs(xpt-x) = 1 and ypt=y ) or ( x=xpt and abs(ypt-y) = 1) then
a := a+1 ;
fi;
od:
fi:
RETURN(a) ;
end:
# loop over generations/steps
for stp from 1 to 10 do
Lnew := [] ;
for x from xmin-1 to xmax+1 do
for y from ymin-1 to ymax+1 do
if cntnei(x,y,Lse) = 1 then
Lnew := [op(Lnew),[x,y]] ;
fi;
od:
od:
for p in Lnew do
xpt := op(1,p) ;
ypt := op(2,p) ;
xmin := min(xmin,xpt) ;
xmax := max(xmax,xpt) ;
ymin := min(ymin,ypt) ;
ymax := max(ymax,ypt) ;
od:
Lse := [op(Lse),op(Lnew)] ;
print(nops(Lse)) ;
  • Mathematica
    Join[{0},Map[Function[Apply[Plus,Flatten[ #1]]],CellularAutomaton[{686,{2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},200]]] (* Nadia Heninger and N. J. A. Sloane, Aug 11 2009; modified by Paolo Xausa, Aug 12 2022 to include the a(0) term *)
ArrayPlot /@ CellularAutomaton[{686, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 16] (* N. J. A. Sloane, Nov 08 2014 *)
A147562list[nmax_]:=Accumulate[Join[{0,1},4*3^(DigitCount[Range[nmax-1],2,1]-1)]];A147562list[100] (* Paolo Xausa, May 21 2023 *)
  • PARI
    a(n) = if (n, 1 + 4*sum(k=1, n-1, 3^(hammingweight(k)-1)), 0); \\ Michel Marcus, Jul 05 2022

Formula

a(n) = 1 + 4*Sum_{k=1..n-1} 3^(wt(k)-1) for n>1, where wt() = A000120(). [Corrected by Paolo Xausa, Aug 12 2022]
For asymptotics see the discussion in the comments in A006046. - N. J. A. Sloane, Mar 11 2021
From Omar E. Pol, Mar 13 2011: (Start)
a(n) = 2*A151917(n) - 1, for n >= 1.
a(n) = 1 + 4*A151920(n-2), for n >= 2.
(End)
It appears that a(n) = A162795(n) = A169707(n), if n is a member of A048645, otherwise a(n) < A162795(n) < A169707(n). - Omar E. Pol, Feb 20 2015
It appears that a(n) = A151920(2n-2), n >= 1. - Omar E. Pol, Mar 06 2015
It appears that a(n) = (A130665(2n-1) - 1)/3, n >= 1. - Omar E. Pol, Mar 07 2015
a(n) = 1 + 4*(A130665(n-1) - 1)/3, n >= 1. Omar E. Pol, Mar 07 2015
a(n) = A323650(2n)/3. - Omar E. Pol, Mar 04 2019

Extensions

Offset and initial terms changed by N. J. A. Sloane, Jun 07 2009
Numbers in the comment adapted to the offset by R. J. Mathar, Mar 03 2010

A160120 Y-toothpick sequence (see Comments lines for definition).

Original entry on oeis.org

0, 1, 4, 7, 16, 19, 28, 37, 58, 67, 76, 85, 106, 121, 142, 169, 220, 247, 256, 265, 286, 301, 322, 349, 400, 433, 454, 481, 532, 583, 640, 709, 826, 907, 928, 937, 958, 973, 994, 1021, 1072, 1105, 1126, 1153, 1204, 1255, 1312, 1381, 1498, 1585, 1618, 1645
Offset: 0

Views

Author

Omar E. Pol, May 02 2009

Keywords

Comments

A Y-toothpick (or Y-shaped toothpick) is formed from three toothpicks of length 1, like a star with three endpoints and only one middle-point.
On the infinite triangular grid, we start at round 0 with no Y-toothpicks.
At round 1 we place a Y-toothpick anywhere in the plane.
At round 2 we add three more Y-toothpicks. After round 2, in the structure there are three rhombuses and a hexagon.
At round 3 we add three more Y-toothpicks.
And so on ... (see illustrations).
The sequence gives the number of Y-toothpicks after n rounds. A160121 (the first differences) gives the number added at the n-th round.
The Y-toothpick pattern has a recursive, fractal (or fractal-like) structure.
Note that, on the infinite triangular grid, a Y-toothpick can be represented as a polyedge with three components. In this case, at the n-th round, the structure is a polyedge with 3*a(n) components.
This structure is more complex than the toothpick structure of A139250. For example, at some rounds we can see inward growth.
The structure contains distinct polygons which have side length equal to 1.
Observation: It appears that the region of the structure where all grid points are covered is formed only by three distinct polygons:
- Triangles
- Rhombuses
- Concave-convex hexagons
Holes in the structure: Also, we can see distinct concave-convex polygons which contains a region where there are no grid points that are covered, for example:
- Decagons (with 1 non-covered grid point)
- Dodecagons (with 4 non-covered grid points)
- 18-gons (with 7 non-covered grid points)
- 30-gons (with 26 non-covered grid points)
- ...
Observation: It appears that the number of distinct polygons that contain non-covered grid points is infinite.
This sequence appears to be related to powers of 2. For example:
Conjecture: It appears that if n = 2^k, k>0, then, between the other polygons, there appears a new centered hexagon formed by three rhombuses with side length = 2^k/2 = n/2.
Conjecture: Consider the perimeter of the structure. It appears that if n = 2^k, k>0, then the structure is a triangle-shaped polygon with A000225(k)*6 sides and a half toothpick in each vertice of the "triangle".
Conjecture: It appears that if n = 2^k, k>0, then the ratio of areas between the Y-toothpick structure and the unitary triangle is equal to A006516(k)*6.
See the entry A139250 for more information about the growth of "standard" toothpicks.
See also A160715 for another version of this structure but without internal growth of Y-toothpicks. [Omar E. Pol, May 31 2010]
For an alternative visualization replace every single toothpick with a rhombus, or in other words, replace every Y-toothpick with the "three-diamond" symbol, so we have a cellular automaton in which a(n) gives the total number of "three-diamond" symbols after n-th stage and A160167(n) counts the total number of "ON" diamonds in the structure after n-th stage. See also A253770. - Omar E. Pol, Dec 24 2015
The behavior is similar to A153006 (see the graph). - Omar E. Pol, Apr 03 2018

Links

Crossrefs

Toothpick sequence: A139250.
Cf. A000079, A000225, A006516, A147562, A153006, A160121, A160123, A160715, A161206, A161328, A161330, A161430, A173066, A173068, A253770.

Programs

  • Mathematica
    YTPFunc[lis_, step_] := With[{out = Extract[lis, {{1, 2}, {2, 1}, {-1, -1}}], in = lis[[2, 2]]}, Which[in == 0 && Count[out, 2] >= 2, 1, in == 0 && Count[out, 2] == 1, 2, True, in]]; A160120[0] = 0; A160120[n_] := With[{m = n - 1}, Count[CellularAutomaton[{YTPFunc, {}, {1, 1}}, {{{2}}, 0}, {{{m}}}], 2, 2]] (* JungHwan Min, Jan 28 2016 *)
A160120[0] = 0; A160120[n_] := With[{m = n - 1}, Count[CellularAutomaton[{435225738745686506433286166261571728070, 3, {{-1, 0}, {0, -1}, {0, 0}, {1, 1}}}, {{{2}}, 0}, {{{m}}}], 2, 2]] (* JungHwan Min, Jan 28 2016 *)

Extensions

More terms from David Applegate, Jun 14 2009, Jun 18 2009

A235461 Primes whose base-4 representation also is the base 2-representation of a prime.

Original entry on oeis.org

5, 17, 257, 277, 337, 1093, 1109, 1297, 1361, 4357, 5189, 16453, 16657, 16661, 17489, 17669, 17681, 17749, 21521, 21569, 21589, 65537, 65557, 65617, 65809, 66821, 70657, 70981, 70997, 81937, 82241, 83221, 83269, 86017, 86357, 87317, 263429, 263489, 267541, 278549
Offset: 1

Views

Author

M. F. Hasler, Jan 11 2014

Keywords

Comments

This sequence is part of the two-dimensional array of sequences based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.
For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
When the smaller base is b=2 such that only digits 0 and 1 are allowed, these are primes that are the sum of distinct powers of the larger base, here c=4, thus a subsequence of A077718 and therefore also of A000695, the Moser-de Bruijn sequence.

Examples

			5 = 11_4 and 11_2 = 3 are both prime, so 5 is a term.
17 = 101_4 and 101_2 = 5 are both prime, so 17 is a term.

Links

Crossrefs

Cf. A090707 - A091924, A235462 - A235482. See the LINK for further cross-references.

Programs

  • PARI
    is(p,b=2,c=4)=vecmax(d=digits(p,c))
  • Python
    from itertools import islice
from sympy import nextprime, isprime
def A235461_gen(): # generator of terms
    p = 1
    while (p:=nextprime(p)):
        if isprime(m:=int(bin(p)[2:],4)):
            yield m
A235461_list = list(islice(A235461_gen(),20)) # Chai Wah Wu, Aug 21 2023

Extensions

a(37)-a(40) from Robert Price, Nov 01 2023

A048883 a(n) = 3^wt(n), where wt(n) = A000120(n).

Original entry on oeis.org

1, 3, 3, 9, 3, 9, 9, 27, 3, 9, 9, 27, 9, 27, 27, 81, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 243, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 243, 9, 27, 27, 81, 27, 81, 81, 243, 27, 81, 81, 243, 81, 243, 243, 729, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81
Offset: 0

Views

Author

John W. Layman

Keywords

Comments

Or, a(n)=number of 1's ("live" cells) at stage n of a 2-dimensional cellular automata evolving by the rule: 1 if NE+NW+S=1, else 0.
This is the odd-rule cellular automaton defined by OddRule 013 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). - N. J. A. Sloane, Feb 25 2015
Or, start with S=[1]; replace S by [S, 3*S]; repeat ad infinitum.
Fixed point of the morphism 1 -> 13, 3 -> 39, 9 -> 9(27), ... = 3^k -> 3^k 3^(k+1), ... starting from a(0) = 1; 1 -> 13 -> 1339 -> = 1339399(27) -> 1339399(27)399(27)9(27)(27)(81) -> ..., . - Robert G. Wilson v, Jan 24 2006
Equals row sums of triangle A166453 (the square of Sierpiński's gasket, A047999). - Gary W. Adamson, Oct 13 2009
First bisection of A169697=1,5,3,19,3,. a(2n+2)+a(2n+3)=12,12,36,=12*A147610 ? Distribution of terms (in A000244): A011782=1,A000079 for first array, A000079 for second. - Paul Curtz, Apr 20 2010
a(A000225(n)) = A000244(n) and a(m) != A000244(n) for m < A000225(n). - Reinhard Zumkeller, Nov 14 2011
This sequence pertains to phenotype Punnett square mathematics. Start with X=1. Each hybrid cross involves the equation X:3X. Therefore, the ratio in the first (mono) hybrid cross is X=1:3X=3(1) or 3; or 3:1. When you move up to the next hybridization level, replace the previous cross ratio with X. X now represents 2 numbers-1:3. Therefore, the ratio in the second (di) hybrid cross is X=(1:3):3X=[3(1):3(3)] or (3:9). Put it together and you get 1:3:3:9. Each time you move up a hybridization level, replace the previous ratio with X, and use the same equation-X:3X to get its ratio. - John Michael Feuk, Dec 10 2011
Number of odd values in the n-th layer of Pascal's tetrahedron (see A268240). - Caden Le, Mar 03 2025
a(x*y) <= a(x)^A000120(y). - Joe Amos, Mar 28 2025

Examples

			From _Omar E. Pol_, Jun 07 2009: (Start)
Triangle begins:
  1;
  3;
  3,9;
  3,9,9,27;
  3,9,9,27,9,27,27,81;
  3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243;
  3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243,9,27,27,81,27,81,81,243,27,...
Or
  1;
  3,3;
  9,3,9,9;
  27,3,9,9,27,9,27,27;
  81,3,9,9,27,9,27,27,81,9,27,27,81,27,81,81;
  243,3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243,9,27,27,81,27,81,81,243,27...
(End)

Links

Crossrefs

For generating functions Product_{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.
A generalization of A001316. Cf. A102376.
Partial sums give A130665. - David Applegate, Jun 11 2009
Cf. A000079, A122018, A166453, A268240.

Programs

  • Haskell
    a048883 = a000244 . a000120  -- Reinhard Zumkeller, Nov 14 2011
  • Mathematica
    Nest[ Join[#, 3#] &, {1}, 6] (* Robert G. Wilson v, Jan 24 2006 and modified Jul 27 2014*)
a[n_] := 3^DigitCount[n, 2, 1]; Array[a, 80, 0] (* Jean-François Alcover, Nov 15 2017 *)
  • PARI
    a(n)=n=binary(n);3^sum(i=1,#n,n[i])

Formula

a(n) = Product_{k=0..log_2(n)} 3^b(n,k), where b(n,k) = coefficient of 2^k in binary expansion of n (offset 0). - Paul D. Hanna
a(n) = 3*a(n/2) if n is even, otherwise a(n) = a((n+1)/2).
G.f.: Product_{k>=0} (1+3*x^(2^k)). The generalization k^A000120 has generating function (1 + kx)*(1 + kx^2)*(1 + kx^4)*...
a(n+1) = Sum_{i=0..n} (binomial(n, i) mod 2) * Sum_{j=0..i} (binomial(i, j) mod 2). - Benoit Cloitre, Nov 16 2003
a(0)=1, a(n) = 3*a(n-A053644(n)) for n > 0. - Joe Slater, Jan 31 2016
G.f. A(x) satisfies: A(x) = (1 + 3*x) * A(x^2). - Ilya Gutkovskiy, Jul 09 2019

Extensions

Corrected by Ralf Stephan, Jun 19 2003
Entry revised by N. J. A. Sloane, May 30 2009
Offset changed to 0, Jun 11 2009

A147582 First differences of A147562.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 12, 36, 4, 12, 12, 36, 12, 36, 36, 108, 4, 12, 12, 36, 12, 36, 36, 108, 12, 36, 36, 108, 36, 108, 108, 324, 4, 12, 12, 36, 12, 36, 36, 108, 12, 36, 36, 108, 36, 108, 108, 324, 12, 36, 36, 108, 36, 108, 108, 324, 36, 108, 108, 324, 108, 324, 324, 972, 4
Offset: 1

Views

Author

N. J. A. Sloane, Apr 29 2009

Keywords

Comments

Bisection of A323651. - Omar E. Pol, Mar 04 2019

Examples

			From _Omar E. Pol_, Jun 14 2009: (Start)
When written as a triangle:
.1;
.4;
.4,12;
.4,12,12,36;
.4,12,12,36,12,36,36,108;
.4,12,12,36,12,36,36,108,12,36,36,108,36,108,108,324;
.4,12,12,36,12,36,36,108,12,36,36,108,36,108,108,324,12,36,36,108,36,108,...
The rows converge to A161411. (End)

References

  • D. Singmaster, On the cellular automaton of Ulam and Warburton, M500 Magazine of the Open University, #195 (December 2003), pp. 2-7.
  • S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962.

Links

Crossrefs

Cf. A147562, A147610 (the sequence divided by 4), A048881, A000120.
Cf. A000079, A161411, A151779, A139250.
Cf. A048883, A139251, A160121, A162349. [Omar E. Pol, Nov 02 2009]
Cf. A323651.

Programs

  • Maple
    A000120 := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end: wt := A000120; A147582 := n-> if n <= 1 then n else 4*3^(wt(n-1)-1); fi; [seq(A147582(n),n=0..1000)]; # N. J. A. Sloane, Apr 07 2010
  • Mathematica
    s = Plus @@ Flatten@ # & /@ CellularAutomaton[{686, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 200]; f[n_] = If[n == 0, 1, s[[n + 1]] - s[[n]]]; Array[f, 120, 0] (* Michael De Vlieger, Apr 09 2015, after Nadia Heninger and N. J. A. Sloane at A147562 *)

Formula

a(1) = 1; for n > 1, a(n) = 4*3^(wt(n-1)-1) where wt() = A000120(). - R. J. Mathar, Apr 30 2009
This formula is (essentially) given by Singmaster. - N. J. A. Sloane, Aug 06 2009
G.f.: x + 4*x*(Product_{k >= 0} (1 + 3*x^(2^k)) - 1)/3. - N. J. A. Sloane, Jun 10 2009

Extensions

Extended by R. J. Mathar, Apr 30 2009

A153006 Toothpick sequence starting at the outside corner of an infinite square from which protrudes a half toothpick.

Original entry on oeis.org

0, 1, 3, 6, 9, 13, 20, 28, 33, 37, 44, 53, 63, 78, 100, 120, 129, 133, 140, 149, 159, 174, 196, 217, 231, 246, 269, 297, 332, 384, 448, 496, 513, 517, 524, 533, 543, 558, 580, 601, 615, 630, 653, 681, 716, 768, 832, 881, 903, 918, 941
Offset: 0

Views

Author

Omar E. Pol, Dec 17 2008, Dec 19 2008, Apr 28 2009

Keywords

Comments

a(n) is the total number of integer toothpicks after n steps.
It appears that this sequence is related to triangular numbers, Mersenne primes and even perfect numbers. Conjecture: a(A000668(n))=A000217(A000668(n)). Conjecture: a(A000668(n))=A000396(n), assuming there are no odd perfect numbers.
The main entry for this sequence is A139250. See also A152980 (the first differences) and A147646.
The Mersenne prime conjectures are true, but aren't really about Mersenne primes. a(2^i-1) = 2^i (2^i-1)/2 for all i (whether or not i or 2^i-1 is prime). This follows from the formulas for A139250(2^i-1) and A139250(2^i). - David Applegate, May 11 2009
Then we can write a(A000225(k)) = A006516(k), for k > 0. - Omar E. Pol, May 23 2009
Equals A151550 convolved with [1, 2, 2, 2, ...]. (This is equivalent to the observation that the g.f. is x((1+x)/(1-x)) * Product_{n >= 1} (1 + x^(2^n-1) + 2*x^(2^n)).) Equivalently, equals A151555 convolved with A151575. - Gary W. Adamson, May 25 2009
It appears that a(n) is also 1/4 of the total path length of a toothpick structure as A139250 after n-th stage which is constructed following a special rule: toothpicks of the new generation have length 4 when are placed on the square grid (every toothpick has four components of length 1), but after every stage, one (or two) of the four components of every toothpick of the new generation is removed, if such component contains a endpoint of the toothpick and if such endpoint is touching the midpoint or the endpoint of another toothpick. The truncated endpoints of the toothpicks remain exposed forever. Note that there are three sizes of toothpicks in the structure: toothpicks of length 4, 3 and 2. a(n) is also 1/4 of the number of grid points that are covered after n-th stage, except the central point of the structure. A159795 gives the total path length and also the total number of components in the structure after n-th stage. - Omar E. Pol, Oct 24 2011

Links

Crossrefs

Cf. A000079, A000217, A000396, A000668, A139250, A139251, A139560, A152978, A152979, A152980, A153000, A153001, A153002.
Cf. A000225, A006516.
Cf. A153007, A078008, A151555.

Programs

  • Maple
    G:=x*((1 + x)/(1 - x)) * mul( (1 + x^(2^n-1) + 2*x^(2^n)), n=1..20); # N. J. A. Sloane, May 20 2009

Formula

G.f.: x*((1 + x)/(1 - x)) * Product_{n >= 1} (1 + x^(2^n-1) + 2*x^(2^n)). - N. J. A. Sloane, May 20 2009

Extensions

Edited by N. J. A. Sloane, Dec 19 2008

A152980 First differences of toothpick corner sequence A153006.

Original entry on oeis.org

1, 2, 3, 3, 4, 7, 8, 5, 4, 7, 9, 10, 15, 22, 20, 9, 4, 7, 9, 10, 15, 22, 21, 14, 15, 23, 28, 35, 52, 64, 48, 17, 4, 7, 9, 10, 15, 22, 21, 14, 15, 23, 28, 35, 52, 64, 49, 22, 15, 23, 28, 35, 52, 65, 56, 43, 53, 74, 91, 122, 168, 176, 112, 33, 4, 7, 9, 10, 15, 22, 21, 14, 15, 23, 28, 35, 52
Offset: 0

Views

Author

Omar E. Pol, Dec 16 2008, Dec 19 2008, Jan 02 2009

Keywords

Comments

Rows of A152978 when written as a triangle converge to this sequence. - Omar E. Pol, Jul 19 2009

Examples

			Triangle begins:
.1;
.2;
.3,3;
.4,7,8,5;
.4,7,9,10,15,22,20,9;
.4,7,9,10,15,22,21,14,15,23,28,35,52,64,48,17;
....
Rows converge to A153001. - _N. J. A. Sloane_, Jun 07 2009

Links

Crossrefs

Equals A151688 divided by 2. - N. J. A. Sloane, Jun 03 2009
For generating functions of the form Product_{k>=c} (1+a*x^(2^k-1)+b*x^2^k) for the following values of (a,b,c) see: (1,1,0) A160573, (1,1,1) A151552, (1,1,2) A151692, (2,1,0) A151685, (2,1,1) A151691, (1,2,0) A151688 and A152980, (1,2,1) A151550, (2,2,0) A151693, (2,2,1) A151694.
Equals A147646/4. - N. J. A. Sloane, May 01 2009
Cf. A139250, A139251, A152968, A152978, A153006, A153001, A159785, A153004.

Programs

  • Maple
    Maple code from N. J. A. Sloane, May 18 2009. First define old version with offset 1:
S:=proc(n) option remember; local i,j;
if n <= 0 then RETURN(0); fi;
if n <= 2 then RETURN(2^(n-1)); fi;
i:=floor(log(n)/log(2));
j:=n-2^i;
if j=0 then RETURN(n/2+1); fi;
if j<2^i-1 then RETURN(2*S(j)+S(j+1)); fi;
if j=2^i-1 then RETURN(2*S(j)+S(j+1)-1); fi;
-1;
end;
# Now change the offset:
T:=n->S(n+1);
G := (1 + x) * mul(1 + x^(2^k-1) + 2*x^(2^k),k=1..20);
  • Mathematica
    nmax = 78;
G = x*((1 + x)/(1 - x)) * Product[ (1 + x^(2^n - 1) + 2*x^(2^n)), {n, 1, Log2[nmax] // Ceiling}];
CoefficientList[G + O[x]^nmax, x] // Differences (* Jean-François Alcover, Jul 21 2022 *)

Formula

G.f.: (1 + x) * Prod_{ n >= 1} (1 + x^(2^n-1) + 2*x^(2^n)). - N. J. A. Sloane, May 20 2009, corrected May 21 2009
For formula see A147646 (or, better, see the Maple code below).

Extensions

More terms (based on A147646) from N. J. A. Sloane, May 01 2009
Offset changed by N. J. A. Sloane, May 18 2009

A160121 First differences of A160120.

Original entry on oeis.org

1, 3, 3, 9, 3, 9, 9, 21, 9, 9, 9, 21, 15, 21, 27, 51, 27, 9, 9, 21, 15, 21, 27, 51, 33, 21, 27, 51, 51, 57, 69, 117, 81, 21, 9, 21, 15, 21, 27, 51, 33, 21, 27, 51, 51, 57, 69, 117, 87, 33, 27, 51, 51, 57, 75, 129, 117, 75, 69, 117, 135, 141, 171, 279, 231, 69, 9, 21, 15, 21, 27
Offset: 1

Views

Author

Omar E. Pol, May 02 2009

Keywords

Comments

Number of Y-toothpicks added at n-th stage to the Y-toothpick structure of A160120.
For a simpler version, see A151710. - Omar E. Pol, Dec 18 2012

Examples

			Contribution from _Omar E. Pol_, Jun 18 2009: (Start)
May be written as a triangle:
1,
3,
3,
9,
3,9,
9,21,9,9,
9,21,15,21,27,51,27,9,
9,21,15,21,27,51,33,21,27,51,51,57,69,117,81,21,
9,21,15,21,27,51,33,21,27,51,51,57,69,117,87,33,27,51,51,57,75,129,117,75,69,117,135,141,171,279,231,69;
Rows converge to A161326.
(End)
Contribution from _Omar E. Pol_, Dec 18 2012: (Start):
Also this sequence may be written as another triangle (according to the structure of triangle A151710):
1;
3;
3,  9;
3,  9,9,21;
9,  9,9,21,15,21,27,51;
27, 9,9,21,15,21,27,51,33,21,27,51,51,57,69,117;
81,21,9,21,15,21,27,51,33,21,27,51,51,57,69,117,87,33,27,51,51,57,75,129,117,75,69,117,135,141,171,279;
(End)

Links

Crossrefs

Cf. A139250, A139251, A151710, A160171, A160173, A160120, A160715, A161207, A161329, A161331, A161326, A161431, A147582.

Programs

  • Mathematica
    YTPFunc[lis_, step_] := With[{out = Extract[lis, {{1, 2}, {2, 1}, {-1, -1}}], in = lis[[2, 2]]}, Which[in == 1, 3, in == 0 && Count[out, 1] >= 2, 2, in == 0 && Count[out, 1] == 1, 1, True, in]]; A160121[n_] := Count[CellularAutomaton[{YTPFunc, {}, {1, 1}}, {{{1}}, 0}, {{{n}}}], 1, 2] (* JungHwan Min, Jan 28 2016 *)
A160121[n_] := Count[CellularAutomaton[{13390417258775213635414055181254541831894674613399006361662885886563211940509571858857491972104491013971547937418035084866785430974106432144737472376143620, 4, {{-1, 0}, {0, -1}, {0, 0}, {1, 1}}}, {{{1}}, 0}, {{{n}}}], 1, 2] (* JungHwan Min, Jan 28 2016 *)

Extensions

More terms from David Applegate, Jun 14 2009

A001196 Double-bitters: only even length runs in binary expansion.

Original entry on oeis.org

0, 3, 12, 15, 48, 51, 60, 63, 192, 195, 204, 207, 240, 243, 252, 255, 768, 771, 780, 783, 816, 819, 828, 831, 960, 963, 972, 975, 1008, 1011, 1020, 1023, 3072, 3075, 3084, 3087, 3120, 3123, 3132, 3135, 3264, 3267, 3276, 3279, 3312, 3315, 3324, 3327, 3840, 3843
Offset: 0

Views

Author

N. J. A. Sloane, based on an email from Bart la Bastide (bart(AT)xs4all.nl)

Keywords

Comments

Numbers whose set of base 4 digits is {0,3}. - Ray Chandler, Aug 03 2004
n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 4 for every i. - Ray Chandler, Aug 03 2004
The first 2^n terms of the sequence could be obtained using the Cantor-like process for the segment [0, 4^n-1]. E.g., for n=1 we have [0, {1, 2}, 3] such that numbers outside of braces are the first 2 terms of the sequence; for n=2 we have [0, {1, 2}, 3, {4, 5, 6, 7, 8, 9, 10, 11}, 12, {13, 14}, 15] such that the numbers outside of braces are the first 4 terms of the sequence, etc. - Vladimir Shevelev, Dec 17 2012
From Emeric Deutsch, Jan 26 2018: (Start)
Also, the indices of the compositions having only even parts. For the definition of the index of a composition, see A298644. For example, 195 is in the sequence since its binary form is 11000011 and the composition [2,4,2] has only even parts. 132 is not in the sequence since its binary form is 10000100 and the composition [1,4,1,2] also has odd parts.
The command c(n) from the Maple program yields the composition having index n. (End)
After the k-th step of generating the Koch snowflake curve, label the edges of the curve consecutively 0..3*4^k-1 starting from a vertex of the originating triangle. a(0), a(1), ... a(2^k-1) are the labels of the edges contained in one edge of the originating triangle. Add 4^k to each label to get the labels for the next edge of the triangle. Compare with A191108 in respect of the Sierpinski arrowhead curve. - Peter Munn, Aug 18 2019

Links

Crossrefs

3 times the Moser-de Bruijn sequence A000695.
Two digits in other bases: A005823, A097252-A097262.
Digit duplication in other bases: A338086, A338754.
Main diagonal of A054238.
Cf. A191108.

Programs

  • C
    int a_next(int a_n) { int t = a_n << 1; return a_n ^ t ^ (t + 3); } /* Falk Hüffner, Jan 24 2022 */
  • Haskell
    a001196 n = if n == 0 then 0 else 4 * a001196 n' + 3 * b
            where (n',b) = divMod n 2
-- Reinhard Zumkeller, Feb 21 2014
  • Maple
    Runs := proc (L) local j, r, i, k: j := 1: r[j] := L[1]: for i from 2 to nops(L) do if L[i] = L[i-1] then r[j] := r[j], L[i] else j := j+1: r[j] := L[i] end if end do: [seq([r[k]], k = 1 .. j)] end proc: RunLengths := proc (L) map(nops, Runs(L)) end proc: c := proc (n) ListTools:-Reverse(convert(n, base, 2)): RunLengths(%) end proc: A := {}: for n to 3350 do if type(product(1+c(n)[j], j = 1 .. nops(c(n))), odd) = true then A := `union`(A, {n}) else  end if end do: A; # most of the Maple  program is due to W. Edwin Clark. - Emeric Deutsch, Jan 26 2018
# second Maple program:
a:= proc(n) option remember;
     `if`(n<2, 3*n, 4*a(iquo(n, 2, 'r'))+3*r)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 24 2022
  • Mathematica
    fQ[n_] := Union@ Mod[Length@# & /@ Split@ IntegerDigits[n, 2], 2] == {0}; Select[ Range@ 10000, fQ] (* Or *)
fQ[n_] := Union@ Join[IntegerDigits[n, 4], {0, 3}] == {0, 3}; Select[ Range@ 10000, fQ] (* Robert G. Wilson v, Dec 24 2012 *)
  • PARI
    a(n) = 3*fromdigits(binary(n),4); \\ Kevin Ryde, Nov 07 2020
  • Python
    def inA001196(n):
    while n != 0:
        if n%4 == 1 or n%4 == 2:
            return 0
        n = n//4
    return 1
n, a = 0, 0
while n < 20:
    if inA001196(a):
        print(n,a)
        n = n+1
    a = a+1 # A.H.M. Smeets, Aug 19 2019
  • Python
    from itertools import groupby
def ok2lb(n):
  if n == 0: return True  # by convention
  return all(len(list(g))%2 == 0 for k, g in groupby(bin(n)[2:]))
print([i for i in range(3313) if ok2lb(i)]) # Michael S. Branicky, Jan 04 2021
  • Python
    def A001196(n): return 3*int(bin(n)[2:],4) # Chai Wah Wu, Aug 21 2023

Formula

a(2n) = 4*a(n), a(2n+1) = 4*a(n) + 3.
a(n) = 3 * A000695(n).
Sum_{n>=1} 1/a(n) = 0.628725478158702414849086504025451177643560169366348272891020450593453403709... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022

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

Views

Author

Paul Barry, Jan 05 2005

Keywords

Comments

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

Examples

			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)

Links

Crossrefs

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.

Programs

  • Haskell
    a102376 = (4 ^) . a000120  -- Reinhard Zumkeller, Feb 13 2015
  • Maple
    seq(4^convert(convert(n,base,2),`+`),n=0..100); # Robert Israel, Apr 30 2017
  • Mathematica
    Table[4^DigitCount[n, 2, 1], {n, 0, 100}] (* Indranil Ghosh, Apr 30 2017 *)
  • PARI
    {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
  • Python
    def a(n): return 4**bin(n)[2:].count("1") # Indranil Ghosh, Apr 30 2017
  • Python
    def A102376(n): return 1<<(n.bit_count()<<1) # Chai Wah Wu, Nov 15 2022

Formula

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
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