A008611 -id:A008611 - cp's OEIS frontend

A060553 a(n) is the number of distinct (modulo geometric D3-operations) patterns which can be formed by an equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal's triangle modulo 2, where n is the number of cells in each edge of the arrangement. The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.

2, 2, 4, 6, 10, 16, 32, 52, 104, 192, 376, 720, 1440, 2800, 5600, 11072, 22112, 43968, 87936, 175296, 350592, 700160, 1400192, 2798336, 5596672, 11188992, 22377984, 44747776, 89495040, 178973696, 357947392, 715860992, 1431721984, 2863378432, 5726754816

Offset: 1

André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001

Harry J. Smith, Table of n, a(n) for n=1..500
A. Barbé, Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2, Discr. Appl. Math. 105(2000), 1-38.
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-4,-4,8).

Cf. A000079, A060547, A060546, A008611, A008619.

a(n) = { (2^(n-1) + 2^(floor(n/3) + (n%3)%2))/3 + 2^floor((n-1)/2) } \\ Harry J. Smith, Jul 07 2009

a(n) = (2^(n-1)+2^(floor(n/3) + (n mod 3)mod 2))/3 + 2^floor((n-1)/2).
a(n) = (A000079(n-1) + A060547(n))/3 + A060546(n)/2.
a(n) = (A000079(n-1) + 2^A008611(n-1))/3 + 2^(A008619(n-1) - 1), for n >= 1.
G.f.: -2*x*(4*x^5 + x^4 - x^3 - 2*x^2 - x + 1) / ((2*x-1)*(2*x^2-1)*(2*x^3-1)). - Colin Barker, Aug 29 2013

More terms from Colin Barker, Aug 29 2013

A106250 Expansion of (1-x+x^2+x^3)/(1-x-x^5+x^6).

Original entry on oeis.org

1, 0, 1, 2, 2, 3, 2, 3, 4, 4, 5, 4, 5, 6, 6, 7, 6, 7, 8, 8, 9, 8, 9, 10, 10, 11, 10, 11, 12, 12, 13, 12, 13, 14, 14, 15, 14, 15, 16, 16, 17, 16, 17, 18, 18, 19, 18, 19, 20, 20, 21, 20, 21, 22, 22, 23, 22, 23, 24, 24, 25, 24, 25, 26, 26, 27, 26, 27, 28, 28, 29, 28, 29, 30, 30, 31, 30

Offset: 0

Paul Barry, Apr 27 2005

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A008611.

CoefficientList[Series[(1-x+x^2+x^3)/(1-x-x^5+x^6),{x,0,120}],x] (* or *) LinearRecurrence[{1,0,0,0,1,-1},{1,0,1,2,2,3},121] (* Harvey P. Dale, Apr 26 2011 *)

G.f.: (1-x+x^2+x^3)/(1-x-x^5+x^6)=(1+x^2+2x^3+2x^4+x^5+2x^6+x^7)/(1-x^5)^2; a(n)=sum{k=0..n, -mu(k mod 5)}.

A173177 Numbers k such that 2k+3 is a prime of the form 3*A034936(m) + 4.

Original entry on oeis.org

2, 5, 8, 14, 17, 20, 29, 32, 35, 38, 47, 50, 53, 62, 68, 74, 77, 80, 89, 95, 98, 104, 110, 113, 119, 134, 137, 140, 152, 155, 164, 167, 173, 182, 185, 188, 197, 203, 209, 215, 218, 227, 230, 242, 248, 260, 269, 272, 284, 287, 299

Offset: 1

Eric Desbiaux, Feb 11 2010

With Bachet-Bézout theorem implicating Gauss Lemma and the Fundamental Theorem of Arithmetic,
for k > 1, k = 2*a + 3*b (a and b integers)
first type
A001477 = (2*A080425) + (3*A008611)
A000040 = (2*A039701) + (3*A157966)
A024893 Numbers k such that 3*k + 2 is prime
A034936 Numbers k such that 3*k + 4 is prime
OR
second type
A001477 = (2*A028242) + (3*A059841)
A000040 = (2*A067076) + (3*1)
A067076 Numbers k such that 2*k + 3 is prime
   k   a b OR a b
  --   - -    - -
   0   0 0    0 0
   1   - -    - -
   2   1 0    1 0
   3   0 1    0 1
   4   2 0    2 0
   5   1 1    1 1
   6   0 2    3 0
   7   2 1    2 1
   8   1 2    4 0
   9   0 3    3 1
  10   2 2    5 0
  11   1 3    4 1
  12   0 4    6 0
  13   2 3    5 1
  14   1 4    7 0
  15   0 5    6 1
  ...
  2* 2 + 3 OR 3* 1 + 4 =  7;
  2* 5 + 3 OR 3* 3 + 4 = 13;
  2* 8 + 3 OR 3* 5 + 4 = 19;
  2*14 + 3 OR 3* 9 + 4 = 31;
  2*17 + 3 OR 3*11 + 4 = 37;
  2*20 + 3 OR 3*13 + 4 = 43;
  2*29 + 3 OR 3*19 + 4 = 61;
  2*32 + 3 OR 3*21 + 4 = 67;
  2*35 + 3 OR 3*23 + 4 = 73.
A034936 Numbers k such that 3k+4 is prime.
A002476 Primes of the form 6k+1.
A024899 Nonnegative integers k such that 6k+1 is prime.
2, 5, 8, 14, 17, 20, ... = (3*(4*A024899 - A034936) - 5)/2.

Chris K. Caldwell, FAQ: Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?

Cf. A067076, A034936, A002476, A024899.

Select[Range[300],PrimeQ[2#+3]&&Divisible[2#-1,3]&] (* Harvey P. Dale, Aug 25 2016 *)

More terms from Harvey P. Dale, Aug 25 2016

A181633 Irregular triangle read by rows, where row n contains the pairs [q,q'] of all compositions n=q+q' with q,q'>0 and q == q' (mod 3).

Original entry on oeis.org

1, 1, 2, 2, 4, 1, 1, 4, 3, 3, 5, 2, 2, 5, 7, 1, 4, 4, 1, 7, 6, 3, 3, 6, 8, 2, 5, 5, 2, 8, 10, 1, 7, 4, 4, 7, 1, 10, 9, 3, 6, 6, 3, 9, 11, 2, 8, 5, 5, 8, 2, 11, 13, 1, 10, 4, 7, 7, 4, 10, 1, 13, 12, 3, 9, 6, 6, 9, 3, 12, 14, 2, 11, 5, 8, 8, 5, 11, 2, 14, 16, 1, 13, 4, 10, 7, 7, 10, 4, 13, 1, 16, 15, 3, 12, 6, 9, 9, 6, 12, 3, 15

Offset: 2

Florentin Smarandache (smarand(AT)unm.edu), Nov 03 2010

If (s,t) is a pair in the sequence, then (s+3u,t-3u) is also a pair in the sequence for any integer u such that both s+3u > 0 and t-3u > 0.

			The table starts with rows of even length at n=2 as:
(1,1)
(empty)
(2,2)
(4,1),(1,4)
(3,3)
(5,2),(2,5)

Cf. A181634 (where q and q' may be zero), A008611 (half of the row lengths).

A181633_row := proc(n)
    local L,a,b;
    L := [] ;
    for a from n-1 to 1 by -1 do
        b := n-a ;
        if modp(a,3) = modp(b,3) then
            L := [op(L),a,b] ;
        end if;
    end do:
    L ;
end proc: # R. J. Mathar, May 14 2016

Table[Select[Transpose@{#, n - #}, Mod[First@ #, 3] == Mod[Last@ #, 3] &] &@ Reverse@ Range[1, n - 1], {n, 18}] // Flatten (* Michael De Vlieger, May 15 2016 *)

i) If n is even, n=2k, then its pairs are: (k+3p,k-3p), where p is an integer such that both k+3p > 0 and k-3p > 0. ii) If n is odd, n=2k+1, then its pairs are (k+3p+2,k-3p-1), where p is an integer such that both k+3p+2 > 0 and k-3p-1 > 0.

Edited by R. J. Mathar, May 14 2016

A181634 Irregular triangle T(n,k) where row n contains all pairs [a,b] of the compositions n=a+b into nonnegative pairs where a == b (mod 3).

Original entry on oeis.org

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

Offset: 2

Florentin Smarandache (smarand(AT)unm.edu), Nov 03 2010

Row lengths are 2*A008611(n). Row sums are n*A008611(n).- R. J. Mathar, May 13 2016

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

A181634_row := proc(n)
    local L,a,b;
    L := [] ;
    for a from n to 0 by -1 do
        b := n-a ;
        if modp(a,3) = modp(b,3) then
            L := [op(L),a,b] ;
        end if;
    end do:
    L ;
end proc:
for n from 2 to 18 do
    print(op(A181634_row(n))) ;
end do: # R. J. Mathar, May 13 2016

If[First@ # == Last@ # &@ Take[#, 2], Join[Reverse@ Drop[#, 2], #], Join[Reverse@ #, #]] & /@ Function[n, Flatten@ Select[Transpose@ {n - #, #}, Mod[First@ #, 3] == Mod[Last@ #, 3] &] &@ Range[Ceiling[n/2], n]] /@ Range[2, 16] // Flatten (* Michael De Vlieger, May 13 2016 *)

Edited by R. J. Mathar, May 13 2016

A181635 Expansion of 36x^2(1+36x^2-6x) / ((36x^2+6x+1)(1-6x)^2).

Original entry on oeis.org

0, 36, 0, 1296, 15552, 46656, 559872, 5038848, 20155392, 181398528, 1451188224, 6530347008, 52242776064, 391820820480, 1880739938304, 14105549537280, 101559956668416, 507799783342080, 3656158440062976, 25593109080440832, 131621703842267136, 921351926895869952, 6317841784428822528, 33168669368251318272, 227442304239437611008

Offset: 1

Florentin Smarandache (smarand(AT)unm.edu), Nov 03 2010

The previous definition was: Let n=q+q' define any state with q quarks and q' antiquarks, q,q'>0 and q==q' (mod 3). Then a(n) = sum_{q,q'} 6^q*6^q' counts all states allowing q and q' to be any of the 6 quarks or 6 antiquarks. - Colin Barker, May 14 2016
In the following q and a represent any of the 6 quarks or antiquarks.
For n = 1, we have no combination.
For combinations of 2 we have: qa, [mesons and antimesons]; the number of all possible combinations will be 6^2 = 36.
For combinations of n= 7 we have: qqqqqaa, qqaaaaa; the number of all possible combinations will be 6^5*6^2 + 6^2*6^5 =559872.
For combinations of n=8 we have: qqqqaaaa, qqqqqqqa, qaaaaaaa; the number of all possible combinations will be 6^4*6^4 + 6^7*6^1 + 6^1*6^7 = 5038848
For combinations of n=9 we have: qqqqqqaaa, qqqaaaaaa; the number of all possible combinations will be 6^6*6^3 + 6^3*6^6 = 2*6^9 = 20155392.
For combinations of n=10 we have: qqqqqqqqaa, qqqqqaaaaa, qqaaaaaaaa; the number of all possible combinations will be 3*6^10 = 181398528.
If n is even, n=2k, then its pairs are: (k+3p,k-3p), where p is an integer such that both k+3p > 0 and k-3p > 0.
If n is odd, n=2k+1, then its pairs are(k+3p+2,k-3p-1), where p is an integer such that both k+3p+2 > 0 and k-3p-1 > 0.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,0,216,-1296).

Cf. A181633, A181685.

A181635 := proc(n)
    res := 0 ;
    for q from 1 to n-1 do
        a := n-q ;
        if modp(a,3) = modp(q,3) then
            res := res+6^n;
        end if;
    end do:
   res;
end proc:
seq(A181635(n),n=1..40) ; # R. J. Mathar, May 13 2016

LinearRecurrence[{6,0,216,-1296},{0,36,0,1296},40] (* Harvey P. Dale, Jul 18 2024 *)

a(n) = round((-2^n*3^(1+n)+(-3-I*sqrt(3))*(-3-3*I*sqrt(3))^n-3*(-3+3*I*sqrt(3))^n+I*sqrt(3)*(-3+3*I*sqrt(3))^n+2^n*3^(1+n)*n)/9) \\ Colin Barker, May 14 2016

a(n) = Sum_{q>0, q'>0, q+q'=n, q==q' (mod 3)} 6^(q+q').
G.f.: 36*x^2*(1+36*x^2-6*x) / ( (36*x^2+6*x+1)*(1-6*x)^2 ). - Joerg Arndt, Mar 16 2013
From Colin Barker, May 14 2016: (Start)
a(n) = (-2^n*3^(1+n)+(-3-i*sqrt(3))*(-3-3*i*sqrt(3))^n-3*(-3+3*i*sqrt(3))^n+i*sqrt(3)*(-3+3*i*sqrt(3))^n+2^n*3^(1+n)*n)/9 where i is the imaginary unit. - Colin Barker, May 14 2016
a(n) = 6*a(n-1)+216*a(n-3)-1296*a(n-4) for n>4.
(End)
E.g.f.: 1 + ((18*x - 3)*exp(9*x) - 4*sqrt(3)*cos(Pi/6-3*sqrt(3)*x))*exp(-3*x)/9. - Ilya Gutkovskiy, May 14 2016
a(n) = 6^n*A008611(n-2). - R. J. Mathar, May 14 2016

Edited by R. J. Mathar, May 13 2016

Name changed by Colin Barker, May 14 2016

A287794 Nine steps forward, eight steps back.

Original entry on oeis.org

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

Offset: 0

Wesley Ivan Hurt, May 31 2017

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A008611 (one step back, two steps forward).
Cf. A058207 (three steps forward, two steps back).
Cf. A260644 (four steps forward, three steps back).
Cf. A271800 (five steps forward, four steps back).
Cf. A271859 (six steps forward, five steps back).
Cf. A287655 (seven steps forward, six steps back).
Cf. A287793 (eight steps forward, seven steps back).

a:=n->add((-1)^floor((2*i-2)/17), i=1..n): seq(a(n), n=0..200);

Table[Sum[(-1)^Floor[(2 i - 2)/17], {i, n}], {n, 0, 100}]
LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1},{0,1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1},100] (* Harvey P. Dale, Aug 25 2024 *)

a(n) = Sum_{i=1..n} (-1)^floor((2*i-2)/17).

a(n) = a(n-1) + a(n-17) - a(n-18) for n > 17.

A287796 Ten steps forward, nine steps back.

Original entry on oeis.org

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

Offset: 0

Wesley Ivan Hurt, May 31 2017

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A008611 (one step back, two steps forward).
Cf. A058207 (three steps forward, two steps back).
Cf. A260644 (four steps forward, three steps back).
Cf. A271800 (five steps forward, four steps back).
Cf. A271859 (six steps forward, five steps back).
Cf. A287655 (seven steps forward, six steps back).
Cf. A287793 (eight steps forward, seven steps back).
Cf. A287794 (nine steps forward, eight steps back).

A287796:=n->add((-1)^floor((2*i-2)/19), i=1..n): seq(A287796(n), n=0..200);

Table[Sum[(-1)^Floor[(2 i - 2)/19], {i, n}], {n, 0, 100}]

a(n) = Sum_{i=1..n} (-1)^floor((2*i-2)/19).

a(n) = a(n-1) + a(n-19) - a(n-20) for n > 19.

A337879 a(n) is the length of the n-th line segment to draw the squares of the Fibonacci spiral without lifting the pencil, including superpositions.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 3, 5, 3, 5, 8, 5, 8, 13, 8, 13, 21, 13, 21, 34, 21, 34, 55, 34, 55, 89, 55, 89, 144, 89, 144, 233, 144, 233, 377, 233, 377, 610, 377, 610, 987, 610, 987, 1597, 987, 1597, 2584, 1597, 2584, 4181, 2584, 4181, 6765, 4181, 6765, 10946

Offset: 1

Victor Kelly, Sep 22 2020

To draw the Fibonacci squares without lifting your pencil, go a(n) units and turn 90 degrees (same direction every time), and so on (see the animation).

As an array read by rows, with two columns, in which column 1 lists three copies of every nonzero term of A001906 in nondecreasing order, and the column 2 lists 1 together with three copies of every term > 1 of A001519 in nondecreasing order. - Omar E. Pol, Sep 29 2020

Victor Kelly, Animated GIF
Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,1).

Cf. A000045, A001519, A001906, A008611.

a(n) = Fibonacci(A008611(n+2)). - David A. Corneth, Sep 28 2020
G.f.: -x*(x^5+x^3+x^2+x+1)/(x^6+x^3-1). - Alois P. Heinz, Sep 29 2020
a(n) = a(n-3) + a(n-6) for n > 6. - Jinyuan Wang, Sep 30 2020

More terms from Alois P. Heinz, Sep 29 2020

A093041 Expansion of (1-4x+6x^2-3x^3)/(1-5x+9x^2-8x^3+4x^4).

Original entry on oeis.org

1, 1, 2, 6, 16, 38, 86, 192, 426, 938, 2048, 4438, 9558, 20480, 43690, 92842, 196608, 415062, 873814, 1835008, 3844778, 8039082, 16777216, 34952534, 72701270, 150994944, 313174698, 648719018, 1342177280, 2773833046, 5726623062

Offset: 0

Paul Barry, Mar 18 2004

Binomial transform of A008611.

Index entries for linear recurrences with constant coefficients, signature (5,-9,8,-4)

CoefficientList[Series[(1-4x+6x^2-3x^3)/(1-5x+9x^2-8x^3+4x^4),{x,0,30}],x] (* or *) LinearRecurrence[ {5,-9,8,-4},{1,1,2,6},40] (* Harvey P. Dale, Feb 27 2024 *)

a(n)=2(cos(pi*n/3)-sqrt(3)sin(pi*n/3)/3)/3+2^n/3+n*2^(n-1)/3; a(n)=2*A010892(n+1)/3+A000079(n)/3+A001787(n)/3.

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A106250 Expansion of (1-x+x^2+x^3)/(1-x-x^5+x^6).

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A173177 Numbers k such that 2k+3 is a prime of the form 3*A034936(m) + 4.

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A181633 Irregular triangle read by rows, where row n contains the pairs [q,q'] of all compositions n=q+q' with q,q'>0 and q == q' (mod 3).

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A181634 Irregular triangle T(n,k) where row n contains all pairs [a,b] of the compositions n=a+b into nonnegative pairs where a == b (mod 3).

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A181635 Expansion of 36*x^2*(1+36*x^2-6*x) / ((36*x^2+6*x+1)*(1-6*x)^2).

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A287794 Nine steps forward, eight steps back.

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A287796 Ten steps forward, nine steps back.

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A181635 Expansion of 36x^2(1+36x^2-6x) / ((36x^2+6x+1)(1-6x)^2).