A177702 -id:A177702 - cp's OEIS frontend

A004773 Numbers congruent to {0, 1, 2} mod 4: a(n) = floor(4*n/3).

0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 21, 22, 24, 25, 26, 28, 29, 30, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 46, 48, 49, 50, 52, 53, 54, 56, 57, 58, 60, 61, 62, 64, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 89, 90

Offset: 0

N. J. A. Sloane

The sequence b(n) = floor((4/3)*(n+2)) appears as an upper bound in Fijavz and Wood.
Binary expansion does not end in 11.
From Guenther Schrack, May 04 2023: (Start)
The sequence is the interleaving of the sequences A008586, A016813, A016825, in that order.
Let S(n) = a(n) + a(n+1) + a(n+2). Then floor(S(n)/3) = A042968(n+1), round(S(n)/3) = a(n+1), ceiling(S(n)/3) = A042965(n+2). (End)

Daniel Starodubtsev, Table of n, a(n) for n = 0..10000
Gasper Fijavz and David R. Wood, Graph Minors and Minimum Degree, arXiv:0812.1064 [math.CO], 2008.
Niall Graham and Frank Harary, Edge Sums of Hypercubes, Bull. Irish Math. Soc., Vol. 21 (1988), pp. 8-12.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A177702 (first differences), A000969 (partial sums).
Cf. A032766, this sequence, A001068, A047226, A047368, A004777.
Cf. similar sequences with formula n+i*floor(n/3) listed in A281899.
Cf. A002264, A004396, A004523, A004767, A004772, A008586, A010872, A016813, A016825, A042965, A042968.

[n: n in [0..100] | n mod 4 in [0..2]]; // Vincenzo Librandi, Dec 23 2010

seq(floor(n/3)+n,n=0..68); # Gary Detlefs, Mar 20 2010

f[n_] := Floor[4 n/3]; Array[f, 69, 0] (* Robert G. Wilson v, Dec 24 2010 *)
fQ[n_] := Mod[n, 4] != 3; Select[ Range[0, 90], fQ] (* Robert G. Wilson v, Dec 24 2010 *)
a[0] = 0; a[n_] := a[n] = a[n - 1] + 2 - If[ Mod[a[n - 1], 4] < 2, 1, 0]; Array[a, 69, 0] (* Robert G. Wilson v, Dec 24 2010 *)
CoefficientList[ Series[x (1 + x + 2 x^2)/((1 - x) (1 - x^3)), {x, 0, 68}], x] (* Robert G. Wilson v, Dec 24 2010 *)

a(n)=4*n\3 \\ Charles R Greathouse IV, Sep 27 2012

G.f.: x*(1+x+2*x^2)/((1-x)*(1-x^3)).
a(0) = 0, a(n+1) = a(n) + a(n) mod 4 + 0^(a(n) mod 4). - Reinhard Zumkeller, Mar 23 2003
a(n) = A004396(n) + A004523(n); complement of A004767. - Reinhard Zumkeller, Aug 29 2005
a(n) = floor(n/3) + n. - Gary Detlefs, Mar 20 2010
a(n) = (12*n-3+3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9. - Wesley Ivan Hurt, Sep 30 2017
E.g.f.: (3*exp(x)*(4*x - 1) + exp(-x/2)*(3*cos((sqrt(3)*x)/2) + sqrt(3)*sin((sqrt(3)*x)/2)))/9. - Stefano Spezia, Jun 09 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)-1)*Pi/8 + sqrt(2)*log(sqrt(2)+2)/4 + (2-sqrt(2))*log(2)/8. - Amiram Eldar, Dec 05 2021
From Guenther Schrack, May 04 2023: (Start)
a(n) = (12*n - 3 +  w^(2*n)*(w + 2) - w^n*(w - 1))/9 where w = (-1 + sqrt(-3))/2.
a(n) = 2*floor(n/3) + floor((n+1)/3) + floor((n+2)/3).
a(n) = (4*n - n mod 3)/3.
a(n) = a(n-3) + 4.
a(n) = a(n-1) + a(n-3) - a(n-4).
a(n) = 4*A002264(n) + A010872(n).
a(n) = A042968(n+1) - 1.
(End)

A300322 Number T(n,k) of Dyck paths of semilength n such that 2k is the difference between the area under the right half of the path and the area under the left half of the path; triangle T(n,k), n>=0, -floor(n(n-1)/6) <= k <= floor(n*(n-1)/6), read by rows.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 1, 3, 6, 3, 1, 2, 5, 8, 12, 8, 5, 2, 1, 4, 9, 16, 22, 28, 22, 16, 9, 4, 1, 1, 4, 11, 21, 34, 49, 60, 69, 60, 49, 34, 21, 11, 4, 1, 2, 7, 15, 31, 53, 82, 114, 147, 171, 186, 171, 147, 114, 82, 53, 31, 15, 7, 2, 1, 5, 13, 30, 56, 95, 150, 216, 293, 371, 445, 495, 522, 495, 445, 371, 293, 216, 150, 95, 56, 30, 13, 5, 1

Offset: 0

Alois P. Heinz, Mar 02 2018

			               /\
T(3,-1) = 1:  /  \/\
.
                /\
               /  \     /\/\
T(3,0) = 3:   /    \   /    \   /\/\/\
.
                 /\
T(3,1) = 1:   /\/  \
.
Triangle T(n,k) begins:
:                             1                            ;
:                             1                            ;
:                             2                            ;
:                         1,  3,  1                        ;
:                     1,  3,  6,  3,  1                    ;
:                 2,  5,  8, 12,  8,  5,  2                ;
:         1,  4,  9, 16, 22, 28, 22, 16,  9,  4,  1        ;
:  1, 4, 11, 21, 34, 49, 60, 69, 60, 49, 34, 21, 11, 4, 1  ;

Alois P. Heinz, Rows n = 0..60, flattened
Wikipedia, Counting lattice paths

Row sums give A000108.
Column k=0 gives A300323.
Cf. A130518, A129182, A177702, A239927, A298645, A300953.

b:= proc(x, y, v) option remember; expand(
     `if`(min(y, v, x-max(y, v))<0, 0, `if`(x=0, 1, (l-> add(add(
      b(x-1, y+i, v+j)*z^((y-v)/2+(i-j)/4), i=l), j=l))([-1, 1]))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=ldegree(p)..degree(p)))(
             add(b(n, (n-2*j)$2), j=0..n/2)):
seq(T(n), n=0..12);

b[x_, y_, v_] := b[x, y, v] = Expand[If[Min[y, v, x - Max[y, v]] < 0, 0, If[x == 0, 1, Function[l, Sum[Sum[b[x - 1, y + i, v + j] z^((y - v)/2 + (i - j)/4), {i, l}], {j, l}]][{-1, 1}]]]];
T[n_] := Function[p, Table[Coefficient[p, z, i], {i, Range[Exponent[p, z, Reverse @@ # &], Exponent[p, z]]}]][Sum[b[n, n-2j, n-2j], {j, 0, n/2}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 31 2018, from Maple *)

T(n,k) = T(n,-k).

T(n,A130518(n)) = A177702(n).

A245477 Period 6: repeat [1, 1, 1, 1, 1, 2].

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 0

Hailey R. Olafson, Jul 23 2014

First differences of A047368. The first differences of this sequence are in A131533. - Wesley Ivan Hurt, Jul 24 2014

Binomial Transform of a(n) gives: 1, 2, 4, 8, 16, 33, 70, 149, 312, 638, 1276, 2511, ... - Wesley Ivan Hurt, Aug 13 2014

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

Cf. A024643, A047368, A097325, A130782, A131534, A172051, A177702, A177704, A177706.

[Floor((n+1)*7/6) - Floor((n)*7/6) : n in [0..100]]; // Wesley Ivan Hurt, Aug 06 2014

A:= n -> piecewise(n mod 6 = 5, 2, 1);
seq(A(n), n=0..100); # Robert Israel, Jul 23 2014

Table[2 - Sign[Mod[n + 1, 6]], {n, 0, 100}] (* Wesley Ivan Hurt, Jul 24 2014 *)
PadRight[{},120,{1,1,1,1,1,2}] (* Harvey P. Dale, Jun 02 2016 *)

a(n) = 7*(n+1)\6 - 7*n\6; \\ Michel Marcus, Jul 23 2014

[floor((n+1)*7/6) - floor((n)*7/6) for n in [0..200]]

a(n) = floor((n+1)*7/6) - floor((n)*7/6).
G.f.: 1/(1-x) + x^5/(1-x^6). - Robert Israel, Jul 23 2014
From Wesley Ivan Hurt, Jul 24 2014, Aug 06-29 2014: (Start)
  a(n) = 2 - sign((n+1) mod 6).
  a(n) = 3 - 2^sign((n+1) mod 6).
  a(n) = A172051(n) + 1.
  a(2n) = 1, a(2n+1) = A177702(n).
  Sum_{i=0..n-2} a(i) = A047368(n), n>0.
  a(n) = 1 + mod(n, 1 + mod(n-1, 3)).
  a(n) = 1 + binomial(mod(5n + 10, 6), 5). (End)
From Wesley Ivan Hurt, Jun 23 2016: (Start)
a(n) = a(n-6) for n>5.
a(n) = (7 - cos(n*Pi) + cos(n*Pi/3) - cos(2*n*Pi/3) - sqrt(3)*sin(n*Pi/3) - sqrt(3)*sin(2*n*Pi/3))/6. (End)

A177703 Decimal expansion of (2+sqrt(10))/3.

Original entry on oeis.org

1, 7, 2, 0, 7, 5, 9, 2, 2, 0, 0, 5, 6, 1, 2, 6, 4, 4, 3, 9, 9, 9, 6, 3, 1, 1, 8, 1, 4, 7, 7, 5, 7, 2, 8, 4, 4, 5, 7, 3, 1, 8, 5, 0, 4, 6, 4, 4, 1, 7, 3, 8, 9, 4, 2, 2, 8, 5, 8, 3, 4, 9, 5, 0, 9, 3, 0, 8, 6, 4, 8, 1, 2, 8, 7, 9, 7, 4, 6, 0, 7, 3, 7, 8, 1, 4, 1, 6, 0, 3, 6, 1, 2, 6, 4, 3, 3, 4, 3, 1, 7, 2, 9, 1, 1

Offset: 1

Klaus Brockhaus, May 11 2010

Continued fraction expansion of (2+sqrt(10))/3 is A177702.

			(2+sqrt(10))/3 = 1.72075922005612644399...

Cf. A010467 (decimal expansion of sqrt(10)), A177702 (repeat 1, 1, 2).

RealDigits[(2+Sqrt[10])/3,10,120][[1]] (* Harvey P. Dale, Jun 21 2011 *)

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A004773 Numbers congruent to {0, 1, 2} mod 4: a(n) = floor(4*n/3).

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A300322 Number T(n,k) of Dyck paths of semilength n such that 2*k is the difference between the area under the right half of the path and the area under the left half of the path; triangle T(n,k), n>=0, -floor(n*(n-1)/6) <= k <= floor(n*(n-1)/6), read by rows.

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A245477 Period 6: repeat [1, 1, 1, 1, 1, 2].

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A177703 Decimal expansion of (2+sqrt(10))/3.

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A300322 Number T(n,k) of Dyck paths of semilength n such that 2k is the difference between the area under the right half of the path and the area under the left half of the path; triangle T(n,k), n>=0, -floor(n(n-1)/6) <= k <= floor(n*(n-1)/6), read by rows.