A077859 -id:A077859 - cp's OEIS frontend

A021823 Decimal expansion of 1/819.

0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1

Offset: 0

N. J. A. Sloane

Partial sums of A010892. - Paul Barry, Jun 06 2003
Expansion in any base b >= 3 of 1/((b-1)*(b^2-b+1)) = 1/(b^3-2b^2+2b-1). E.g., 1/14 in base 3, 1/39 in base 4, 1/84 in base 5, etc. - Franklin T. Adams-Watters, Nov 07 2006
a(n) is the second least significant digit in the ternary representation of 2^n (cf. A004642). - Alexandre Herrera, Oct 09 2023

			0.0012210012210012210...

Index entries for linear recurrences with constant coefficients, signature (2,-2,1).

Cf. A077859, A027444.

Cf. A004642, A153130 (2^n mod 9).

Join[{0,0},RealDigits[1/819,10,120][[1]]] (* or *) PadRight[{},120,{0,0,1,2,2,1}] (* or *) LinearRecurrence[{2,-2,1},{0,0,1},120] (* Harvey P. Dale, Aug 19 2012 *)

a(n)=1/819. \\ Charles R Greathouse IV, Sep 24 2015

a(n) = a(n-1)-a(n-2)+1 = 2-a(n-3) = a(n-6). - Henry Bottomley, Apr 12 2000
a(n) = Sum_{k=1..floor(n/2)} (-1)^(k+1)*binomial(n-k, k) = 1-((-1)^floor(n/3)+(-1)^(floor((n+1)/3)))/2. - Vladeta Jovovic, Feb 10 2003
G.f.: x^2/(1-2x+2x^2-x^3)=x^2/((1-x)(x^2-x+1)). - Paul Barry, Jun 06 2003
a(n+2) = sum{k=0..n, binomial(n-2k, n-k)}. - Paul Barry, Jan 15 2005
a(0)=0, a(1)=0, a(2)=1, a(n)=2*a(n-1)-2*a(n-2)+a(n-3). - Harvey P. Dale, Aug 19 2012

A279230 Expansion of 1/((1-x)^2(1-2x+2*x^2)).

Original entry on oeis.org

1, 4, 9, 14, 15, 8, -7, -22, -21, 12, 77, 142, 143, 16, -239, -494, -493, 20, 1045, 2070, 2071, 24, -4071, -8166, -8165, 28, 16413, 32798, 32799, 32, -65503, -131038, -131037, 36, 262181, 524326, 524327, 40, -1048535, -2097110, -2097109, 44, 4194349, 8388654, 8388655

Offset: 0

Philippe Deléham, Dec 08 2016

Partial sums of A077860.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-7,6,-2).

Cf. A001477, A045618, A077859, A077860, A279231.

Vec(1 / ((1 - x)^2*(1 - 2*x + 2*x^2)) + O(x^50)) \\ Colin Barker, Aug 04 2017

{a(n) = sum(k=0, n\2, (-1)^k*binomial(n+3, 2*k+3))} \\ Seiichi Manyama, Apr 07 2019

a(n) = 4*a(n-1) - 7*a(n-2) + 6*a(n-3) - 2*a(n-4) for n>3.
a(n) = 2*a(n-1) - 2*a(n-2) + n + 1, with a(-1) = a(-2) = 0.
a(n) = (3 - (1-i)^(1+n) - (1+i)^(1+n) + n) where i=sqrt(-1). - Colin Barker, Aug 04 2017
From Seiichi Manyama, Apr 07 2019: (Start)
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n+3,2*k+3).
a(n) = Sum_{i=0..n} Sum_{j=0..n-i} (-1)^j * binomial(i+1,j+1) * binomial(n-i+1,j+1). (End)

A279231 Expansion of g.f. 1/((1 - x)^2(1 - 3x + 3*x^2)).

Original entry on oeis.org

1, 5, 15, 34, 62, 90, 91, 11, -231, -716, -1444, -2172, -2171, 17, 6579, 19702, 39386, 59070, 59071, 23, -177123, -531416, -1062856, -1594296, -1594295, 29, 4782999, 14348938, 28697846, 43046754, 43046755, 35, -129140127, -387420452, -774840940, -1162261428

Offset: 0

Philippe Deléham, Dec 08 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,9,-3).

Cf. A001477, A045618, A077859.

CoefficientList[Series[1/((1-x)^2(1-3x+3x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{5,-10,9,-3},{1,5,15,34},40] (* Harvey P. Dale, Aug 03 2025 *)

Vec(1/((1-x)^2*(1-3*x+3*x^2)) + O(x^30)) \\ Colin Barker, Dec 08 2016

a(n) = 5*a(n-1) - 10*a(n-2) + 9*a(n-3) - 3*a(n-4) for n > 3.
a(n) = 3*a(n-1) - 3*a(n-2) + n + 1, with a(-1) = a(-2) = 0.
a(n) = 4 + n + 3^(1+n/2)*(sqrt(3)*sin(n*Pi/6) - cos(n*Pi/6)). - Stefano Spezia, Feb 11 2023
a(n) = Sum_{k=0..floor(n/3)} (-1)^k*binomial(n+4,3*k+4). - Taras Goy, Jan 03 2025
E.g.f.: exp(x)*(4 + x - 3*exp(x/2)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))). - Stefano Spezia, Jan 03 2025

Incorrect term corrected by Colin Barker, Dec 09 2016

A122434 Expansion of (1+x)^3/(1+x+x^2).

Original entry on oeis.org

1, 2, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0

Offset: 0

Paul Barry, Sep 04 2006

Row sums of number triangle A122433. Binomial transform is A077859.

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

CoefficientList[Series[(1+x)^3/(1+x+x^2),{x,0,70}],x] (* or *) PadRight[ {1,2,0},71,{-1,1,0}] (* Harvey P. Dale, Apr 25 2012 *)

a(n) = 2*sqrt(3)*cos(2*Pi*n/3+Pi/6)/3+C(1,n)+C(0,n).

a(n) = -A049347(n) if n>=2 . - R. J. Mathar, Feb 08 2008

A144083 Triangle read by rows: partial sums from the right of an A010892 subsequences decrescendo triangle.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 1, 0, 0, 1, 2, 2, 1, 2, 1, 0, 0, 1, 2, 2, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1

Offset: 0

Gary W. Adamson, Sep 10 2008

n-th row = (n+1) terms of an infinitely periodic cycle: (..., 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1), shifting to the right one place for the next row.

Construct an A010892 decrescendo triangle: (1; 1,1; 0,1,1; -1,0,1,1; ...) and take partial sums starting from the right.

			First few rows of the triangle:
  1;
  2, 1;
  2, 2, 1;
  1, 2, 2, 1;
  0, 1, 2, 2, 1;
  0, 0, 1, 2, 2, 1;
  1, 0, 0, 1, 2, 2, 1;
  2, 1, 0, 0, 1, 2, 2, 1;
  2, 2, 1, 0, 0, 1, 2, 2, 1;
  1, 2, 2, 1, 0, 0, 1, 2, 2, 1;
  0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1;
  ...
Row 3 = (1, 2, 2, 1) = partial sums of (-1, 0, 1, 1).

Cf. A010892, A077859 (row sums), A164965 (1st column).

A010892[n_]:={1, 1, 0, -1, -1,0}[[Mod[n, 6]+1]]; T[n_,k_]:=1+A010892[n-k-1]; Table[T[n,k], {n,0, 11},{k,0,n}]//Flatten (* Stefano Spezia, Feb 11 2023 *)

T(n, k) = 1 + A010892(n-k-1), with 0 <= k <= n. - Stefano Spezia, Feb 11 2023

A359626 a(n) is equal to the number of filled unit triangles in a regular triangle whose coloring scheme is given in the comments.

Original entry on oeis.org

1, 4, 9, 15, 21, 27, 34, 43, 54, 66, 78, 90, 103, 118, 135, 153, 171, 189, 208, 229, 252, 276, 300, 324, 349, 376, 405, 435, 465, 495, 526, 559, 594, 630, 666, 702, 739, 778, 819, 861, 903, 945, 988, 1033, 1080, 1128, 1176, 1224, 1273, 1324, 1377, 1431, 1485, 1539, 1594, 1651, 1710, 1770, 1830, 1890, 1951, 2014, 2079

Offset: 1

Nicolay Avilov, Apr 20 2023

A regular triangle with side n is divided by segments parallel to the sides of the triangle into n^2 unit triangles. In it, you can select triangular frames nested inside each other. Coloring them through one, starting from the outer one, we obtain a coloring of unit triangles corresponding to the given sequence. See link.

			a(7) = 7^2 - 4^2 + 1^2 = 34;
a(8) = 8^2 - 5^2 + 2^2 = 43;
a(9) = 9^2 - 6^2 + 3^2 = 54.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Nicolay Avilov, Triangle Coloring Scheme
Index entries for linear recurrences with constant coefficients, signature (4,-7,7,-4,1).

Cf. A000096, A077859 (first differences).

A359626list[nmax_]:=LinearRecurrence[{4,-7,7,-4,1},{1, 4, 9, 15, 21},nmax];A359626list[100] (* Paolo Xausa, Aug 05 2023 *)

Let r = n (mod 6), then we get
  a(n) = n*(n+3)/2 - 1 if r = 1 or r = 2;
         n*(n+3)/2     if r = 0 or r = 3;
         n*(n+3)/2 + 1 if r = 4 or r = 5.
From Stefano Spezia, Apr 20 2023: (Start)
O.g.f.: x/((1 - x)^3*(1 - x + x^2)).
E.g.f.: exp(x)*x*(4 + x)/2 - 2*exp(x/2)*sin(sqrt(3)*x/2)/sqrt(3). (End)
a(n) - a(n-1) = A077859(n-1). - R. J. Mathar, Apr 20 2023

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A021823 Decimal expansion of 1/819.

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A279230 Expansion of 1/((1-x)^2*(1-2*x+2*x^2)).

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A279231 Expansion of g.f. 1/((1 - x)^2*(1 - 3*x + 3*x^2)).

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A122434 Expansion of (1+x)^3/(1+x+x^2).

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A144083 Triangle read by rows: partial sums from the right of an A010892 subsequences decrescendo triangle.

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A359626 a(n) is equal to the number of filled unit triangles in a regular triangle whose coloring scheme is given in the comments.

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A279230 Expansion of 1/((1-x)^2(1-2x+2*x^2)).

A279231 Expansion of g.f. 1/((1 - x)^2(1 - 3x + 3*x^2)).