A022998 -id:A022998 - cp's OEIS frontend

A174007 a(2n+1)=2. a(2n)= 1-n.

2, 0, 2, -1, 2, -2, 2, -3, 2, -4, 2, -5, 2, -6, 2, -7, 2, -8, 2, -9, 2, -10, 2, -11, 2, -12, 2, -13, 2, -14, 2, -15, 2, -16, 2, -17, 2, -18, 2, -19, 2, -20, 2, -21, 2, -22, 2, -23, 2, -24, 2, -25, 2, -26, 2, -27, 2, -28, 2, -29, 2, -30, 2, -31, 2, -32, 2, -33, 2, -34, 2, -35, 2

Offset: 1

Paul Curtz, Mar 05 2010

A064680(n)+A022998(n-1) =c(n) = 2, 2, 10, 5, 18, 8, 26, 11,.. has differences c(2n)-c(2n-1) = -5*(n-1) = -A008587(n-1).

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A147657.

a(n) = n/4+3/2-(-1)^n*(n/4+1/2).
a(n)= +2*a(n-2) -a(n-4). G.f.: -x*(-2+2*x^2+x^3) / ( (x-1)^2*(1+x)^2 ).
a(n+1)-a(n) = (-1)^n*A008619(n+1).
a(n) = A064680(n)-A022998(n-1).

A261790 Regular triangle read by rows: T(n,k) is the least positive number m such that km and km*(m+1)/2 are both divisible by n, with 0<=k<=n and T(0,0)=1.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 3, 3, 1, 1, 8, 4, 8, 1, 1, 5, 5, 5, 5, 1, 1, 12, 3, 4, 3, 12, 1, 1, 7, 7, 7, 7, 7, 7, 1, 1, 16, 8, 16, 4, 16, 8, 16, 1, 1, 9, 9, 3, 9, 9, 3, 9, 9, 1, 1, 20, 5, 20, 5, 4, 5, 20, 5, 20, 1, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 1, 24, 12, 8, 3, 24, 4, 24, 3, 8, 12, 24, 1

Offset: 0

Michel Marcus, Sep 01 2015

T(360,k) is the number of steps for a Logo turtle to return to the same orientation and same heading when using the INSPIR program with starting angle and angular increment k.

			Triangle starts:
1;
1, 1;
1, 4, 1;
1, 3, 3, 1;
1, 8, 4, 8, 1;
1, 5, 5, 5, 5, 1;
1, 12, 3, 4, 3, 12, 1;
1, 7, 7, 7, 7, 7, 7, 1;
1, 16, 8, 16, 4, 16, 8, 16, 1;
...

Harold Abelson and Andrea diSessa, Turtle Geometry, Artificial Intelligence Series, MIT Press, July 1986, pp. 20 and 36.
Brian Hayes, La tortue vagabonde, in Récréations Informatiques, Pour La Science, Belin, Paris, 1987, pp. 24-28, in French, translation from Computer Recreations, February 1984, Scientific American Volume 250, Issue 2.

Cf. A011772, A022998 (2nd column).

{1}~Join~Table[m = 1; While[Nand[Mod[k m, n] == 0, Mod[k m (m + 1)/2, n] == 0], m++]; m, {n, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Sep 01 2015 *)

T(n, k) = {if (n==0, return (1)); m=1; while(((k*m*(m+1)/2) % n) || (k*m % n), m++); m;}
row(n) = vector(n+1, k, k--; T(n,k));
tabl(nn) = for(n=0, nn, print(row(n)));

A267942 Interleave (n-1)^2 + 2 and (n+1)^2 + 2.

Original entry on oeis.org

3, 3, 2, 6, 3, 11, 6, 18, 11, 27, 18, 38, 27, 51, 38, 66, 51, 83, 66, 102, 83, 123, 102, 146, 123, 171, 146, 198, 171, 227, 198, 258, 227, 291, 258, 326, 291, 363, 326, 402, 363, 443, 402, 486, 443, 531, 486, 578, 531, 627, 578, 678, 627, 731, 678, 786, 731

Offset: 0

Paul Curtz, Jan 22 2016

Trisections:
3,  6,  6,  27, 27, 66,  66, ... = 3*(1, 2, 2, 9, 9, 22, 22, ... ). See A056105.
3,  3,  18, 18, 51, 51, 102, ... = 3*(1, 1, 6, 6, 17, 17, ... ). See A056109.
2, 11,  11, 38, 38, 83,  83, ...   (== 2 (mod 9)).
The trisections also have the signature (1,2,-2,-1,1). The corresponding main sequence is 0, 0, 0, 0, 1, 1, 3, 3, ... = A161680(n) with each term duplicated.

			a(0) = (2+13)/5, a(1) = (13+2)/5, a(2) = (5+5)/5, a(3) = (29+1)/5, ... (using first formula).

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

Cf. A000290, A007395, A010701, A022998, A056105, A056109, A059100, A161680, A174474, A193356, A255844, A261327.

&cat [[(n-1)^2+2, (n+1)^2+2]: n in [0..50]]; // Vincenzo Librandi, Jan 23 2016

Flatten[Table[{n^2 - 2 n + 3, n^2 + 2 n + 3}, {n, 0, 30}]] (* Vincenzo Librandi, Jan 23 2016 *)
CoefficientList[Series[(3 - 7 x^2 + 4 x^3 + 2 x^4)/((1 - x)^3 (1 + x)^2), {x, 0, 56}], x] (* Michael De Vlieger, Jan 24 2016 *)

Vec((3-7*x^2+4*x^3+2*x^4)/((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 22 2016

a(n) = (A261327(n+2) + A261327(n-3))/5.
a(n+1) = a(n) + (-1)^n * A022998(n), a(0)=3.
a(n+3) = a(n) + 3*A193356(n), a(0)=a(1)=3, a(2)=2.
a(n) = 3 + A174474(n).
a(2n) + a(2n+1) = A255844(n).
From Colin Barker, Jan 22 2016: (Start)
a(n) = (2*n^2 - 6*(-1)^n*n - 2*n + 3*(-1)^n + 21)/8.
a(n) = (n^2 - 4*n + 12)/4 for n even.
a(n) = (n^2 + 2*n + 9)/4 for n odd.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4.
G.f.: (3 - 7*x^2 + 4*x^3 + 2*x^4) / ((1-x)^3*(1+x)^2).
(End)

More terms from Colin Barker, Jan 22 2016

cp's OEIS Frontend

A174007 a(2n+1)=2. a(2n)= 1-n.

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A261790 Regular triangle read by rows: T(n,k) is the least positive number m such that km and km*(m+1)/2 are both divisible by n, with 0<=k<=n and T(0,0)=1.

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A267942 Interleave (n-1)^2 + 2 and (n+1)^2 + 2.

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cp's OEIS Frontend

A174007 a(2n+1)=2. a(2n)= 1-n.

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Author

Keywords

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Crossrefs

Formula

A261790 Regular triangle read by rows: T(n,k) is the least positive number m such that k*m and k*m*(m+1)/2 are both divisible by n, with 0<=k<=n and T(0,0)=1.

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Author

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Comments

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References

Crossrefs

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Mathematica

PARI

A267942 Interleave (n-1)^2 + 2 and (n+1)^2 + 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A261790 Regular triangle read by rows: T(n,k) is the least positive number m such that km and km*(m+1)/2 are both divisible by n, with 0<=k<=n and T(0,0)=1.