A066345 -id:A066345 - cp's OEIS frontend

A008353 2^n*(2^(n+1) - n - 1).

1, 4, 20, 96, 432, 1856, 7744, 31744, 128768, 519168, 2085888, 8364032, 33501184, 134103040, 536625152, 2146959360, 8588820480, 34357379072, 137433972736, 549745328128, 2199001235456, 8796046884864, 35184275619840, 140737287028736, 562949533990912

Offset: 0

N. J. A. Sloane and J. H. Conway

Vincenzo Librandi, Table of n, a(n) for n = 0..172
Index entries for linear recurrences with constant coefficients, signature (8,-20,16).

a(n) = A066345(2*n+1).

a(n) = 1 for n=0, 4 * A008464(n-2) else.

[2^n*(2^(n+1) - n - 1): n in [0..50]]; // Vincenzo Librandi, Apr 25 2011

Vec(-(8*x^2-4*x+1)/((2*x-1)^2*(4*x-1)) + O(x^100)) \\ Colin Barker, Feb 26 2015

a(n) = sum{k=0..n+1, C(2n+2, 2k)-2k(n-k+1)*C(n+1, k)/n}, n>0. - Paul Barry, Feb 24 2005
a(n) = 8*a(n-1)-20*a(n-2)+16*a(n-3). - Colin Barker, Feb 26 2015
G.f.: -(8*x^2-4*x+1) / ((2*x-1)^2*(4*x-1)). - Colin Barker, Feb 26 2015

A066346 Number of winning binary "same game" templates with ternary digits totaling n.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 4, 9, 13, 28, 46, 84, 146, 252, 433, 736, 1242, 2087, 3482, 5791, 9587, 15823, 26038, 42743, 70016, 114485, 186903, 304728, 496260, 807395, 1312504, 2132102, 3461407, 5616609, 9109732

Offset: 0

Frank Ellermann, Dec 23 2001

Equivalently, templates whose minimum matching string has length n.

			a(1)..a(5) correspond to the winning templates -;2;-; 121,22; 122,221.
a(6) = 4 winning templates 11211,1212,2121 and 222 have a total of 6.

Sean A. Irvine, Java program (github)

Cf. A066345 (definition), A007931 (templates). A035615 (binary same game).

a(17)-a(35) from Sean A. Irvine, Oct 09 2023

A066709 Triangle T(r,c) of winning binary "same game" templates.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 0, 2, 4, 1, 1, 5, 8, 5, 1, 0, 3, 14, 15, 6, 1, 1, 9, 25, 32, 21, 7, 1, 0, 4, 32, 62, 56, 28, 8, 1, 1, 14, 56, 109, 122, 84, 36, 9, 1, 0, 5, 60, 170, 242, 210, 120, 45, 10, 1, 1, 20, 105, 275, 436, 457, 330, 165, 55, 11, 1, 0, 6, 100, 375, 732, 912, 792, 495, 220, 66, 12, 1

Offset: 1

Frank Ellermann, Dec 31 2001

T(r,c) is the number of winning templates with length r and minimum matching string length c; equivalently, ternary digits totaling r+c. For a definition and row sums 1,1,4,7,20, etc. see A066345. For antidiagonal sums 1,0,2,2,4,9, etc. see A066346.

			Rows:
1;
0,1;
1,2,1;
0,2,4,1;
1,5,8,5,1;
0,3,14,15,6,1; ...
a(17) = T(6,2) = 3 winning templates with length 6 and total 8 = 6+2: 211211, 121121, 112112.
A035615(6) = 2*( 1*1+0*1+1*3+1*1+2*2+1*1+1*1+0*1+2*1+1*1 ) = 2*13 = 26.

Sean A. Irvine, Java program (github)

Cf. A035615, A066345, A066346, A007318.

A035615(n) = 2 * Sum_{r=1..n-1, c=1..min(r,n-r)} T(r,c) * P(n-r,c) where P(n-r,c) = C(n-r-1,c-1) = (n-r-1)!/((n-r-c-2)!*(c-1)!).

More terms from Sean A. Irvine, Nov 03 2023

cp's OEIS Frontend

A008353 2^n*(2^(n+1) - n - 1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

PARI

Formula

A066346 Number of winning binary "same game" templates with ternary digits totaling n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A066709 Triangle T(r,c) of winning binary "same game" templates.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions