A006958 -id:A006958 - cp's OEIS frontend

A300122 Number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions connected skew partitions.

1, 4, 13, 51, 183, 771, 3087, 13601, 59933, 278797, 1311719, 6453606, 32179898, 166075956, 871713213, 4704669005, 25831172649, 145260890323

Offset: 1

Gus Wiseman, Feb 25 2018

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns. A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers.

			The a(3) = 13 tableaux:
1 1 1   1 1 2   1 2 2   1 2 3
.
1 1   1 1   1 2   1 2   1 3
1     2     1     3     2
.
1   1   1   1
1   1   2   2
1   2   2   3

Cf. A000085, A000898, A006958, A138178, A238690, A259479, A259480, A296561, A297388, A299699, A299925, A299926, A300118, A300120, A300121, A300123, A300124.

undcon[y_]:=Select[Tuples[Range[0,#]&/@y],Function[v,GreaterEqual@@v&&With[{r=Select[Range[Length[y]],y[[#]]=!=v[[#]]&]},Or[Length[r]<=1,And@@Table[v[[i]]

A300123 Number of ways to tile the diagram of the integer partition with Heinz number n using connected skew partitions.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 8, 4, 10, 8, 16, 8, 32, 16, 20, 8, 64, 20, 128, 16, 40, 32, 256, 16, 52, 64, 52, 32, 512, 40, 1024, 16, 80, 128, 104, 40, 2048, 256, 160, 32, 4096, 80, 8192, 64, 104, 512, 16384, 32, 272, 104

Offset: 1

Gus Wiseman, Feb 25 2018

nonn

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Solomon W. Golomb, Tiling with polyominoes, Journal of Combinatorial Theory, 1-2 (1966), 280-296.
Gus Wiseman, The a(25) = 52 connected tilings of (3,3).

Cf. A000085, A000898, A056239, A006958, A238690, A259479, A259480, A296150, A296561, A297388, A299699, A299925, A299926, A300060, A300118, A300120, A300121, A300122, A300124.

A300124 Number of ways to tile the diagram of an integer partition of n using connected skew partitions.

Original entry on oeis.org

1, 4, 12, 42, 120, 416, 1184, 3888

Offset: 1

Gus Wiseman, Feb 25 2018

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns.

Solomon W. Golomb, Tiling with polyominoes, Journal of Combinatorial Theory, 1-2 (1966), 280-296.
Gus Wiseman, Connected tilings n = 1..4.
Gus Wiseman, Connected tilings n = 5.
Gus Wiseman, Connected tilings n = 6.

Cf. A000085, A000898, A006958, A138178, A259479, A259480, A296561, A297388, A299926, A300120, A300122, A300123.

A275762 G.f.: 2 - x/(1+2x - x^3/(1+2x^2 - x^5/(1+2x^3 - x^7/(1+2x^4 - x^9/(1+2*x^5 - x^11/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

2, -1, 2, -4, 7, -12, 22, -41, 74, -133, 243, -444, 806, -1465, 2669, -4859, 8840, -16087, 29282, -53296, 96994, -176527, 321290, -584755, 1064251, -1936952, 3525296, -6416092, 11677369, -21252993, 38680798, -70399646, 128128414, -233195704, 424419826, -772450633, 1405872057, -2558708924, 4656889892, -8475611623, 15425744240, -28075093283, 51097104306, -92997520459, 169256926243, -308050225082, 560656176744, -1020402917484, 1857149100126, -3380040101304, 6151725289638

Offset: 0

Paul D. Hanna, Aug 10 2016

sign

a(n) ~ c/r^n, where r = -0.54944587773859960333406076695895194626366374257497442830... and c = 0.6098779103867259353642411483841966048261178594794555738...

The g.f. of related triangle A275760 satisfies: G(x,y) = x*y + 1/G(x,x*y) with G(0,y) = 1.

			G.f.: A(x) = 2 - x + 2*x^2 - 4*x^3 + 7*x^4 - 12*x^5 + 22*x^6 - 41*x^7 + 74*x^8 - 133*x^9 + 243*x^10 - 444*x^11 + 806*x^12 - 1465*x^13 + 2669*x^14 - 4859*x^15 +...

Seiichi Manyama, Table of n, a(n) for n = 0..3842 (terms 0..500 from Paul D. Hanna)

Cf. A275760, A275761, A006958, A227309, A291940 (column 1).

m = 51;
2 + ContinuedFractionK[-x^(2i-1), 1+2x^i, {i, 1, Sqrt[m]//Floor}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 02 2019 *)

{a(n) = my(A=1 +x*O(x^n)); for(k=0, n, A = 1/(A + y*x^(n+1-k))); polcoeff(1 + subst(A,y,1), n)}
for(n=0, 50, print1(a(n), ", "))

Equals the diagonal sums of the irregular triangle A275760.
G.f.: 1/F(x) + 1, where F(x) is the g.f. of A275761, the row sums of triangle A275760.
G.f.: G(x,1/x), where G(x,y) = x*y + 1/G(x,x*y) with G(0,y) = 1, where G(x,y) is the g.f. of A275760.
G.f.: 2 - x/(1+x + x/(1+x^2 - x^4/(1+x^3 + x^2/(1+x^4 - x^7/(1+x^5 + x^3/(1+x^6 - x^10/(1+x^7 + x^4/(1+x^8 - x^13/(1+x^9 + x^5/(1+x^10 - x^16/(1 + ...))))))))))), a continued fraction.
G.f.: 1/(1 - 1/(1 + (1+x) - x^2/(1 + x*(1+x) - x^4/(1 + x^2*(1+x) - x^6/(1 + x^3*(1+x) - x^8/(1 + x^4*(1+x) - x^10/(1 + x^5*(1+x) - x^12/(1 - ...)))))))), a continued fraction.
G.f.: 1/(1 - 1/(1+x + 1/(1+x^2 - x^3/(1+x^3 + x/(1+x^4 - x^6/(1+x^5 + x^2/(1+x^6 - x^9/(1+x^7 + x^3/(1+x^8 - x^12/(1+x^9 + x^4/(1+x^10 - x^15/(1 + ...)))))))))))), a continued fraction.
G.f.: 1 + 1/(1 + x/(1 + x/(1 + x^2/(1 + x^2/(1 + x^3/(1 + x^3/(1 + ...))))))) since the odd part of this continued fraction equals the defining continued fraction given above. Cf. A006958 and A227309. - Peter Bala, Oct 29 2017

A291929 Triangle read by rows: T(n,k) = T(n-k,k-1) + 2*T(n-k,k) + T(n-k,k+1) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 4, 1, 0, 9, 2, 0, 20, 6, 0, 46, 13, 1, 0, 105, 32, 2, 0, 242, 73, 6, 0, 557, 171, 15, 0, 1285, 394, 36, 1, 0, 2964, 914, 85, 2, 0, 6842, 2109, 201, 6, 0, 15793, 4877, 467, 15, 0, 36463, 11261, 1086, 38, 0, 84187, 26014, 2517, 89, 1, 0, 194388

Offset: 0

Seiichi Manyama, Sep 06 2017

			First few rows are:
  1;
  0,    1;
  0,    2;
  0,    4,   1;
  0,    9,   2;
  0,   20,   6;
  0,   46,  13,  1;
  0,  105,  32,  2;
  0,  242,  73,  6;
  0,  557, 171, 15;
  0, 1285, 394, 36, 1.

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A291930.
Columns 0-1 give A000007, A006958 (for n>0).
Cf. A008289, A291895.

A300118 Number of skew partitions whose quotient diagram is connected and whose numerator is the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 4, 6, 5, 6, 5, 7, 6, 7, 5, 8, 7, 9, 6, 8, 7, 10, 6, 10, 8, 10, 7, 11, 8, 12, 6, 9, 9, 11, 8, 13, 10, 10, 7, 14, 9, 15, 8, 11, 11, 16, 7, 15, 11, 11, 9, 17, 11, 12, 8, 12, 12, 18, 9, 19, 13, 12, 7, 13, 10, 20, 10, 13, 12, 21, 9, 22, 14, 15, 11

Offset: 1

Gus Wiseman, Feb 25 2018

nonn

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns.

			The a(15) = 7 denominators are (), (1), (11), (22), (3), (31), (32) with diagrams:
o o o   . o o   . o o   . . o   . . .   . . .   o o o
o o     o o     . o     . .     o o     . o     o o
Missing are the two disconnected skew partitions:
. . o   . . o
o o     . o

Cf. A000085, A000898, A056239, A006958, A138178, A153452, A238690, A259479, A259480, A296150, A297388, A299925, A299926, A300056, A300060, A300120, A300122, A300123, A300124.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
undcon[y_]:=Select[Tuples[Range[0,#]&/@y],Function[v,GreaterEqual@@v&&With[{r=Select[Range[Length[y]],y[[#]]=!=v[[#]]&]},Or[Length[r]<=1,And@@Table[v[[i]]

A075125 Number of parallelogram polyominoes of site-perimeter n (also called staircase polyominoes, although that term is overused).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 2, 5, 10, 21, 46, 102, 230, 526, 1216, 2838, 6678, 15825, 37734, 90469, 217962, 527418, 1281250, 3123603, 7639784, 18740795, 46096732, 113666820, 280928470, 695796891, 1726744166, 4293121609, 10692145390, 26671959375, 66634602702

Offset: 1

Andrew Rechnitzer (a.rechnitzer(AT)ms.unimelb.edu.au), Sep 09 2002

nonn

a(n) is the number of Dyck n-paths with no UDU's and no DUD's (A004148) whose first ascent is of length 3. For example, a(5)=2 counts UUUDDUUDDD, UUUDDDUUDD. - David Callan, May 08 2007
From Emeric Deutsch, Nov 07 2009: (Start)
a(n) = Sum_{k>=0} k*A166299(n-2,k).
Number of UUDD's starting at level 0 in all Dyck paths of semilength n-2 that have no ascents and no descents of length 1. Example: a(6)=2 because in UUDDUUDD and UUUUDDDD we have 2 + 0 = 2 UUDD's starting at level 0. (The Dyck paths having no ascents and no descents of length 1 are enumerated by the secondary structure numbers A004148).
(End)

M. P. Delest, D. Gouyou-Beauchamps and B. Vauquelin, Enumeration of parallelogram polyominoes with given bond and site parameter, Graphs and Combinatorics, 3(1987),325-339. [From Emeric Deutsch, Nov 07 2009]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Cf. A004148, A006958, A075126, A166299.

G := 4*z^4/(1+z-z^2+sqrt((1+z+z^2)*(1-3*z+z^2)))^2: Gser := series(G, z = 0, 32): seq(coeff(Gser, z, n), n = 1 .. 30); # Emeric Deutsch, Nov 07 2009

Rest[CoefficientList[Series[4 x^4/(1 + x - x^2 + Sqrt[(1 + x + x^2) (1 - 3 x + x^2)])^2, {x, 0, 40}], x]] (* Vaclav Kotesovec, Mar 21 2014 *)

a(n):=2*sum((binomial(k-2,2*k-n+2)*binomial(k+1,n-k-3))/(k+1),k,floor((n-2)/2),n-3); /* Vladimir Kruchinin, Oct 12 2020 */

G.f.: p^2/2*(1-p^2-2*p^3+p^4-(1+p-p^2)*sqrt((1+p+p^2)*(1-3*p+p^2)));
a(n) ~ sqrt(2) * ((3+sqrt(5))/2)^n / (sqrt(377 + 843/sqrt(5)) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 21 2014. Equivalently, a(n) ~ 5^(1/4) * phi^(2*n - 7) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021
Conjecture: -(2*n-11)*(n-2)*(2*n-9)*a(n) +4*(2*n-11)*(n-3)*(n-5)*a(n-1) +(4*n^3-60*n^2+317*n-582)*a(n-2) +2*(2*n-7)*(2*n^2-26*n+81)*a(n-3) -(n-10)*(2*n-7)*(2*n-9)*a(n-4)=0. - R. J. Mathar, May 30 2016
a(n) = 2 * Sum_{k=floor((n-2)/2)..n-3} C(k-2,2*k-n+2)*C(k+1,n-k-3)/(k+1). - Vladimir Kruchinin, Oct 12 2020

Offset changed to 1 by Emeric Deutsch, Nov 07 2009
More terms from Vincenzo Librandi, Mar 22 2014
Name modified by Alois P. Heinz, Sep 21 2016

A226728 G.f.: 1/G(0), where G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ).

Original entry on oeis.org

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

Offset: 0

Joerg Arndt, Jun 29 2013

sign

Cf. A049346 (g.f.: 1 - 1/G(0), G(k)= 1 + q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
Cf. A226729 (g.f.: 1/G(0), G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A006958 (g.f.: 1/G(0), G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
Cf. A227309 (g.f.: 1/G(0), G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ) ).

N = 166;  q = 'q + O('q^N);
G(k) = if(k>N, 1, 1 + q^(k+1) / (1 - q^(k+1) / G(k+2) ) );
gf = 1 / G(0);
Vec(gf)

G.f.: 1/(1+q/(1-q/(1+q^3/(1-q^3/(1+q^5/(1-q^5/(1+q^7/(1-q^7/(1+ ... ))))))))).

G.f.: 1/W(0), where W(k)= 1 + x^(2*k+1)/(1 - x^(2*k+1)/W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 16 2013

A259481 T(n,m) counts of border strips in skew tabloids of shape lambda/mu, with lambda and mu partitions of n and m (0<=m<=n).

Original entry on oeis.org

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

Offset: 0

Wouter Meeussen, Jul 01 2015

Border strips are defined as connected skew tabloids free of 2-by-2 cells.

Row sums are the partition numbers (A000041), diagonals sum to 2^n (A000079).

			T(8,2) = 6, the pairs of partitions are ((5,3)/(2)), ((4,3,1)/(2)), ((4,2,2)/(1,1)), ((3,3,1,1)/(2)), ((3,2,2,1)/(1,1)) and ((2,2,2,1,1)/(1,1)); the diagrams are:
  x x 0 0 0 , x x 0 0 , x 0 0 0 , x x 0 , x 0 0 , x 0
  0 0 0       0 0 0     x 0       0 0 0   x 0     x 0
              0         0 0       0       0 0     0 0
                                  0       0       0
                                                  0
Triangle begins:
      k=0  1  2  3  4  5  6  7
  n=0;  0
  n=1;  1  0
  n=2;  2  0  0
  n=3;  3  0  0  0
  n=4;  4  1  0  0  0
  n=5;  5  2  0  0  0  0
  n=6;  6  3  2  0  0  0  0
  n=7;  7  4  4  0  0  0  0  0

I. G. MacDonald: "Symmetric functions and Hall polynomials"; Oxford University Press, 1979. Page 4.

Cf. A259478, A259479, A259480, A161492, A227309, A006958.

(* see A259479 *) Table[Sum[Boole[majorsweak[\[Lambda],\[Mu]]&&( Tr[\[Lambda]]-Tr[\[Mu]]==Length[\[Lambda]]+First[\[Lambda]]-1 )&& redu[\[Lambda],\[Mu]]==factor[\[Lambda],\[Mu]]=={\[Lambda],\[Mu]}],{\[Lambda],Partitions[n]},{\[Mu],Partitions[k]}],{n,0,12},{k,0,n}]

A275761 G.f.: 1/(1 - x/(1+2x - x^3/(1+2x^2 - x^5/(1+2x^3 - x^7/(1+2x^4 - x^9/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

1, 1, -1, 1, 0, -1, 0, 2, -1, -2, 1, 3, -3, -1, 3, 1, -7, 3, 7, -2, -12, 10, 5, -10, -8, 27, -8, -23, 2, 46, -38, -20, 30, 45, -100, 27, 71, 12, -183, 141, 65, -71, -213, 384, -100, -202, -145, 729, -545, -172, 93, 993, -1497, 430, 452, 962, -2982, 2188, 250, 451, -4527, 6014, -2119, -296, -5456, 12440, -9197, 1206, -5307, 20547, -24963, 11156, -5513, 28712, -53013, 40590, -15529, 36553, -93599, 107065, -60129, 52093, -145383, 231326, -186656, 113800, -214705, 429584, -474454, 323536

Offset: 0

Paul D. Hanna, Aug 08 2016

sign

Row sums of triangle A275760.

Limit a(n)/a(n+1) = -0.83683607462189175014302689979307768909437126147437...

			G.f.: A(x) = 1 + x - x^2 + x^3 - x^5 + 2*x^7 - x^8 - 2*x^9 + x^10 + 3*x^11 - 3*x^12 - x^13 + 3*x^14 + x^15 - 7*x^16 + 3*x^17 + 7*x^18 - 2*x^19 - 12*x^20 +...
such that
A(x) = 1/(1 - x/(1 + 2*x - x^3/(1 + 2*x^2 - x^5/(1 + 2*x^3 - x^7/(1 + 2*x^4 - x^9/(1 + 2*x^5 - x^11/(1 + 2*x^6 - x^13/(1 - ...)))))))).
RELATED SERIES.
1/A(x) = 1 - x + 2*x^2 - 4*x^3 + 7*x^4 - 12*x^5 + 22*x^6 - 41*x^7 + 74*x^8 - 133*x^9 + 243*x^10 - 444*x^11 + 806*x^12 - 1465*x^13 + 2669*x^14 - 4859*x^15 + 8840*x^16 - 16087*x^17 + 29282*x^18 - 53296*x^19 + 96994*x^20 - 176527*x^21 + 321290*x^22 - 584755*x^23 + 1064251*x^24 +...+ A275762(n)*x^n +...

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A275760, A275762, A006958, A227309.

{a(n) = my(A=1 +x*O(x^n)); for(k=0, n, A = 1/A + y*x^(n+1-k)); subst(polcoeff(A, n),y,1)}
for(n=0,100,print1(a(n),", "))

G.f.: 1/(1 - x/(1+x + x/(1+x^2 - x^4/(1+x^3 + x^2/(1+x^4 - x^7/(1+x^5 + x^3/(1+x^6 - x^10/(1+x^7 + x^4/(1+x^8 - x^13/(1+x^9 + x^5/(1+x^10 - x^16/(1 + ...)))))))))))), a continued fraction.
G.f.: G(x,1) where G(x,y) = x*y + 1/G(x,x*y) with G(0,y) = 1 (cf. A275760).
G.f.: 1 + x/(1 + x/(1 + x^2/(1 + x^2/(1 + x^3/(1 + x^3/(1 + ...)))))). Cf. A006958 and A227309. - Peter Bala, Oct 29 2017

cp's OEIS Frontend

A300122 Number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions connected skew partitions.

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A300123 Number of ways to tile the diagram of the integer partition with Heinz number n using connected skew partitions.

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A300124 Number of ways to tile the diagram of an integer partition of n using connected skew partitions.

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A275762 G.f.: 2 - x/(1+2*x - x^3/(1+2*x^2 - x^5/(1+2*x^3 - x^7/(1+2*x^4 - x^9/(1+2*x^5 - x^11/(1 - ...)))))), a continued fraction.

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A291929 Triangle read by rows: T(n,k) = T(n-k,k-1) + 2*T(n-k,k) + T(n-k,k+1) with T(0,0) = 1 for 0 <= k <= A003056(n).

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A300118 Number of skew partitions whose quotient diagram is connected and whose numerator is the integer partition with Heinz number n.

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A075125 Number of parallelogram polyominoes of site-perimeter n (also called staircase polyominoes, although that term is overused).

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A226728 G.f.: 1/G(0), where G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ).

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A259481 T(n,m) counts of border strips in skew tabloids of shape lambda/mu, with lambda and mu partitions of n and m (0<=m<=n).

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A275762 G.f.: 2 - x/(1+2x - x^3/(1+2x^2 - x^5/(1+2x^3 - x^7/(1+2x^4 - x^9/(1+2*x^5 - x^11/(1 - ...)))))), a continued fraction.

A275761 G.f.: 1/(1 - x/(1+2x - x^3/(1+2x^2 - x^5/(1+2x^3 - x^7/(1+2x^4 - x^9/(1 - ...)))))), a continued fraction.