A275760 -id:A275760 - cp's OEIS frontend

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.

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

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

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.

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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.

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cp's OEIS Frontend

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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A275761 G.f.: 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 - ...)))))), a continued fraction.

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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.