A228645 -id:A228645 - cp's OEIS frontend

A185646 Square array A(n,m), n>=0, m>=0, read by antidiagonals, where column m is the expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))).

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, -1, 1, 1, 1, 2, 2, 1, 0, 1, 1, 1, 2, 3, 3, 1, 0, 1, 1, 1, 2, 3, 4, 5, 1, -1, 1, 1, 1, 2, 3, 5, 7, 6, 1, 0, 1, 1, 1, 2, 3, 5, 8, 11, 10, 1, 0, 1, 1, 1, 2, 3, 5, 9, 13, 17, 14, 1, 0, 1, 1, 1, 2, 3, 5, 9, 14, 22, 28, 21, 1, 0

Offset: 0

Alois P. Heinz, Aug 29 2013

			Square array A(n,m) begins:
   1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   0,  1,  1,  1,  1,  1,  1,  1,  1, ...
   0,  1,  2,  2,  2,  2,  2,  2,  2, ...
   0,  1,  2,  3,  3,  3,  3,  3,  3, ...
  -1,  1,  3,  4,  5,  5,  5,  5,  5, ...
   0,  1,  5,  7,  8,  9,  9,  9,  9, ...
   0,  1,  6, 11, 13, 14, 15, 15, 15, ...
  -1,  1, 10, 17, 22, 24, 25, 26, 26, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Paul D. Hanna et al., Formula Needed for a Family of Continued Fractions and follow-up messages on the SeqFan list, Jul 28 2013

Columns m=0-10 give: A143064, A000012, A227360, A173173(n+1), A227374, A227375, A228646, A228644, A185648, A228645, A185649.

Diagonal gives: A005169.

nMax = 12; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x]&; A = Table[col[m][[1 ;; nMax + 1]], {m, 0, nMax}] // Transpose; a[n_ /; 0 <= n <= nMax, m_ /; 0 <= m <= nMax] := With[{n1 = n + 1, m1 = m + 1}, A[[n1, m1]]]; Table[a[n - m, m], {n, 0, nMax}, {m, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2016 *)

A227375 G.f.: 1/(1 - x(1-x^6)/(1 - x^2(1-x^7)/(1 - x^3(1-x^8)/(1 - x^4(1-x^9)/(1 - x^5*(1-x^10)/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 14, 24, 41, 69, 118, 200, 340, 579, 985, 1677, 2854, 4858, 8270, 14078, 23966, 40798, 69453, 118235, 201280, 342655, 583328, 993046, 1690543, 2877949, 4899369, 8340598, 14198887, 24171937, 41149884, 70052848, 119256753, 203020631, 345618810, 588375486, 1001640259

Offset: 0

Paul D. Hanna, Jul 09 2013

nonn

Radius of convergence r is a root of 1 - r - r^2 - r^3 + r^5 + r^6 + r^7 = 0,
where r = Limit a(n)/a(n+1) = 0.587411973105598587998520092901249815195963...
Compare to sequence A227376, generated by 1/(1-x-x^2-x^3+x^5+x^6+x^7).

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 9*x^6 + 14*x^7 + 24*x^8 +...

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

Cf. A173173, A227374, A227360, A227376, A228644, A228645.

Column m=5 of A185646.

nMax = 42; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x]&; A227375 = col[5][[1 ;; nMax]] (* Jean-François Alcover, Nov 03 2016 *)
LinearRecurrence[{1,1,1,0,0,-2,-2,-1,0,1,1,1},{1,1,1,2,3,5,9,14,24,41,69,118},50] (* Harvey P. Dale, Jul 08 2023 *)

a(n)=local(CF); CF=1+x; for(k=0, n, CF=1/(1 - x^(n-k+1)*(1 - x^(n-k+6))*CF+x*O(x^n))); polcoeff(CF, n)
for(n=0,50,print1(a(n),", "))

/* From R. J. Mathar's g.f. formula: */
{a(n)=polcoeff((1-x-x^4)*(1+x-x^3-x^4-x^5)/((1-x^5)*(1-x-x^2-x^3+x^5+x^6+x^7) +x*O(x^n)),n)}
for(n=0,50,print1(a(n),", ")) \\ Paul D. Hanna, Jul 18 2013

Conjecture: G.f. -(x^4+x-1)*(x^5+x^4+x^3-x-1) / ( (x-1)*(x^4+x^3+x^2+x+1)*(x^7+x^6+x^5-x^3-x^2-x+1) ). - R. J. Mathar, Jul 17 2013

A227360 G.f.: 1/(1 - x(1-x^3)/(1 - x^2(1-x^4)/(1 - x^3(1-x^5)/(1 - x^4(1-x^6)/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 10, 14, 21, 32, 46, 71, 104, 157, 235, 350, 527, 785, 1179, 1763, 2639, 3954, 5915, 8861, 13262, 19857, 29731, 44507, 66640, 99765, 149366, 223625, 334795, 501247, 750434, 1123518, 1682076, 2518314, 3770306, 5644701, 8450977, 12652376

Offset: 0

Paul D. Hanna, Jul 08 2013

nonn

Compare to the continued fraction representation for the g.f. of A173173, where A173173(n) = ceiling(Fibonacci(n)/2).

Limit a(n)/a(n+1) = 0.6679357039724580760720733281356826861233293827578332775311...

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 6*x^7 + 10*x^8 +...

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A173173, A227374, A227375, A228644, A228645.

Column m=2 of A185646.

nMax = 44; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x] &; A227360 = col[2][[1 ;; nMax]] (* Jean-François Alcover, Nov 03 2016 *)

{a(n)=local(CF); CF=1+x; for(k=0, n, CF=1/(1 - x^(n-k+1)*(1 - x^(n-k+3))*CF+x*O(x^n))); polcoeff(CF, n)}
for(n=0,50,print1(a(n),", "))

A227374 G.f.: 1/(1 - x(1-x^5)/(1 - x^2(1-x^6)/(1 - x^3(1-x^7)/(1 - x^4(1-x^8)/(1 - x^5*(1-x^9)/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 22, 36, 61, 101, 169, 283, 473, 793, 1325, 2220, 3715, 6220, 10413, 17431, 29185, 48856, 81797, 136937, 229257, 383813, 642564, 1075762, 1800995, 3015171, 5047886, 8451001, 14148368, 23686705, 39655467, 66389797, 111147511, 186079299, 311527531, 521548600

Offset: 0

Paul D. Hanna, Jul 09 2013

nonn

Limit a(n)/a(n+1) = 0.597312551712707899432116871133154503665320273329853...

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 13*x^7 + 22*x^8 +...

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A173173, A227360, A227375, A228644, A228645.

Column m=4 of A185646.

nMax = 42; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x]&; A227374 = col[4][[1 ;; nMax]] (* Jean-François Alcover, Nov 03 2016 *)

{a(n)=local(CF); CF=1+x; for(k=0, n, CF=1/(1 - x^(n-k+1)*(1 - x^(n-k+5))*CF+x*O(x^n))); polcoeff(CF, n)}
for(n=0,50,print1(a(n),", "))

G.f.: T(0), where T(k) = 1 - x^(k+1)*(1-x^(k+5))/(x^(k+1)*(1-x^(k+5)) - 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 18 2013

A228644 Expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))) for m=7.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 15, 26, 44, 76, 131, 225, 389, 670, 1156, 1994, 3439, 5934, 10236, 17661, 30470, 52569, 90699, 156483, 269985, 465811, 803677, 1386609, 2392357, 4127611, 7121498, 12286951, 21199078, 36575462, 63104849, 108876873, 187848862, 324101847

Offset: 0

Alois P. Heinz, Aug 28 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul D. Hanna et al., Formula Needed for a Family of Continued Fractions and follow-up messages on the SeqFan list, Jul 28 2013
Index entries for linear recurrences with constant coefficients, signature (1,1,1,0,0,-1,-1,-3,-2,-1,0,2,2,3,3,1,0,0,-2,-1,-1,-1).

Cf. A143064 (m=0), A227360 (m=2), A173173 (m=3), A227374 (m=4), A227375 (m=5), A228646(m=6), A228645 (m=9).

Column m=7 of A185646.

a:= n-> coeff(series(-(x^18 +x^17 +x^16 +2*x^15 +x^14 -2*x^11 -2*x^10 -2*x^9 -2*x^8 +x^5 +x^4 +x^3 +x^2-1) / ((x-1)*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)*(x^15 +x^14 +x^13 +2*x^12 -x^9 -2*x^8 -2*x^7 -x^6 +x^3 +x^2 +x-1)), x, n+1), x, n): seq(a(n), n=0..50);

nMax = 39; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x]&; A228644 = col[7][[1 ;; nMax]] (* Jean-François Alcover, Nov 03 2016 *)

G.f.: -(x^18 +x^17 +x^16 +2*x^15 +x^14 -2*x^11 -2*x^10 -2*x^9 -2*x^8 +x^5 +x^4 +x^3 +x^2-1) / ((x-1)*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)*(x^15 +x^14 +x^13 +2*x^12 -x^9 -2*x^8 -2*x^7 -x^6 +x^3 +x^2 +x-1)).

A228646 Expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))) for m=6.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 15, 25, 43, 74, 126, 217, 372, 638, 1096, 1881, 3230, 5546, 9524, 16353, 28083, 48224, 82811, 142208, 244204, 419360, 720144, 1236670, 2123670, 3646879, 6262611, 10754485, 18468174, 31714525, 54461873, 93524824, 160605817, 275800867

Offset: 0

Alois P. Heinz, Aug 28 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul D. Hanna et al., Formula Needed for a Family of Continued Fractions and follow-up messages on the SeqFan list, Jul 28 2013

Cf. A143064 (m=0), A227360 (m=2), A173173 (m=3), A227374 (m=4), A227375 (m=5), A228644 (m=7), A228645 (m=9).

Column m=6 of A185646.

nMax = 39; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x]&; A228646 = col[6][[1 ;; nMax]] (* Jean-François Alcover, Nov 03 2016 *)

cp's OEIS Frontend

A185646 Square array A(n,m), n>=0, m>=0, read by antidiagonals, where column m is the expansion of g.f. 1/ (1-x^1*(1-x^(m+1))/ (1-x^2*(1-x^(m+2))/ (1- ... ))).

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A227375 G.f.: 1/(1 - x*(1-x^6)/(1 - x^2*(1-x^7)/(1 - x^3*(1-x^8)/(1 - x^4*(1-x^9)/(1 - x^5*(1-x^10)/(1 - ...)))))), a continued fraction.

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A227360 G.f.: 1/(1 - x*(1-x^3)/(1 - x^2*(1-x^4)/(1 - x^3*(1-x^5)/(1 - x^4*(1-x^6)/(1 - ...))))), a continued fraction.

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A227374 G.f.: 1/(1 - x*(1-x^5)/(1 - x^2*(1-x^6)/(1 - x^3*(1-x^7)/(1 - x^4*(1-x^8)/(1 - x^5*(1-x^9)/(1 - ...)))))), a continued fraction.

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A228644 Expansion of g.f. 1/ (1-x^1*(1-x^(m+1))/ (1-x^2*(1-x^(m+2))/ (1- ... ))) for m=7.

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A228646 Expansion of g.f. 1/ (1-x^1*(1-x^(m+1))/ (1-x^2*(1-x^(m+2))/ (1- ... ))) for m=6.

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A185646 Square array A(n,m), n>=0, m>=0, read by antidiagonals, where column m is the expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))).

A227375 G.f.: 1/(1 - x(1-x^6)/(1 - x^2(1-x^7)/(1 - x^3(1-x^8)/(1 - x^4(1-x^9)/(1 - x^5*(1-x^10)/(1 - ...)))))), a continued fraction.

A227360 G.f.: 1/(1 - x(1-x^3)/(1 - x^2(1-x^4)/(1 - x^3(1-x^5)/(1 - x^4(1-x^6)/(1 - ...))))), a continued fraction.

A227374 G.f.: 1/(1 - x(1-x^5)/(1 - x^2(1-x^6)/(1 - x^3(1-x^7)/(1 - x^4(1-x^8)/(1 - x^5*(1-x^9)/(1 - ...)))))), a continued fraction.

A228644 Expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))) for m=7.

A228646 Expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))) for m=6.