cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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- ... ))).

Original entry on oeis.org

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

Views

Author

Alois P. Heinz, Aug 29 2013

Keywords

Examples

			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, ...
		

Crossrefs

Diagonal gives: A005169.

Programs

  • Mathematica
    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 *)

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

Views

Author

Paul D. Hanna, Jul 08 2013

Keywords

Comments

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...

Examples

			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 +...
		

Crossrefs

Column m=2 of A185646.

Programs

  • Mathematica
    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 *)
  • PARI
    {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

Views

Author

Paul D. Hanna, Jul 09 2013

Keywords

Comments

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

Examples

			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 +...
		

Crossrefs

Column m=4 of A185646.

Programs

  • Mathematica
    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 *)
  • PARI
    {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),", "))

Formula

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

Views

Author

Alois P. Heinz, Aug 28 2013

Keywords

Crossrefs

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.

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

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)).

A228645 Expansion of g.f. 1/ (1-x^1*(1-x^(m+1))/ (1-x^2*(1-x^(m+2))/ (1- ... ))) for m=9.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 15, 26, 45, 78, 134, 232, 402, 695, 1205, 2086, 3613, 6259, 10841, 18780, 32531, 56354, 97621, 169111, 292954, 507488, 879136, 1522947, 2638242, 4570298, 7917253, 13715281, 23759370, 41159039, 71300984, 123516755, 213971647, 370669282
Offset: 0

Views

Author

Alois P. Heinz, Aug 28 2013

Keywords

Crossrefs

Cf. A143064 (m=0), A227360 (m=2), A173173 (m=3), A227374 (m=4), A227375 (m=5), A228646 (m=6), A228644 (m=7).
Column m=9 of A185646.

Programs

  • Maple
    a:= n-> coeff(series(-(x^30 +x^29 +x^28 +2*x^27 +2*x^26 +2*x^25 +x^24 +x^23 -x^22 -2*x^21 -2*x^20 -4*x^19 -4*x^18 -3*x^17 -2*x^16 -x^15 +2*x^13 +2*x^12 +3*x^11 +3*x^10 +x^9 +x^8 -x^5 -x^4 -x^3 -x^2+1) / ((x-1)*(x^2 +x+1)*(x^6 +x^3+1)*(x^26 +x^25 +x^24 +2*x^23 +2*x^22 +x^21 +x^20 -2*x^18 -2*x^17 -3*x^16 -3*x^15 -3*x^14 -x^13 -x^12 +x^11 +2*x^10 +2*x^9 +2*x^8 +x^7 +x^6 -x^3 -x^2 -x+1)), x, n+1), x, n): seq(a(n), n=0..50);
  • Mathematica
    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] &; A228645 = col[9][[1 ;; nMax]] (* Jean-François Alcover, Nov 03 2016 *)

Formula

G.f.: -(x^30 +x^29 +x^28 +2*x^27 +2*x^26 +2*x^25 +x^24 +x^23 -x^22 -2*x^21 -2*x^20 -4*x^19 -4*x^18 -3*x^17 -2*x^16 -x^15 +2*x^13 +2*x^12 +3*x^11 +3*x^10 +x^9 +x^8 -x^5 -x^4 -x^3 -x^2+1) / ((x-1)*(x^2 +x+1)*(x^6 +x^3+1)*(x^26 +x^25 +x^24 +2*x^23 +2*x^22 +x^21 +x^20 -2*x^18 -2*x^17 -3*x^16 -3*x^15 -3*x^14 -x^13 -x^12 +x^11 +2*x^10 +2*x^9 +2*x^8 +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

Views

Author

Alois P. Heinz, Aug 28 2013

Keywords

Crossrefs

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.

Programs

  • Mathematica
    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 *)

A227376 G.f.: 1/(1 - x - x^2 - x^3 + x^5 + x^6 + x^7).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 21, 36, 62, 106, 181, 309, 527, 898, 1530, 2606, 4438, 7557, 12867, 21907, 37297, 63497, 108100, 184032, 313298, 533359, 907988, 1545751, 2631469, 4479778, 7626309, 12982911, 22101900, 37625912, 64053725, 109043981, 185634620, 316021206, 537989084, 915863373
Offset: 0

Views

Author

Paul D. Hanna, Jul 09 2013

Keywords

Comments

Compare to sequence A227375, generated by a continued fraction.
Lim_{n->infinity} a(n)/a(n+1) = 0.587411973105598587998520092901249815195963...

Examples

			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 + ...
		

Crossrefs

Cf. A227375.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-x-x^2-x^3+x^5+x^6+x^7),{x,0,50}],x] (* Harvey P. Dale, Mar 09 2021 *)
  • PARI
    {a(n)=polcoeff(1/(1 - x - x^2 - x^3 + x^5 + x^6 + x^7 +x*O(x^n)),n)}
    for(n=0,50,print1(a(n),", "))

Formula

a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-5) - a(n-6) - a(n-7). - Wesley Ivan Hurt, May 22 2021
Showing 1-7 of 7 results.