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 1-10 of 12 results. Next

A027949 a(n) = T(2n,n+1), T given by A027948.

Original entry on oeis.org

1, 4, 25, 97, 309, 894, 2462, 6610, 17519, 46135, 121115, 317484, 831660, 2177872, 5702389, 14929789, 39087537, 102333450, 267913514, 701407870, 1836310955, 4807525939, 12586267895, 32951278872, 86267569944, 225851432284, 591286728337, 1548008754265
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000045, A027948.

Programs

  • GAP
    Concatenation([1], List([2..40], n-> Fibonacci(2*n+4) -(2*n^2 +3*n +3)) ); # G. C. Greubel, Sep 29 2019
  • Magma
    [1] cat [Fibonacci(2*n+4) -(2*n^2 +3*n +3): n in [2..40]]; // G. C. Greubel, Sep 29 2019
  • Maple
    with(combinat); seq(`if`(n=1, 1, fibonacci(2*n+4) -(2*n^2 +3*n +3)), n=1..40); # G. C. Greubel, Sep 29 2019
  • Mathematica
    Join[{1},Table[Fibonacci[2n+4]-2n^2-3n-3,{n,2,40}]] (* or *) Join[ {1}, LinearRecurrence[{6,-13,13,-6,1}, {4,25,97,309,894}, 40]] (* Harvey P. Dale, Apr 20 2012 *)
CoefficientList[Series[(x^5-6x^4+14x^3-14x^2+2x-1)/((x-1)^3(x^2-3x+1)), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 20 2014 *)
  • PARI
    Vec(x*(x^5-6*x^4+14*x^3-14*x^2+2*x-1)/((x-1)^3*(x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Nov 19 2014
  • PARI
    vector(40, n, if(n==1,1,fibonacci(2*n+4) -(2*n^2 +3*n +3)) ) \\ G. C. Greubel, Sep 29 2019
  • Sage
    [1]+[fibonacci(2*n+4) -(2*n^2 +3*n +3) for n in (2..40)] # G. C. Greubel, Sep 29 2019

Formula

For n>1, a(n) = Fibonacci(2*n+4) - (2*n^2 + 3*n + 3).
a(1)=1, a(2)=4, a(3)=25, a(4)=97, a(5)=309, a(6)=894, a(n) = 6*a(n-1) - 13*a(n-2) +13*a(n-3) -6*a(n-4) +a(n-5). - Harvey P. Dale, Apr 20 2012
G.f.: x*(1 -2*x +14*x^2 -14*x^3 +6*x^4 -x^5)/((1-x)^3*(1-3*x+x^2)). - Colin Barker, Nov 19 2014
a(n) = Sum_{j=0..n-1} binomial(2*n-j, j+3), with a(1)=1. - G. C. Greubel, Sep 29 2019

Extensions

More terms from Harvey P. Dale, Apr 20 2012

A027950 a(n) = T(2n,n+2), T given by A027948.

Original entry on oeis.org

1, 6, 63, 344, 1383, 4685, 14323, 41119, 113590, 306605, 816410, 2157046, 5674578, 14893364, 39040633, 102273950, 267839033, 701315739, 1836198205, 4807389285, 12586103720, 32951083211, 86267338468, 225851160284, 591286410708, 1548008385490
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Bisection of A053739.
Cf. A000045, A027948.

Programs

  • GAP
    Concatenation([1], List([3..40], n-> Fibonacci(2*n+6) -(48 +47*n +23*n^2 +4*n^3 +4*n^4)/6) ); # G. C. Greubel, Sep 29 2019
  • Magma
    [1] cat [Fibonacci(2*n+6) -(48 +47*n +23*n^2 +4*n^3 +4*n^4)/6: n in [3..40]]; // G. C. Greubel, Sep 29 2019
  • Maple
    with(combinat); seq(`if`(n=2,1, fibonacci(2*n+6) -(48 +47*n +23*n^2 +4*n^3 +4*n^4)/6), n=2..40); # G. C. Greubel, Sep 29 2019
  • Mathematica
    Table[If[n==2, 1, Fibonacci[2*n+6] -(48 +47*n +23*n^2 +4*n^3 +4*n^4)/6], {n,2,40}] (* G. C. Greubel, Sep 29 2019 *)
CoefficientList[Series[x^2(1-2x+41x^2-49x^3+44x^4-26x^5+8x^6-x^7)/ ((1-3x+x^2)(1-x)^5),{x,0,30}],x] (* or *) LinearRecurrence[{8,-26,45,-45,26,-8,1},{1,6,63,344,1383,4685,14323,41119},30] (* Harvey P. Dale, Aug 15 2021 *)
  • PARI
    Vec(x^2*(x^7-8*x^6+26*x^5-44*x^4+49*x^3-41*x^2+2*x-1)/((x-1)^5* (x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Nov 19 2014
  • PARI
    vector(40, n, my(m=n+1); if(m==2, 1, fibonacci(2*m+6) -(48 +47*m +23*m^2 +4*m^3 +4*m^4)/6) ) \\ G. C. Greubel, Sep 29 2019
  • Sage
    [1]+[fibonacci(2*n+6) -(48 +47*n +23*n^2 +4*n^3 +4*n^4)/6 for n in (3..40)] # G. C. Greubel, Sep 29 2019

Formula

G.f.: x^2*(1-2*x+41*x^2-49*x^3+44*x^4-26*x^5+8*x^6-x^7)/((1-3*x+x^2)*(1-x)^5). - Ralf Stephan, Apr 24 2004
From G. C. Greubel, Sep 29 2019: (Start)
a(n) = Sum_{j=0..n-2} binomial(2*n-j, j+5), with a(2) = 1 for n >= 2.
a(n) = Fibonacci(2*n+6) - (48 + 47*n + 23*n^2 + 4*n^3 + 4*n^4)/6 for n >= 3. (End)

Extensions

More terms from Colin Barker, Nov 19 2014

A027951 a(n) = T(2n,n+3), T given by A027948.

Original entry on oeis.org

1, 8, 129, 967, 4950, 20175, 70954, 226007, 672959, 1914166, 5280288, 14275838, 38102976, 100888126, 265838881, 698489013, 1832277574, 4802042229, 12578921258, 32941567397, 86254888591, 225835057708, 591265802288, 1547982265500, 4052706300752
Offset: 3

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000045, A027948.

Programs

  • GAP
    Concatenation([1], List([4..40], n-> Fibonacci(2*n+8) -(8*n^6 -12*n^5 +110*n^4 +255*n^3 +872*n^2 +1827*n +1890)/90) ); # G. C. Greubel, Sep 29 2019
  • Magma
    [1] cat [Fibonacci(2*n+8) -(8*n^6 -12*n^5 +110*n^4 +255*n^3 +872*n^2 +1827*n +1890)/90: n in [4..40]]; // G. C. Greubel, Sep 29 2019
  • Maple
    with(combinat); seq(`if`(n=3,1, fibonacci(2*n+8) -(8*n^6 -12*n^5 +110*n^4 +255*n^3 +872*n^2 +1827*n +1890)/90), n=3..40); # G. C. Greubel, Sep 29 2019
  • Mathematica
    CoefficientList[Series[(x^9 -10x^8 +43x^7 -105x^6 +162x^5 -148x^4 +84x^3 -92x^2 +2x -1)/((x-1)^7(x^2-3x+1)), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 20 2014 *)
Table[If[n==3, 1, Fibonacci[2*n+8] -(8*n^6 -12*n^5 +110*n^4 +255*n^3 +872*n^2 +1827*n +1890)/90], {n,3,40}] (* G. C. Greubel, Sep 29 2019 *)
  • PARI
    Vec(x^3*(x^9-10*x^8+43*x^7-105*x^6+162*x^5-148*x^4+84*x^3-92*x^2 +2*x-1)/((x-1)^7*(x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Nov 19 2014
  • PARI
    vector(40, n, my(m=n+2); if(m==3, 1, fibonacci(2*m+8) -(8*m^6 -12*m^5 +110*m^4 +255*m^3 +872*m^2 +1827*m +1890)/90) ) \\ G. C. Greubel, Sep 29 2019
  • Sage
    [1]+[fibonacci(2*n+8) -(8*n^6 -12*n^5 +110*n^4 +255*n^3 +872*n^2 +1827*n +1890)/90 for n in (4..40)] # G. C. Greubel, Sep 29 2019

Formula

G.f.: x^3*(1 -2*x +92*x^2 -84*x^3 +148*x^4 -162*x^5 +105*x^6 -43*x^7 +10*x^8 -x^9)/((1-x)^7*(1-3*x+x^2)). - Colin Barker, Nov 19 2014
From G. C. Greubel, Sep 29 2019: (Start)
a(n) = Sum_{j=0..n-3} binomial(2*n-j, j+7), with a(3) = 1 for n >= 3.
a(n) = Fibonacci(2*n+8) - (8*n^6 -12*n^5 +110*n^4 +255*n^3 +872*n^2 +1827*n +1890)/90 for n >= 4. (End)

Extensions

More terms from Colin Barker, Nov 19 2014

A027952 a(n) = T(2n,n+4), T given by A027948.

Original entry on oeis.org

1, 10, 231, 2300, 14820, 72905, 298925, 1077748, 3540913, 10871723, 31775031, 89633545, 246575109, 666605513, 1781049298, 4721874921, 12456394685, 32758238316, 85985810716, 225446971141, 590714939822, 1547211717890, 4051642877482, 10608719012366, 27775885869046
Offset: 4

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000045, A027948.

Programs

  • GAP
    Concatenation([1], List([5..40], n-> Fibonacci(2*n+10) -(138600 +133530*n +63999*n^2 + 20286*n^3 +5929*n^4 +616*n^6 -96*n^7 +16*n^8)/2520) ); # G. C. Greubel, Sep 29 2019
  • Magma
    [1] cat [Fibonacci(2*n+10) -(138600 +133530*n +63999*n^2 + 20286*n^3 +5929*n^4 +616*n^6 -96*n^7 +16*n^8)/2520: n in [5..40]]; // G. C. Greubel, Sep 29 2019
  • Maple
    with(combinat); seq(`if`(n=4,1, fibonacci(2*n+10) -(138600 +133530*n +63999*n^2 + 20286*n^3 +5929*n^4 +616*n^6 -96*n^7 +16*n^8)/2520), n=4..40); # G. C. Greubel, Sep 29 2019
  • Mathematica
    Table[If[n==4, 1, Fibonacci[2*n+10] - (138600 +133530*n +63999*n^2 + 20286*n^3 +5929*n^4 +616*n^6 -96*n^7 +16*n^8)/2520], {n, 4, 40}] (* G. C. Greubel, Sep 29 2019 *)
  • PARI
    vector(40, n, my(m=n+3); if(m==4, 1, fibonacci(2*m+10) -(138600 +133530*m +63999*m^2 + 20286*m^3 +5929*m^4 +616*m^6 -96*m^7 +16*m^8)/2520) ) \\ G. C. Greubel, Sep 29 2019
  • Sage
    [1]+[fibonacci(2*n+10) -(138600 +133530*n +63999*n^2 + 20286*n^3 +5929*n^4 +616*n^6 -96*n^7 +16*n^8)/2520 for n in (5..40)] # G. C. Greubel, Sep 29 2019

Formula

From G. C. Greubel, Sep 29 2019: (Start)
a(n) = Sum_{j=0..n-3} binomial(2*n-j, j+9), with a(4) = 1.
a(n) = Fibonacci(2*n+10) - (138600 +133530*n +63999*n^2 + 20286*n^3 + 5929*n^4 + 616*n^6 - 96*n^7 + 16*n^8)/2520.
G.f.: x^4*(1 - 2*x + 175*x^2 - 33*x^3 + 408*x^4 - 614*x^5 + 587*x^6 - 414*x^7 + 201*x^8 - 64*x^9 + 12*x^10 - x^11)/((1-x)^9*(1-3*x+x^2)). (End)

Extensions

Terms a(23) onward added by G. C. Greubel, Sep 29 2019

A027954 a(n) = T(2n+1, n+2), T given by A027948.

Original entry on oeis.org

1, 5, 41, 189, 674, 2098, 6050, 16703, 44995, 119575, 315460, 829060, 2174596, 5698329, 14924829, 39081553, 102326310, 267905078, 701397990, 1836299475, 4807512695, 12586252715, 32951261576, 86267550344, 225851410184, 591286703533, 1548008726545, 4052739505253, 10610209821610
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000045, A027948.

Programs

  • GAP
    Concatenation([1], List([2..40], n-> Fibonacci(2*n+6) -(24 + 23*n + 12*n^2 + 4*n^3)/3 )); # G. C. Greubel, Sep 30 2019
  • Magma
    [1] cat [Fibonacci(2*n+6) -(24 + 23*n + 12*n^2 + 4*n^3)/3: n in [2..40]]; // G. C. Greubel, Sep 30 2019
  • Maple
    with(combinat); seq(`if`(n=1,1, fibonacci(2*n+6) -(24 +23*n +12*n^2 +4*n^3)/3), n=1..40); # G. C. Greubel, Sep 30 2019
  • Mathematica
    Table[If[n==1, 1, Fibonacci[2*n+6] - (24 +23*n +12*n^2 +4*n^3)/3], {n, 1, 40}] (* G. C. Greubel, Sep 30 2019 *)
  • PARI
    vector(40, n, if(n==1, 1, fibonacci(2*n+6) -(24 +23*n +12*n^2 + 4*n^3)/3 )) \\ G. C. Greubel, Sep 30 2019
  • Sage
    [1]+[fibonacci(2*n+6) -(24 + 23*n + 12*n^2 + 4*n^3)/3 for n in (2..40)] # G. C. Greubel, Sep 30 2019

Formula

G.f.: x*(1 -2*x +25*x^2 -29*x^3 +19*x^4 -7*x^5 +x^6)/((1-x)^4*(1 -3*x +x^2)). - Colin Barker, Nov 25 2014
From G. C. Greubel, Sep 30 2019: (Start)
a(n) = Sum_{j=0..n-1} binomial(2*n-j+1, j+4) for n >= 2.
a(n) = Fibonacci(2*n+6) - (24 + 23*n + 12*n^2 + 4*n^3)/3 for n >= 2. (End)

Extensions

Name corrected and terms a(22) onward by G. C. Greubel, Sep 30 2019

A027955 a(n) = T(2n+1, n+3), T given by A027948.

Original entry on oeis.org

1, 7, 92, 591, 2683, 9955, 32551, 98086, 280271, 773906, 2091266, 5576298, 14750858, 38839257, 101995694, 267462041, 700813797, 1835540197, 4806538617, 12585017712, 32949712457, 86265626164, 225849041524, 591283811748, 1548005222980, 4052735290427, 10610204784368
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000045, A027948.

Programs

  • GAP
    Concatenation([1], List([3..40], n-> Fibonacci(2*n+8) -(630 +607*n +295*n^2 +90*n^3 +20*n^4 +8*n^5)/30) ); # G. C. Greubel, Sep 30 2019
  • Magma
    [1] cat [Fibonacci(2*n+8) -(630 +607*n +295*n^2 +90*n^3 +20*n^4 +8*n^5)/30: n in [3..40]]; // G. C. Greubel, Sep 30 2019
  • Maple
    with(combinat); seq(`if`(n=2,1, fibonacci(2*n+8) -(630 +607*n +295*n^2 +90*n^3 +20*n^4 +8*n^5)/30), n=2..40); # G. C. Greubel, Sep 30 2019
  • Mathematica
    Table[If[n==2, 1, Fibonacci[2*n+8] - (630 +607*n +295*n^2 +90*n^3 +20*n^4 +8*n^5)/30], {n,2,40}] (* G. C. Greubel, Sep 30 2019 *)
  • PARI
    vector(40, n, my(m=n+1); if(m==2, 1, fibonacci(2*m+8) -(630 +607*m +295*m^2 +90*m^3 +20*m^4 +8*m^5)/30) ) \\ G. C. Greubel, Sep 30 2019
  • Sage
    [1]+[fibonacci(2*n+8) -(630 +607*n +295*n^2 +90*n^3 +20*n^4 +8*n^5)/30 for n in (3..40)] # G. C. Greubel, Sep 30 2019

Formula

G.f.: x^2*(1 -2*x +63*x^2 -70*x^3 +85*x^4 -71*x^5 +34*x^6 -9*x^7 +x^8)/( (1-x)^6*(1-3*x+x^2)). - Colin Barker, Nov 25 2014
From G. C. Greubel, Sep 30 2019: (Start)
a(n) = Sum_{j=0..n-2} binomial(2*n-j+1, j+6) for n >= 3.
a(n) = Fibonacci(2*n+8) - (630 +607*n +295*n^2 +90*n^3 +20*n^4 +8*n^5)/30 for n >= 3. (End)

Extensions

Name corrected and terms a(22) onward by G. C. Greubel, Sep 30 2019

A027956 a(n) = T(2n+1, n+4), T given by A027948.

Original entry on oeis.org

1, 9, 175, 1518, 8735, 39130, 148487, 502415, 1568062, 4622488, 13091798, 36067176, 97522270, 260459265, 690141333, 1819657318, 4783398669, 12551942930, 32903246829, 86201363911, 225761428636, 591165917888, 1547848480940, 4052529200192, 10609936578716, 27777538280521
Offset: 3

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000045, A027948.

Programs

  • GAP
    Concatenation([1], List([4..40], n-> Fibonacci(2*n+10) -(34650 +33360*n +16065*n^2 +5089*n^3 +1260*n^4 +280*n^5 +16*n^7)/630) ); # G. C. Greubel, Sep 30 2019
  • Magma
    [1] cat [Fibonacci(2*n+10) -(34650 +33360*n +16065*n^2 +5089*n^3 +1260*n^4 +280*n^5 +16*n^7)/630: n in [4..40]]; // G. C. Greubel, Sep 30 2019
  • Maple
    with(combinat); seq(`if`(n=3,1, fibonacci(2*n+10) -(34650 +33360*n +16065*n^2 +5089*n^3 +1260*n^4 +280*n^5 +16*n^7)/630), n=3..40); # G. C. Greubel, Sep 30 2019
  • Mathematica
    Table[If[n==3, 1, Fibonacci[2*n+10] -(34650 +33360*n +16065*n^2 +5089*n^3 +1260*n^4 +280*n^5 +16*n^7)/630], {n, 3, 40}] (* G. C. Greubel, Sep 30 2019 *)
LinearRecurrence[{11,-53,148,-266,322,-266,148,-53,11,-1},{1,9,175,1518,8735,39130,148487,502415,1568062,4622488,13091798},40] (* Harvey P. Dale, Sep 12 2021 *)
  • PARI
    vector(40, n, my(m=n+2); if(m==3, 1, fibonacci(2*m+10) -(34650 +33360*m +16065*m^2 +5089*m^3 +1260*m^4 +280*m^5 +16*m^7)/630) ) \\ G. C. Greubel, Sep 30 2019
  • Sage
    [1]+[fibonacci(2*n+10) -(34650 +33360*n +16065*n^2 +5089*n^3 +1260*n^4 +280*n^5 +16*n^7)/630 for n in (4..40)] # G. C. Greubel, Sep 30 2019

Formula

From G. C. Greubel, Sep 30 2019: (Start)
a(n) = Sum_{j=0..n-3} binomial(2*n-j+1, j+8) for n >= 4.
a(n) = Fibonacci(2*n+10) - (34650 +33360*n +16065*n^2 +5089*n^3 +1260*n^4 +280*n^5 +16*n^7)/630 for n >= 4.
G.f.: x^3*(1 -2*x +129*x^2 -78*x^3 +246*x^4 -329*x^5 +266*x^6 -148*x^7 +53*x^8 -11*x^9 +x^10)/((1-x)^8*(1-3*x+x^2)). (End)

Extensions

Name corrected and terms a(22) onward by G. C. Greubel, Sep 30 2019

A027957 a(n) = greatest number in row n of array T given by A027948.

Original entry on oeis.org

1, 1, 2, 3, 7, 14, 25, 46, 97, 189, 344, 674, 1383, 2683, 4950, 9955, 20175, 39130, 72905, 148487, 298925, 580328, 1089343, 2233409, 4478413, 8705686, 16438345, 33822205, 67650909, 131688362, 251448212, 515037942, 1028483089, 2004688605, 3860656125, 7878708566, 15715540623, 30670416703, 59451560083
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A027948.

Programs

  • Magma
    A027948:= func< n,k | k eq n select 1 else (&+[Binomial(n-j, 2*(n-k-j)-1): j in [0..n-k]]) >;
b:= func< n | [A027948(n,k): k in [0..n]] >;
A027957:= func< n | Max( b(n) ) >;
[A027957(n): n in [0..50]]; // G. C. Greubel, Jun 08 2025
  • Mathematica
    A027948[n_, k_]:= A027948[n, k]= If[k==n, 1, Sum[Binomial[n-j, 2*(n-k-j)-1], {j,0,n- k}]];
b[n_]:= b[n]= Table[A027948[n,k], {k,0,n}]//Union;
A027957[n_]:= Max[b[n]];
Table[A027957[n], {n,0,50}] (* G. C. Greubel, Jun 07 2025 *)
  • SageMath
    @CachedFunction
def A027948(n, k):
    if (k==n): return 1
    else: return sum(binomial(n-j, 2*(n-k-j)-1) for j in (0..n-k))
def b(n): return sorted(set(flatten([ A027948(n,k) for k in range(n+1)])))
def A027957(n): return max(b(n))
print([A027957(n) for n in range(51)]) # G. C. Greubel, Jun 07 2025

Extensions

More terms added by G. C. Greubel, Jun 07 2025

A027958 a(n) = T(n,m) + T(n,m+1) + ... + T(n,n), where m = floor((n+2)/2), T given by A027948.

Original entry on oeis.org

1, 1, 4, 5, 20, 32, 95, 169, 424, 793, 1816, 3488, 7583, 14789, 31172, 61357, 126892, 251200, 513343, 1019921, 2068496, 4119281, 8313584, 16580800, 33358015, 66594637, 133703500, 267089189, 535524644, 1070217248, 2143959071
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

a(n) is the sum of the terms of the 2nd half of the n-th row of the A027948 triangle. - Michel Marcus, Oct 01 2019

Links

Crossrefs

Cf. A000045, A027948.

Programs

  • GAP
    F:=Fibonacci;; List([1..40], n-> (3 +(-1)^n +2^(n+1) -(-1)^n*F(n+1) -F(n+4))/2); # G. C. Greubel, Sep 30 2019
  • Magma
    F:=Fibonacci; [(3 +(-1)^n +2^(n+1) -(-1)^n*F(n+1) -F(n+4))/2: n in [1..40]]; // G. C. Greubel, Sep 30 2019
  • Maple
    f:= combinat[fibonacci]: seq((3 +(-1)^n +2^(n+1) -(-1)^n*f(n+1) -f(n+4))/2, n=1..40); # G. C. Greubel, Sep 30 2019
  • Mathematica
    Table[(3 +(-1)^n +2^(n+1) -(-1)^n*Fibonacci[n+1] -Fibonacci[n+4])/2, {n,40}] (* G. C. Greubel, Sep 30 2019 *)
  • PARI
    vector(40, n, f=fibonacci; (3 +(-1)^n +2^(n+1) -(-1)^n*f(n+1) -f(n+4))/2 ) \\ G. C. Greubel, Sep 30 2019
  • Sage
    f=fibonacci; [(3 +(-1)^n +2^(n+1) -(-1)^n*f(n+1) -f(n+4))/2 for n in (1..40)] # G. C. Greubel, Sep 30 2019

Formula

G.f.: x*(1 -x -2*x^2 + x^3 +6*x^4 -2*x^6)/((1-2*x)*(1-x^2)(1+x-x^2)*(1-x-x^2)).
a(n) = (3 +(-1)^n +2^(n+1) -(-1)^n*Fibonacci(n+1) -Fibonacci(n+4))/2. - G. C. Greubel, Sep 30 2019

A027959 a(n) = Sum_{k=m..n} T(k,n-k), where m = floor((n+1)/2); a(n) is the n-th diagonal-sum of left justified array T given by A027948.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 16, 27, 37, 59, 85, 129, 192, 285, 428, 634, 949, 1412, 2104, 3140, 4671, 6973, 10378, 15478, 23058, 34362, 51216, 76305, 113736, 169465, 252561, 376362, 560851, 835821, 1245503, 1856132, 2765976, 4121947
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A027948.

Programs

  • GAP
    a:=[1,1,2,3,5,7];; for n in [7..40] do a[n]:=3*a[n-2]+a[n-3] -3*a[n-4]-a[n-5]+a[n-6]; od; a; # G. C. Greubel, Sep 30 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x*(1+x-x^2-x^3+x^4)/((1-x^2)*(1-2*x^2-x^3+x^4)) )); // G. C. Greubel, Sep 30 2019
  • Maple
    seq(coeff(series(x*(1+x-x^2-x^3+x^4)/((1-x^2)*(1-2*x^2-x^3+x^4)), x, n+1), x, n), n = 1..40); # G. C. Greubel, Sep 30 2019
  • Mathematica
    T[n_, k_]:= If[k==n, 1, Sum[Binomial[k+j, 2*j-1], {j, 0, n-k}]]; Table[Sum[T[k, n-k], {k, Floor[(n-1)/2], n}], {n,0,40}] (* G. C. Greubel, Sep 30 2019 *)
  • PARI
    my(x='x+O('x^40)); Vec(x*(1+x-x^2-x^3+x^4)/((1-x^2)*(1-2*x^2-x^3+x^4))) \\ G. C. Greubel, Sep 30 2019
  • Sage
    def A027959_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x-x^2-x^3+x^4)/((1-x^2)*(1-2*x^2-x^3+x^4)) ).list()
a=A027959_list(40); a[1:] # G. C. Greubel, Sep 30 2019

Formula

G.f.: x*(1+x-x^2-x^3+x^4)/((1-x)*(1+x)*(1-2*x^2-x^3+x^4)). - Colin Barker, Nov 25 2014
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