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 14 results. Next

A027936 Uniquification of array T given by A027935.

Original entry on oeis.org

1, 2, 4, 5, 7, 11, 12, 13, 16, 22, 26, 29, 33, 34, 37, 46, 51, 56, 67, 79, 88, 89, 92, 106, 121, 137, 154, 155, 172, 176, 191, 211, 221, 232, 233, 247, 254, 277, 301, 326, 352, 365, 376, 379, 407, 436, 466, 497, 529, 530, 551, 562, 596
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A027935.

Programs

  • Mathematica
    A027935[n_, k_]:= A027935[n, k]= Sum[Binomial[n-j, 2*(n-k-j)], {j,0,Floor[(2*n-2*k+ 1)/2]}];
A027936= Table[A027935[n,k], {n,0,225}, {k,0,n}]//Flatten//Union;
Table[A027936[[n]], {n,100}] (* G. C. Greubel, Jun 06 2025 *)
  • SageMath
    @CachedFunction
def A027935(n,k): return sum(binomial(n-j, 2*(n-k-j)) for j in range(int((2*n-2*k+1)/2+1)) )
A027936 = sorted(set(flatten([[ A027935(n,k) for k in range(n+1)] for n in range(103)])))
print([A027936[n] for n in range(100)]) # G. C. Greubel, Jun 06 2025

A027937 a(n) = T(2*n, n+1), T given by A027935.

Original entry on oeis.org

1, 7, 26, 79, 221, 596, 1581, 4163, 10926, 28635, 75001, 196392, 514201, 1346239, 3524546, 9227431, 24157781, 63245948, 165580101, 433494395, 1134903126, 2971215027, 7778742001, 20365011024, 53316291121
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000045, A027935.

Programs

  • GAP
    List([1..30], n-> Fibonacci(2*n+3) -2*(n+1) ); # G. C. Greubel, Sep 27 2019
  • Magma
    [Fibonacci(2*n+3) - 2*n - 2: n in [1..30]]; // Vincenzo Librandi, Apr 18 2011
  • Maple
    with(combinat); seq(fibonacci(2*n+3) -2*(n+1), n=1..30); # G. C. Greubel, Sep 27 2019
  • Mathematica
    Table[Fibonacci[2*n+3]-2*(n+1), {n,30}] (* G. C. Greubel, Sep 27 2019 *)
  • PARI
    vector(30, n, fibonacci(2*n+3)-2*(n+1)) \\ G. C. Greubel, Sep 27 2019
  • Sage
    [fibonacci(2*n+3) -2*(n+1) for n in (1..30)] # G. C. Greubel, Sep 27 2019

Formula

a(n) = Fibonacci(2*n+3) - 2*n - 2.
From R. J. Mathar, Feb 06 2010: (Start)
a(n) = 5*a(n-1) - 8*a(n-2) + 5*a(n-3) - a(n-4);
G.f.: x*(1 + 2*x - x^2)/((1-3*x+x^2)*(1-x)^2). (End)

A027938 a(n) = T(2n, n+2), T given by A027935.

Original entry on oeis.org

1, 16, 92, 365, 1204, 3588, 10093, 27476, 73440, 194345, 511576, 1342936, 3520457, 9222440, 24151764, 63238773, 165571628, 433484476, 1134891605, 2971201740, 7778726776
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000045, A027935.

Programs

  • GAP
    List([2..30], n-> Fibonacci(2*n+5) - (4*n^3 +6*n^2 +14*n +15)/3 ); # G. C. Greubel, Sep 28 2019
  • Magma
    [Fibonacci(2*n+5) - (4*n^3 +6*n^2 +14*n +15)/3: n in [2..30]]; // G. C. Greubel, Sep 28 2019
  • Maple
    with(combinat); seq(fibonacci(2*n+5) - (4*n^3 +6*n^2 +14*n +15)/3, n=2..30); # G. C. Greubel, Sep 28 2019
  • Mathematica
    Table[Fibonacci[2*n+5] -(4*n^3 +6*n^2 +14*n +15)/3, {n,2,30}] (* G. C. Greubel, Sep 28 2019 *)
  • PARI
    vector(30, n, my(m=n+1); fibonacci(2*m+5) - (4*m^3 +6*m^2 +14*m +15)/3) \\ G. C. Greubel, Sep 28 2019
  • Sage
    [fibonacci(2*n+5) - (4*n^3 +6*n^2 +14*n +15)/3 for n in (2..30)] # G. C. Greubel, Sep 28 2019

Formula

G.f.: x^2*(1+9*x-x^2-x^3) / ((1-x)^4*(1-3*x+x^2)). - Colin Barker, Dec 10 2015
a(n) = Fibonacci(2*n+5) - (4*n^3 + 6*n^2 + 14*n + 15)/3. - G. C. Greubel, Sep 28 2019

A027939 a(n) = T(2*n, n+3), T given by A027935.

Original entry on oeis.org

1, 29, 247, 1300, 5270, 18228, 56967, 166681, 467301, 1274856, 3419252, 9076280, 23945893, 62955061, 165188091, 432974764, 1134224458, 2970340412, 7777628427, 20363608737, 53314542953, 139581703056, 365432651464, 956718812272, 2504726904937, 6557465674125
Offset: 3

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000045, A027935.

Programs

  • GAP
    List([3..30], n-> Fibonacci(2*n+7) - (195 +186*n +90*n^2 +35*n^3 +4*n^5)/15 ); # G. C. Greubel, Sep 28 2019
  • Magma
    [Fibonacci(2*n+7) - (195 +186*n +90*n^2 +35*n^3 +4*n^5)/15: n in [3..30]]; // G. C. Greubel, Sep 28 2019
  • Maple
    with(combinat); seq(fibonacci(2*n+7) - (195 +186*n +90*n^2 +35*n^3 +4*n^5)/15, n=3..30); # G. C. Greubel, Sep 28 2019
  • Mathematica
    Table[Fibonacci[2*n+7] -(195 +186*n +90*n^2 +35*n^3 +4*n^5)/15, {n,3,30}] (* G. C. Greubel, Sep 28 2019 *)
  • PARI
    vector(30, n, my(m=n+2); fibonacci(2*m+7) - (195 +186*m +90*m^2 +35*m^3 +4*m^5)/15) \\ G. C. Greubel, Sep 28 2019
  • Sage
    [fibonacci(2*n+7) - (195 +186*n +90*n^2 +35*n^3 +4*n^5)/15 for n in (3..30)] # G. C. Greubel, Sep 28 2019

Formula

G.f.: x^3*(1+20*x+20*x^2-8*x^3-x^4) / ((1-x)^6*(1-3*x+x^2)). - Colin Barker, Feb 20 2016
a(n) = Fibonacci(2*n+7) - (195 + 186*n + 90*n^2 + 35*n^3 + 4*n^5)/15. - G. C. Greubel, Sep 28 2019

Extensions

Terms a(23) onward added by G. C. Greubel, Sep 28 2019

A027940 a(n) = T(2*n, n+4), T given by A027935.

Original entry on oeis.org

1, 46, 551, 3785, 18955, 77533, 276408, 895103, 2708322, 7811510, 21791338, 59419294, 159571139, 424302452, 1121168305, 2951121095, 7749900701, 20324325571, 53259796514, 139506540045, 365330860180, 956582678652, 2504546934692, 6557230277964, 17167369784405
Offset: 4

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000045, A027935.

Programs

  • GAP
    List([4..40], n-> Fibonacci(2*n+9) - (21420 +20571*n +9961*n^2 +3304*n^3 +490*n^4 +364*n^5 -56*n^6 +16*n^7)/630 ); # G. C. Greubel, Sep 28 2019
  • Magma
    [Fibonacci(2*n+9) - (21420 +20571*n +9961*n^2 +3304*n^3 +490*n^4 +364*n^5 -56*n^6 +16*n^7)/630: n in [4..40]]; // G. C. Greubel, Sep 28 2019
  • Maple
    with(combinat); seq(fibonacci(2*n+9) - (21420 +20571*n +9961*n^2 +3304*n^3 +490*n^4 +364*n^5 -56*n^6 +16*n^7)/630, n=4..40); # G. C. Greubel, Sep 28 2019
  • Mathematica
    Table[Fibonacci[2*n+9] - (21420 +20571*n +9961*n^2 +3304*n^3 +490*n^4 +364*n^5 -56*n^6 +16*n^7)/630, {n,4,40}] (* G. C. Greubel, Sep 28 2019 *)
  • PARI
    vector(40, n, my(m=n+3); fibonacci(2*m+9) - (21420 +20571*m +9961*m^2 +3304*m^3 +490*m^4 +364*m^5 -56*m^6 +16*m^7)/630) \\ G. C. Greubel, Sep 28 2019
  • Sage
    [fibonacci(2*n+9) - (21420 +20571*n +9961*n^2 +3304*n^3 +490*n^4 +364*n^5 -56*n^6 +16*n^7)/630 for n in (4..40)] # G. C. Greubel, Sep 28 2019

Formula

From G. C. Greubel, Sep 28 2019: (Start)
a(n) = Fibonacci(2*n+9) - (21420 + 20571*n + 9961*n^2 + 3304*n^3 + 490*n^4 + 364*n^5 - 56*n^6 + 16*n^7)/630.
G.f.: x^4*(1 + 35*x + 98*x^2 + 14*x^3 - 19*x^4 - x^5)/((1-x)^8*(1 - 3*x + x^2)). (End)

Extensions

Terms a(23) onward added by G. C. Greubel, Sep 28 2019

A027942 a(n) = T(2n+1, n+2), T given by A027935.

Original entry on oeis.org

1, 11, 51, 176, 530, 1490, 4043, 10773, 28445, 74770, 196116, 513876, 1345861, 3524111, 9226935, 24157220, 63245318, 165579398, 433493615, 1134902265, 2971214081, 7778740966, 20365009896, 53316289896, 139583861065, 365435294675, 956722024443, 2504730780248
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000045, A027935.

Programs

  • GAP
    List([1..40], n-> Fibonacci(2*n+5) -(2*n^2+5*n+5) ); # G. C. Greubel, Sep 28 2019
  • Magma
    [Fibonacci(2*n+5)-2*n^2-5*n-5: n in [1..30]]; // Vincenzo Librandi, Apr 18 2011
  • Maple
    with(combinat): seq(fibonacci(2*n+5) -(2*n^2+5*n+5), n=1..40); # G. C. Greubel, Sep 28 2019
  • Mathematica
    CoefficientList[Series[(1+5x-2x^2)/((1-x)^3*(1-3x+x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 18 2013 *)
LinearRecurrence[{6,-13,13,-6,1},{1,11,51,176,530},40] (* Harvey P. Dale, Aug 18 2017 *)
  • PARI
    vector(40, n, fibonacci(2*n+5) -(2*n^2+5*n+5) ) \\ G. C. Greubel, Sep 28 2019
  • Sage
    [fibonacci(2*n+5) -(2*n^2+5*n+5) for n in (1..40)] # G. C. Greubel, Sep 28 2019

Formula

a(n) = Fibonacci(2*n+5) - 2*n^2 - 5*n - 5.
G.f.: x*(1+5*x-2*x^2)/((1-x)^3*(1-3*x+x^2)). - Colin Barker, Sep 20 2012

Extensions

More terms from Vincenzo Librandi, Oct 18 2013

A027943 a(n) = T(2*n+1, n+3), T given by A027935.

Original entry on oeis.org

1, 22, 155, 709, 2587, 8273, 24416, 68595, 187030, 500950, 1327986, 3499982, 9195035, 24115804, 63192397, 165512723, 433410661, 1134800215, 2971089810, 7778591025, 20364830496, 53316076892, 139583609940, 365435000524, 956721681957, 2504730383698, 6557469861231
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000045, A027935.

Programs

  • GAP
    List([2..40], n-> Fibonacci(2*n+7) - (78 +75*n +35*n^2 +12*n^3 +4*n^4)/6 ); # G. C. Greubel, Sep 28 2019
  • Magma
    [Fibonacci(2*n+7) - (78 +75*n +35*n^2 +12*n^3 +4*n^4)/6: n in [2..40]]; // G. C. Greubel, Sep 28 2019
  • Maple
    with(combinat); seq(fibonacci(2*n+7) - (78 +75*n +35*n^2 +12*n^3 +4*n^4)/6, n=2..40); # G. C. Greubel, Sep 28 2019
  • Mathematica
    Table[Fibonacci[2*n+7] - (78 +75*n +35*n^2 +12*n^3 +4*n^4)/6, {n,2,40}]
  • PARI
    vector(30, n, my(m=n+1); fibonacci(2*m+7) - (4*m^4 +12*m^3 +35*m^2 +75*m +78)/6) \\ G. C. Greubel, Sep 28 2019
  • Sage
    [fibonacci(2*n+7) - (78 +75*n +35*n^2 +12*n^3 +4*n^4)/6 for n in (2..40)] # G. C. Greubel, Sep 28 2019

Formula

G.f.: x^2*(1+14*x+5*x^2-4*x^3) / ((1-x)^5*(1-3*x+x^2)). - Colin Barker, Feb 20 2016
From G. C. Greubel, Sep 28 2019: (Start)
a(n) = Sum_{j=0..n-2} binomial(2*n-j+1, 2*(n-j-2)).
a(n) = Fibonacci(2*n+7) - (78 +75*n +35*n^2 +12*n^3 +4*n^4)/6. (End)

Extensions

Terms a(22) onward added by G. C. Greubel, Sep 28 2019

A027944 a(n) = T(2n+1, n+4), T given by A027935.

Original entry on oeis.org

1, 37, 376, 2267, 10220, 38403, 127921, 392688, 1140260, 3189022, 8699540, 23352118, 62048869, 163843187, 431026972, 1131463777, 2966502032, 7772382641, 20356549685, 53305176134, 139569431544, 365416760764, 956698453752, 2504701077772, 6557433205689, 17167634170241
Offset: 3

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000045, A027935.

Programs

  • GAP
    List([3..30], n-> Fibonacci(2*n+9) - (8*n^6 +12*n^5 +110*n^4 +465*n^3 +1412*n^2 + 2943*n +3060)/90 ); # G. C. Greubel, Sep 28 2019
  • Magma
    [Fibonacci(2*n+9) - (8*n^6 +12*n^5 +110*n^4 +465*n^3 +1412*n^2 + 2943*n +3060)/90: n in [3..30]]; // G. C. Greubel, Sep 28 2019
  • Maple
    with(combinat); seq(fibonacci(2*n+9) -(8*n^6 +12*n^5 +110*n^4 +465*n^3 +1412*n^2 + 2943*n +3060)/90, n=3..30); # G. C. Greubel, Sep 28 2019
  • Mathematica
    Table[Fibonacci[2*n+5] -(8*n^6 +12*n^5 +110*n^4 +465*n^3 +1412*n^2 + 2943*n +3060)/90, {n,3,30}] (* G. C. Greubel, Sep 28 2019 *)
  • PARI
    vector(30, n, my(m=n+2); fibonacci(2*m+9) - (8*m^6 +12*m^5 +110*m^4 +465*m^3 +1412*m^2 + 2943*m +3060)/90) \\ G. C. Greubel, Sep 28 2019
  • Sage
    [fibonacci(2*n+9) - (8*n^6 +12*n^5 +110*n^4 +465*n^3 +1412*n^2 + 2943*n +3060)/90 for n in (3..30)] # G. C. Greubel, Sep 28 2019

Formula

From G. C. Greubel, Sep 28 2019: (Start)
a(n) = Sum_{j=0..n-3} binomial(2*n-j+1, 2*(n-j-3)).
a(n) = Fibonacci(2*n+9) - (8*n^6 + 12*n^5 + 110*n^4 + 465*n^3 + 1412*n^2 + 2943*n + 3060)/90.
G.f.: x^3*(1 + 27*x + 49*x^2 - 7*x^3 - 6*x^4)/((1-x)^7*(1-3*x+x^2)). (End)

Extensions

Terms a(22) onward added by G. C. Greubel, Sep 28 2019

A027945 Greatest number in row n of array T given by A027935.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 26, 51, 92, 176, 365, 709, 1300, 2587, 5270, 10220, 18955, 38403, 77533, 150438, 281403, 575333, 1155661, 2245004, 4227273, 8684673, 17390359, 33828704, 64250459, 131901368, 263589730, 513445147, 984880747, 2013363836, 4018052441, 7836832057, 15144704167, 30860244790, 61530661493
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A027935.

Programs

  • Mathematica
    A027935[n_, k_]:= A027935[n,k]= Sum[Binomial[n-j, 2*(n-k-j)], {j,0,Floor[(2*n-2*k+ 1)/2]}];
b[n_]:= b[n]= Table[A027935[n,k], {k,0,n}]//Union;
A027945[n_]:= Max[b[n]];
Table[A027945[n], {n,0,50}] (* G. C. Greubel, Jun 06 2025 *)
  • SageMath
    @CachedFunction
def A027935(n,k): return sum(binomial(n-j, 2*(n-k-j)) for j in range(int((2*n-2*k+1)/2+1)) )
def b(n): return sorted(set(flatten([ A027935(n,k) for k in range(n+1)])))
def A027945(n): return max(b(n))
[A027945(n) for n in range(51)] # G. C. Greubel, Jun 06 2025

A027946 a(n) is the sum of the non-Fibonacci numbers in row n of array T given by A027935, computed as T(n,m) + T(n,m+1) + ... + T(n,n-1), where m = floor((n+2)/2).

Original entry on oeis.org

0, 0, 0, 4, 7, 23, 42, 106, 200, 456, 879, 1903, 3718, 7814, 15396, 31780, 62951, 128487, 255378, 517522, 1030864, 2079440, 4147935, 8342239, 16655822, 33433038, 66791052, 133899916, 267603415, 536038871, 1071563514, 2145305338
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000045, A027935.

Programs

  • GAP
    Concatenation([0], List([1..40], n-> (2^(n+1)-2-Fibonacci(n+3) -(-1)^n*Fibonacci(n))/2)); # G. C. Greubel, Sep 28 2019
  • Magma
    [0] cat [(2^(n+1)-2-Fibonacci(n+3) -(-1)^n*Fibonacci(n))/2: n in [1..40]]; // G. C. Greubel, Sep 28 2019
  • Maple
    with(combinat); seq(`if`(n=0,0, (2^(n+1)-2-fibonacci(n+3) -(-1)^n* fibonacci(n))/2), n=0..40); # G. C. Greubel, Sep 28 2019
  • Mathematica
    Table[If[n==0,0,(2^(n+1) -2 -Fibonacci[n+3] -(-1)^n*Fibonacci[n])/2], {n, 0, 40}] (* G. C. Greubel, Sep 28 2019 *)
  • PARI
    concat([0], vector(40, n, (2^(n+1)-2-fibonacci(n+3) -(-1)^n* fibonacci(n))/2)) \\ G. C. Greubel, Sep 28 2019
  • Sage
    [0]+[(2^(n+1)-2-fibonacci(n+3) -(-1)^n*fibonacci(n))/2 for n in (1..40)] # G. C. Greubel, Sep 28 2019

Formula

G.f.: x^3*(4 - 5*x - 2*x^2 + 2*x^3)/((1-x)*(1-2*x)*(1+x-x^2)*(1-x-x^2)).
From G. C. Greubel, Sep 28 2019: (Start)
a(n) = (2^(n+1) - 2 - Fibonacci(n+3) - (-1)^n*Fibonacci(n))/2, n > 0.
a(2*n) = 4^n - 1 - Fibonacci(2*n+2), n > 0.
a(2*n+1) = 2^(2*n+1) - 1 - Fibonacci(2*n+2). (End)
Showing 1-10 of 14 results. Next

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