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.

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A053296 Partial sums of A053295.

Original entry on oeis.org

1, 8, 37, 129, 376, 967, 2267, 4950, 10220, 20175, 38403, 70954, 127921, 226007, 392688, 672959, 1140260, 1914166, 3189022, 5280288, 8699540, 14275838, 23352118, 38102976, 62048869, 100888126, 163843187, 265838881, 431026972, 698489013, 1131463777, 1832277574, 2966502032, 4802042229
Offset: 0

Views

Author

Barry E. Williams, Mar 04 2000

Keywords

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.

Links

Crossrefs

Cf. A000045, A053739, A014166, A136431, A228074.
Right-hand column 14 of triangle A011794.

Programs

  • Magma
    [(&+[Binomial(n+7-j, n-2*j): j in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, May 24 2018
  • Mathematica
    Table[Sum[Binomial[n+7-j, n-2*j], {j, 0, Floor[n/2]}], {n, 0, 50}] (* G. C. Greubel, May 24 2018 *)
  • PARI
    for(n=0, 30, print1(sum(j=0, floor(n/2), binomial(n+7-j, n-2*j)), ", ")) \\ G. C. Greubel, May 24 2018

Formula

a(n) = Sum_{i=0..floor(n/2)} C(n+7-i, n-2i), n >= 0.
a(n) = a(n-1) + a(n-2) + C(n+6,6); n >= 0, with a(-1) = 0.
From G. C. Greubel, Oct 21 2024: (Start)
a(n) = Fibonacci(n+15) - Sum_{j=0..6} Fibonacci(14-2*j)*binomial(n+j,j).
a(n) = Fibonacci(n+15) - (1/6!)*(n^6 + 39*n^5 + 685*n^4 + 7185*n^3 + 48994*n^2 + 209496*n + 438480).
G.f.: 1/((1-x)^7*(1 - x - x^2)). (End)

Extensions

Terms a(28) onward added by G. C. Greubel, May 24 2018

A027927 Number of plane regions after drawing (in general position) a convex n-gon and all its diagonals.

Original entry on oeis.org

1, 2, 5, 12, 26, 51, 92, 155, 247, 376, 551, 782, 1080, 1457, 1926, 2501, 3197, 4030, 5017, 6176, 7526, 9087, 10880, 12927, 15251, 17876, 20827, 24130, 27812, 31901, 36426, 41417, 46905, 52922, 59501, 66676, 74482, 82955, 92132, 102051, 112751, 124272, 136655, 149942, 164176, 179401
Offset: 2

Views

Author

Clark Kimberling

Keywords

Comments

For n>=1, a(n+1) is the number of Grassmannian permutations that avoid a pattern, sigma, where sigma is a pattern of size 5 with exactly one descent. - Jessica A. Tomasko, Nov 15 2022

Examples

			a(2)=1 (segment traced twice has only exterior).

Links

Crossrefs

Cf. A006522 (does not count exterior of n-gon).
Cf. A000045, A000217, A004006, A027926, A228074.

Programs

  • GAP
    List([2..50], n-> (n^4 -6*n^3 +23*n^2 -42*n +48)/24); # G. C. Greubel, Sep 06 2019
  • Magma
    [(n^4 -6*n^3 +23*n^2 -42*n +48)/24: n in [2..50]]; // G. C. Greubel, Sep 06 2019
  • Maple
    seq((n^4 -6*n^3 +23*n^2 -42*n +48)/24, n=2..50); # G. C. Greubel, Sep 06 2019
  • Mathematica
    LinearRecurrence[{5,-10,10,-5,1 }, {1,2,5,12,26}, 50] (* Vincenzo Librandi, Feb 01 2012 *)
S[n_] :=n*(n+1)/2; Table[S[S[n]+2]/3, {n, 0, 50}] (* Waldemar Puszkarz, Jan 22 2016 *)
  • PARI
    a(n)=n*(n^3-6*n^2+23*n-42)/24+2 \\ Charles R Greathouse IV, Jan 31 2012
  • Sage
    [(n^4 -6*n^3 +23*n^2 -42*n +48)/24 for n in (2..50)] # G. C. Greubel, Sep 06 2019

Formula

a(n) = T(n, 2*n-4), T given by A027926.
a(n) = 1 + binomial(n, 4) + binomial(n-1, 2) = (n^4 - 6*n^3 + 23*n^2 - 42*n + 48)/24.
G.f.: x^2*(1 -3*x +5*x^2 -3*x^3 +x^4)/(1-x)^5. - Colin Barker, Jan 31 2012
a(n) = (1/6)*A152950(n-1)*A152948(n). - Bruno Berselli, Jan 31 2012
a(n) = A000217(A000217(n-2)+2)/3, a(n+1) - a(n) = A004006(n-1) for n > 2. - Waldemar Puszkarz, Jan 22 2016 [Adjusted for offset by Peter Munn, Jan 10 2023]
a(n) = 1 + Sum {i=3..5} binomial(n-1, i-1). - Jessica A. Tomasko, Nov 15 2022

Extensions

New name from Len Smiley, Oct 19 2001

A053308 Partial sums of A053296.

Original entry on oeis.org

1, 9, 46, 175, 551, 1518, 3785, 8735, 18955, 39130, 77533, 148487, 276408, 502415, 895103, 1568062, 2708322, 4622488, 7811510, 13091798, 21791338, 36067176, 59419294, 97522270, 159571139, 260459265, 424302452, 690141333
Offset: 0

Views

Author

Barry E. Williams, Mar 06 2000

Keywords

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.

Links

Crossrefs

Cf. A053296, A053295, A136431.
Cf. A228074.

Programs

  • Magma
    [(&+[Binomial(n+8-j, n-2*j): j in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, May 24 2018
  • Mathematica
    Table[Sum[Binomial[n+8-j, n-2j], {j, 0, Floor[n/2]}], {n, 0, 50}] (* G. C. Greubel, May 24 2018 *)
  • PARI
    for(n=0, 30, print1(sum(j=0, floor(n/2), binomial(n+8-j, n-2*j)), ", ")) \\ G. C. Greubel, May 24 2018

Formula

a(n) = Sum_{i=0..floor(n/2)} C(n+8-i, n-2i), n >= 0.
a(n) = a(n-1) + a(n-2) + C(n+7,7); n >= 0; a(-1)=0.

A053309 Partial sums of A053308.

Original entry on oeis.org

1, 10, 56, 231, 782, 2300, 6085, 14820, 33775, 72905, 150438, 298925, 575333, 1077748, 1972851, 3540913, 6249235, 10871723, 18683233, 31775031, 53566369, 89633545, 149052839, 246575109, 406146248, 666605513, 1090907965
Offset: 0

Views

Author

Barry E. Williams, Mar 06 2000

Keywords

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.

Links

Crossrefs

Cf. A053296, A053295, A136431.
Cf. A228074.

Programs

  • Magma
    [(&+[Binomial(n+9-j, n-2*j): j in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, May 24 2018
  • Mathematica
    Table[Sum[Binomial[n+9-j, n-2j], {j, 0, Floor[n/2]}], {n, 0, 50}] (* G. C. Greubel, May 24 2018 *)
  • PARI
    for(n=0, 30, print1(sum(j=0, floor(n/2), binomial(n+9-j, n-2*j)), ", ")) \\ G. C. Greubel, May 24 2018

Formula

a(n) = Sum_{i=0..floor(n/2)} C(n+9-i, n-2i), n >= 0.
a(n) = a(n-1) + a(n-2) + C(n+8,8); n >= 0; a(-1)=0.
G.f.: 1/((x^2 + x - 1)*(x-1)^9). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009

A027933 a(n) = T(n, 2*n-10), T given by A027926.

Original entry on oeis.org

1, 2, 5, 13, 34, 89, 232, 596, 1490, 3588, 8273, 18228, 38403, 77533, 150438, 281403, 509015, 892926, 1523117, 2532359, 4112704, 6536993, 10186540, 15586342, 23449376, 34731776, 50700937, 73018870, 103843433, 145950389, 202879594, 279108997, 380260541
Offset: 5

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A027926, A228074.

Programs

  • GAP
    List([5..40], n-> Sum([0..5], k-> Binomial(n-k, 10-2*k)) ); # G. C. Greubel, Sep 27 2019
  • Magma
    [&+[Binomial(n-k, 10-2*k): k in [0..5]] : n in [5..40]]; // G. C. Greubel, Sep 27 2019
  • Maple
    seq(add(binomial(n-k, 10-2*k), k=0..5), n=5..40); # G. C. Greubel, Sep 27 2019
  • Mathematica
    Table[Sum[Binomial[n-k, 10-2k], {k,0,5}], {n,5,40}] (* or *)
Drop[#, 5] &@ CoefficientList[Series[x^5(1-x+x^2)(1-5x+9x^2-5x^3+x^4)(1- 3x+5x^2-3x^3+x^4)/(1-x)^11, {x, 0, 37}], x] (* Michael De Vlieger, Feb 17 2016 *)
  • PARI
    Vec(x^5*(1-x+x^2)*(1-5*x+9*x^2-5*x^3+x^4)*(1-3*x+5*x^2-3*x^3+x^4) / (1-x)^11 + O(x^40)) \\ Colin Barker, Feb 17 2016
  • PARI
    vector(40, n, sum(k=0,5, binomial(n+4-k, 10-2*k)) ) \\ G. C. Greubel, Sep 27 2019
  • Sage
    [sum(binomial(n-k, 10-2*k) for k in (0..5)) for n in (5..40)] # G. C. Greubel, Sep 27 2019

Formula

a(n) = Sum_{k=0..5} binomial(n-k, 10-2*k). - Len Smiley, Oct 20 2001
a(n) = 34 -9161*n/280 -101897*n^3/20160 +794293*n^2/50400 -287*n^5/1280 +438209*n^4/362880 +5593*n^6/172800 -47*n^7/13440 -n^9/80640 +n^8/3780 +n^10/3628800. - R. J. Mathar, Oct 05 2009
G.f.: x^5*(1-x+x^2)*(1-5*x+9*x^2-5*x^3+x^4)*(1-3*x+5*x^2-3*x^3+x^4) / (1-x)^11. - Colin Barker, Feb 17 2016

A027988 Greatest number in row n of array T given by A027926.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 26, 51, 97, 189, 365, 709, 1383, 2683, 5270, 10220, 20175, 39130, 77533, 150438, 298925, 580328, 1155661, 2245004, 4478413, 8705686, 17390359, 33828704, 67650909, 131901368, 263589730, 515037942, 1028483089
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Also greatest number in row n of the Fibonacci-Pascal matrix A105809. - Jason Riggle (jriggle(AT)uchicago.edu), Aug 22 2006
For n > 0 also largest term in row n of the triangle in A228074. - Reinhard Zumkeller, Aug 15 2013

Programs

A027928 a(n) = T(n, 2*n-5), T given by A027926.

Original entry on oeis.org

1, 3, 8, 20, 46, 97, 189, 344, 591, 967, 1518, 2300, 3380, 4837, 6763, 9264, 12461, 16491, 21508, 27684, 35210, 44297, 55177, 68104, 83355, 101231, 122058, 146188, 174000, 205901, 242327, 283744, 330649, 383571, 443072
Offset: 3

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A228074, A000045.

Programs

  • GAP
    List([3..40], n-> (n-2)*(n^4 -8*n^3 +39*n^2 -92*n +180)/120); # G. C. Greubel, Sep 06 2019
  • Magma
    [(n-2)*(n^4-8*n^3+39*n^2-92*n+180)/120: n in [3..40]]; // Vincenzo Librandi, Apr 22 2012
  • Maple
    seq(binomial(n,n-1)+binomial(n+1,n-2)+binomial(n+2,n-3), n=1..35); # Zerinvary Lajos, May 29 2007
  • Mathematica
    CoefficientList[Series[(1-3*x+5*x^2-3*x^3+x^4)/(1-x)^6,{x,0,40}],x] (* Vincenzo Librandi, Apr 22 2012 *)
  • PARI
    vector(40, n, m=n+2; n*(m^4 -8*m^3 +39*m^2 -92*m +180)/120) \\ G. C. Greubel, Sep 06 2019
  • Sage
    [(n-2)*(n^4 -8*n^3 +39*n^2 -92*n +180)/120 for n in (3..40)] # G. C. Greubel, Sep 06 2019

Formula

a(n) = (n-2)*(n^4 - 8*n^3 + 39*n^2 - 92*n + 180)/120.
a(n) = C(n,n-1) + C(n+1,n-2) + C(n+2,n-3) with offset 1. - Zerinvary Lajos, May 29 2007
G.f.: x^3*(1 - 3*x + 5*x^2 - 3*x^3 + x^4)/(1-x)^6. - Colin Barker, Mar 18 2012
E.g.f.: 3 + x -(360 - 240*x + 60*x^2 - 20*x^3 - x^5)*exp(x)/120. - G. C. Greubel, Sep 06 2019

A027929 a(n) = T(n, 2*n-6), T given by A027926.

Original entry on oeis.org

1, 2, 5, 13, 33, 79, 176, 365, 709, 1300, 2267, 3785, 6085, 9465, 14302, 21065, 30329, 42790, 59281, 80789, 108473, 143683, 187980, 243157, 311261, 394616, 495847, 617905, 764093, 938093, 1143994, 1386321, 1670065, 2000714
Offset: 3

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A228074.

Programs

  • GAP
    List([3..40], n-> (3600 -3420*n +1684*n^2 -525*n^3 +115*n^4 -15*n^5 +n^6)/720); G. C. Greubel, Sep 06 2019
  • Magma
    [(3600 -3420*n +1684*n^2 -525*n^3 +115*n^4 -15*n^5 +n^6)/720: n in [3..40]]; // G. C. Greubel, Sep 06 2019
  • Maple
    seq((3600 -3420*n +1684*n^2 -525*n^3 +115*n^4 -15*n^5 +n^6)/720, n=3..40); # G. C. Greubel, Sep 06 2019
  • Mathematica
    CoefficientList[Series[(1-x+x^2)(1-4x+7x^2-4x^3+x^4)/(1-x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 18 2013 *)
  • PARI
    vector(40, n, m=n+2; (3600 -3420*m +1684*m^2 -525*m^3 +115*m^4 -15*m^5 +m^6)/720) \\ G. C. Greubel, Sep 06 2019
  • Sage
    [(3600 -3420*n +1684*n^2 -525*n^3 +115*n^4 -15*n^5 +n^6)/720 for n in (3..40)] # G. C. Greubel, Sep 06 2019

Formula

a(n) = Sum_{k=0..3} binomial(n-k, 6-2*k). - Len Smiley, Oct 20 2001
From Colin Barker, May 01 2012: (Start)
a(n) = (3600 -3420*n +1684*n^2 -525*n^3 +115*n^4 -15*n^5 +n^6)/720.
G.f.: x^3*(1-x+x^2)*(1-4*x+7*x^2-4*x^3+x^4)/(1-x)^7. (End)
E.g.f.: (3600 - 2160*x + 720*x^2 - 120*x^3 + 30*x^4 + x^6)*exp(x)/720 - 5 + 2*x - x^2/2. - G. C. Greubel, Sep 06 2019

A027930 a(n) = T(n, 2*n-7), T given by A027926.

Original entry on oeis.org

1, 3, 8, 21, 54, 133, 309, 674, 1383, 2683, 4950, 8735, 14820, 24285, 38587, 59652, 89981, 132771, 192052, 272841, 381314, 524997, 712977, 956134, 1267395, 1662011, 2157858, 2775763, 3539856, 4477949, 5621943, 7008264, 8678329, 10679043, 13063328, 15890685
Offset: 4

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A228074.

Programs

  • GAP
    List([4..40], n-> Binomial(n-1, n-7) + (n-3)*((n-3)^4 +15*(n-3)^2 +104)/120); # G. C. Greubel, Sep 06 2019
  • Magma
    [Binomial(n-1, n-7) + (n-3)*((n-3)^4 +15*(n-3)^2 +104)/120: n in [4..40]]; // G. C. Greubel, Sep 06 2019
  • Maple
    seq(binomial(n-3,n-4)+binomial(n-2,n-5)+binomial(n-1,n-6)+binomial(n,n-7) , n=4..50); # Zerinvary Lajos, May 29 2007
  • Mathematica
    Table[Total[Binomial[First[#],Last[#]]&/@Table[{n+i,n-1-i},{i,0,3}]],{n,35}] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1}, {1,3,8,21,54,133,309,674}, 35] (* Harvey P. Dale, Jun 23 2011 *)
  • PARI
    vector(40, n, binomial(n+3, n-4) + n*(n^4 +15*n^2 +104)/120) \\ G. C. Greubel, Sep 06 2019
  • Sage
    [binomial(n-1, n-7) + (n-3)*((n-3)^4 +15*(n-3)^2 +104)/120 for n in (4..40)] # G. C. Greubel, Sep 06 2019

Formula

a(n) = Sum_{k=0..3} binomial(n-k, 7-2k). - Len Smiley, Oct 20 2001
a(n) = C(n-3,n-4)+C(n-2,n-5)+C(n-1,n-6)+C(n,n-7). - Zerinvary Lajos, May 29 2007
From R. J. Mathar, Oct 05 2009: (Start)
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
G.f.: x^4*(1 - x + x^2)*(1 - 4*x + 7*x^2 - 4*x^3 + x^4)/(1-x)^8. (End)
From G. C. Greubel, Sep 06 2019: (Start)
a(n) = binomial(n-1, n-7) + (n-3)*((n-3)^4 + 15*(n-3)^2 + 104)/120.
E.g.f.: x*(5040 + 2520*x + 1680*x^2 + 630*x^3 + 168*x^4 + 21*x^5 + x^6)*exp(x)/5040. (End)

A027931 T(n, 2n-8), T given by A027926.

Original entry on oeis.org

1, 2, 5, 13, 34, 88, 221, 530, 1204, 2587, 5270, 10220, 18955, 33775, 58060, 96647, 156299, 246280, 379051, 571103, 843944, 1225258, 1750255, 2463232, 3419366, 4686761, 6348772, 8506630, 11282393, 14822249, 19300198, 24922141
Offset: 4

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A027926, A228074.

Programs

  • GAP
    List([4..40], n-> Sum([0..4], k-> Binomial(n-k, 8-2*k)) ); # G. C. Greubel, Sep 27 2019
  • Magma
    [&+[Binomial(n-k, 8-2*k): k in [0..4]] : n in [4..40]]; // G. C. Greubel, Sep 27 2019
  • Maple
    A027931 := proc(n)
    add(binomial(n-k,8-2*k),k=0..4) ;
end proc: # R. J. Mathar, Oct 31 2015
  • Mathematica
    Sum[Binomial[Range[4,40] -k, 8-2*k], {k,0,4}] (* G. C. Greubel, Sep 27 2019 *)
  • PARI
    vector(40, n, sum(k=0,4, binomial(n+3-k, 8-2*k)) ) \\ G. C. Greubel, Sep 27 2019
  • Sage
    [sum(binomial(n-k, 8-2*k) for k in (0..4)) for n in (4..40)] # G. C. Greubel, Sep 27 2019

Formula

a(n) = Sum_{k=0..4} binomial(n-k, 8-2*k). - Len Smiley, Oct 20 2001
G.f.: x^4*(1 -7*x +23*x^2 -44*x^3 +55*x^4 -44*x^5 +23*x^6 -7*x^7+ x^8) / (1-x)^9 . - R. J. Mathar, Oct 31 2015
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