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

A055819 Row sums of array T in A055818; twice the odd-indexed Fibonacci numbers after initial term.

Original entry on oeis.org

1, 2, 4, 10, 26, 68, 178, 466, 1220, 3194, 8362, 21892, 57314, 150050, 392836, 1028458, 2692538, 7049156, 18454930, 48315634, 126491972, 331160282, 866988874, 2269806340, 5942430146, 15557484098, 40730022148, 106632582346
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Comments

Solutions (x, y) = (a(n), a(n+1)) satisfying x^2 + y^2 = 3xy - 4. - Michel Lagneau, Feb 01 2014
Except for the first term, positive values of x (or y) satisfying x^2 - 18xy + y^2 + 256 = 0. - Colin Barker, Feb 16 2014

Links

Crossrefs

Essentially the same as A052995.

Programs

  • GAP
    a:=[2,4];; for n in [3..30] do a[n]:=3*a[n-1]-a[n-2]; od; Concatenation([1], a); # G. C. Greubel, Jan 22 2020
  • Magma
    I:=[2,4]; [1] cat [n le 2 select I[n] else 3*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 22 2020
  • Maple
    seq(`if`(n=0, 1, simplify(2*(ChebyshevU(n, 3/2)-2*ChebyshevU(n-1, 3/2)))), n = 0..30); # G. C. Greubel, Jan 22 2020
  • Mathematica
    CoefficientList[Series[(1-x-x^2)/(1-3*x+x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 05 2014 *)
Join[{1},LinearRecurrence[{3,-1},{2,4},30]] (* Harvey P. Dale, Oct 01 2014 *)
Table[If[n==0, 1, 2*(ChebyshevU[n, 3/2] -2*ChebyshevU[n-1, 3/2])], {n,0,30}] (* G. C. Greubel, Jan 22 2020 *)
  • PARI
    Vec((1-x-x^2)/(1-3*x+x^2) + O(x^40)) \\ Colin Barker, Feb 01 2014
  • Sage
    [1]+[2*(chebyshev_U(n,3/2) -2*chebyshev_U(n-1,3/2)) for n in (1..30)] # G. C. Greubel, Jan 22 2020

Formula

From Colin Barker, Feb 01 2014: (Start)
a(n) = 3*a(n-1) - a(n-2) for n > 0.
G.f.: (1 -x -x^2)/(1-3*x+x^2). (End)
a(n) = 2*A001519(n) for n > 0. - Colin Barker, Feb 04 2014
From G. C. Greubel, Jan 22 2020: (Start)
a(n) = 2*(ChebyshevU(n, 3/2) - 2*ChebyshevU(n-1, 3/2)), with a(0)=1.
E.g.f.: -1 + 2*exp(3*x/2)*( cosh(sqrt(5)*x/2) - sinh(sqrt(5)*x/2)/sqrt(5) ). (End)

A055820 a(n) = T(n,n-3), array T as in A055818.

Original entry on oeis.org

1, 11, 24, 43, 69, 103, 146, 199, 263, 339, 428, 531, 649, 783, 934, 1103, 1291, 1499, 1728, 1979, 2253, 2551, 2874, 3223, 3599, 4003, 4436, 4899, 5393, 5919, 6478, 7071, 7699, 8363, 9064, 9803, 10581, 11399, 12258
Offset: 3

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055818.

Programs

  • GAP
    Concatenation([1], List([4..50], n-> (n^3 +3*n^2 -10*n -6)/6)); # G. C. Greubel, Jan 22 2020
  • Magma
    [1] cat [(n^3 +3*n^2 -10*n -6)/6: n in [4..50]]; // G. C. Greubel, Jan 22 2020
  • Maple
    seq( `if`(n=3, 1, (n^3 +3*n^2 -10*n -6)/6), n=3..50); # G. C. Greubel, Jan 22 2020
  • Mathematica
    Join[{1},Table[(n^3+3n^2-10n-6)/6,{n,4,50}]] (* or *) Join[{1},LinearRecurrence[ {4,-6,4,-1},{11,24,43,69},50]] (* Harvey P. Dale, Sep 18 2011 *)
  • PARI
    vector(50, n, my(m=n+2); if(m==3, 1, (m^3 +3*m^2 -10*m -6)/6)) \\ G. C. Greubel, Jan 22 2020
  • Sage
    [1]+[(n^3 +3*n^2 -10*n -6)/6 for n in (4..50)] # G. C. Greubel, Jan 22 2020

Formula

a(n) = (n^3 +3*n^2 -10*n -6)/6, for n>3, with a(3) = 1.
From G. C. Greubel, Jan 22 2020: (Start)
G.f.: x^3*(1 + 7*x - 14*x^2 + 9*x^3 - 2*x^4)/(1-x)^4.
E.g.f.: (6 +12*x +3*x^2 -2*x^3 - (6 + 6*x - 6*x^2 - x^3)*exp(x))/6. (End)

A055821 a(n) = T(n,n-4), array T as in A055818.

Original entry on oeis.org

1, 23, 60, 122, 217, 354, 543, 795, 1122, 1537, 2054, 2688, 3455, 4372, 5457, 6729, 8208, 9915, 11872, 14102, 16629, 19478, 22675, 26247, 30222, 34629, 39498, 44860, 50747, 57192, 64229, 71893, 80220, 89247, 99012, 109554, 120913, 133130, 146247, 160307, 175354
Offset: 4

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055818.

Programs

  • GAP
    Concatenation([1], List([5..50], n-> (72 -54*n -25*n^2 +6*n^3 +n^4)/24 )); # G. C. Greubel, Jan 22 2020
  • Magma
    I:=[1,23,60,122,217,354,543,795,1122]; [n le 9 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)- 5*Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Dec 30 2016
  • Maple
    seq( `if`(n=4, 1, (72 -54*n -25*n^2 +6*n^3 +n^4)/24), n=4..50); # G. C. Greubel, Jan 22 2020
  • Mathematica
    Join[{1, 23, 60, 122}, LinearRecurrence[{5,-10,10,-5,1}, {217,354,543,795, 1122}, 45]] (* Vincenzo Librandi, Dec 30 2016 *)
Table[If[n==4, 1, (72 -54*n -25*n^2 +6*n^3 +n^4)/24], {n,4,50}] (* G. C. Greubel, Jan 22 2020 *)
  • PARI
    vector(50, n, my(m=n+3); if(m==4, 1, (72 -54*m -25*m^2 +6*m^3 +m^4)/24)) \\ G. C. Greubel, Jan 22 2020
  • Sage
    [1]+[(72 -54*n -25*n^2 +6*n^3 +n^4)/24 for n in (5..50)] # G. C. Greubel, Jan 22 2020

Formula

From Chai Wah Wu, Dec 29 2016: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 9.
G.f.: x^4*(1 + 18*x - 45*x^2 + 42*x^3 - 18*x^4 + 3*x^5)/(1-x)^5. (End)
From G. C. Greubel, Jan 22 2020: (Start)
a(n) = (72 - 54*n - 25*n^2 + 6*n^3 + n^4)/24 for n > 4, with a(4) = 1.
E.g.f.: (-72 +36*x^2 -3*x^4 + (72 - 72*x + 12*x^3 + x^4)*exp(x))/24. (End)

A055822 a(n) = T(n, n-5), array T as in A055818.

Original entry on oeis.org

1, 47, 144, 328, 640, 1131, 1863, 2910, 4359, 6311, 8882, 12204, 16426, 21715, 28257, 36258, 45945, 57567, 71396, 87728, 106884, 129211, 155083, 184902, 219099, 258135, 302502, 352724, 409358, 472995, 544261, 623818, 712365, 810639, 919416, 1039512, 1171784
Offset: 5

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055818.

Programs

  • GAP
    Concatenation([1], List([6..50], n-> (480 +524*n -250*n^2 -45*n^3 +10*n^4 +n^5)/120 )); # G. C. Greubel, Jan 22 2020
  • Magma
    I:=[1,47,144,328,640,1131,1863,2910,4359,6311, 8882]; [n le 11 select I[n] else 6*Self(n-1)-15*Self(n-2) +20*Self(n-3)-15*Self(n-4)+6*Self(n-5)-Self(n-6): n in [1..50]]; // Vincenzo Librandi, Dec 30 2016
  • Maple
    seq( `if`(n=5, 1, (480 +524*n -250*n^2 -45*n^3 +10*n^4 +n^5)/120), n=5..50); # G. C. Greubel, Jan 22 2020
  • Mathematica
    Join[{1,47,144,328,640}, LinearRecurrence[{6,-15,20,-15,6,-1}, {1131,1863,2910, 4359,6311,8882}, 5004]] (* Vincenzo Librandi, Dec 30 2016 *)
Table[If[n==5, 1, (480 +524*n -250*n^2 -45*n^3 +10*n^4 +n^5)/120], {n,5,50}] (* G. C. Greubel, Jan 22 2020 *)
  • PARI
    vector(45, n, my(m=n+4); if(m==5, 1, (480 +524*m -250*m^2 -45*m^3 +10*m^4 +m^5)/120)) \\ G. C. Greubel, Jan 22 2020
  • Sage
    [1]+[(480 +524*n -250*n^2 -45*n^3 +10*n^4 +n^5)/120 for n in (6..50)] # G. C. Greubel, Jan 22 2020

Formula

From Chai Wah Wu, Dec 29 2016: (Start)
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n > 11.
G.f.: x^5*(1 + 41*x -123*x^2 + 149*x^3 - 93*x^4 + 30*x^5 - 4*x^6)/(1-x)^6. (End)
From G. C. Greubel, Jan 22 2020: (Start)
a(n) = (480 +524*n -250*n^2 -45*n^3 +10*n^4 +n^5)/120, for n>5, with a(5) = 1.
E.g.f.: (-480 -720*x -180*x^2 +60*x^3 +30*x^4 -4*x^5 + (480 +240*x -300*x^2 + 40*x^3 +20*x^4 +x^5)*exp(x))/120. (End)

A055823 a(n) = T(n,n-6), array T as in A055818.

Original entry on oeis.org

1, 95, 336, 848, 1800, 3422, 6017, 9974, 15782, 24045, 35498, 51024, 71672, 98676, 133475, 177734, 233366, 302555, 387780, 491840, 617880, 769418, 950373, 1165094, 1418390, 1715561, 2062430, 2465376, 2931368, 3468000, 4083527, 4786902, 5587814, 6496727, 7524920
Offset: 6

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055818, A055819, A055820, A055821, A055822, A055824, A055825, A055826, A055827, A055828, A055829.

Programs

  • GAP
    Concatenation([1], List([7..50], n-> (n^6 +15*n^5 -65*n^4 -795*n^3 +1864*n^2 +6180*n -7200)/720 )); # G. C. Greubel, Jan 22 2020
  • Magma
    I:=[1,95,336,848,1800,3422,6017,9974,15782,24045, 35498,51024,71672]; [n le 13 select I[n] else 7*Self(n-1)- 21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6)+Self(n-7): n in [1..50]]; // Vincenzo Librandi, Dec 30 2016
  • Maple
    seq( `if`(n=6, 1, (n^6 +15*n^5 -65*n^4 -795*n^3 +1864*n^2 +6180*n -7200)/720), n=6..50); # G. C. Greubel, Jan 22 2020
  • Mathematica
    Join[{1, 95, 336, 848, 1800, 3422}, LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {6017, 9974, 15782, 24045, 35498, 51024, 71672}, 50]] (* Vincenzo Librandi, Dec 30 2016 *)
Table[If[n==6, 1, (n^6 +15*n^5 -65*n^4 -795*n^3 +1864*n^2 +6180*n -7200)/720], {n, 6, 50}] (* G. C. Greubel, Jan 22 2020 *)
  • PARI
    vector(45, n, my(m=n+5); if(m==6, 1, (m^6 +15*m^5 -65*m^4 -795*m^3 +1864*m^2 +6180*m -7200)/720)) \\ G. C. Greubel, Jan 22 2020
  • Sage
    [1]+[(n^6 +15*n^5 -65*n^4 -795*n^3 +1864*n^2 +6180*n -7200)/720 for n in (7..50)] # G. C. Greubel, Jan 22 2020

Formula

From Chai Wah Wu, Dec 29 2016: (Start)
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 13.
G.f.: x^6*(1 + 88*x - 308*x^2 + 456*x^3 - 370*x^4 + 174*x^5 - 45*x^6 + 5*x^7)/(1-x)^7. (End)
From G. C. Greubel, Jan 22 2020: (Start)
a(n) = (n^6 + 15*n^5 - 65*n^4 - 795*n^3 + 1864*n^2 + 6180*n -7200)/720, for n > 6, with a(6) = 1.
E.g.f.: (7200 - 2880*x^2 - 960*x^3 + 30*x^4 + 60*x^5 - 5*x^6 + (-7200 + 7200*x - 720*x^2 - 720*x^3 + 150*x^4 + 30*x^5 + x^6)*exp(x))/720. (End)

A055824 a(n) = T(2*n,n), array T as in A055818.

Original entry on oeis.org

1, 2, 9, 43, 217, 1131, 6017, 32467, 177009, 972691, 5378425, 29889531, 166795977, 934039419, 5246059761, 29540072355, 166708076001, 942651407907, 5339465635049, 30291114653131, 172081678284729, 978807205953931
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055818, A055819, A055820, A055821, A055822, A055823, A055825, A055826, A055827, A055828, A055829.

Programs

  • Maple
    T:= proc(i, j) option remember;
      if i=0 or j=0 then 1
    else add(add(T(h,m), m=0..j), h=0..i-1)
fi; end: seq(T(n, n), n=0..30); # G. C. Greubel, Jan 22 2020
  • Mathematica
    T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n, n], {n,0,25}] (* G. C. Greubel, Jan 22 2020 *)
  • Sage
    @CachedFunction
def T(i, j):
    if (i==0 or j==0): return 1
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[T(n, n) for n in (0..30)] # G. C. Greubel, Jan 22 2020

A055825 a(n) = T(2n+1,n), array T as in A055818.

Original entry on oeis.org

1, 5, 24, 122, 640, 3422, 18536, 101362, 558336, 3093302, 17218168, 96214890, 539415552, 3032659086, 17091411912, 96527966178, 546184965120, 3095613086822, 17571039730136, 99868193737306, 568303494617472
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Comments

Also main diagonal of array: A(i,1)=i, A(1,j)=j; A(i,j) = 2*A(i,j-1) - A(i-1,j). - Benoit Cloitre, Feb 26 2003

Links

Crossrefs

Cf. A055818, A055819, A055820, A055821, A055822, A055823, A055824, A055826, A055827, A055828, A055829.

Programs

  • Maple
    T:= proc(i, j) option remember;
      if i=0 or j=0 then 1
    else add(add(T(h,m), m=0..j), h=0..i-1)
  fi; end:
seq(T(n+1, n), n=0..30); # G. C. Greubel, Jan 22 2020
  • Mathematica
    T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n+1, n], {n,0,25}] (* G. C. Greubel, Jan 22 2020 *)
  • Sage
    @CachedFunction
def T(i, j):
    if (i==0 or j==0): return 1
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[T(n+1, n) for n in (0..30)] # G. C. Greubel, Jan 22 2020

A055826 a(n) = T(2n+2,n), array T as in A055818.

Original entry on oeis.org

1, 11, 60, 328, 1800, 9924, 54964, 305680, 1706256, 9554620, 53653996, 302038488, 1703995800, 9631951476, 54539233380, 309296779296, 1756495236128, 9987721546860, 56857004161180, 324008331785320, 1848182861702184
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055818, A055819, A055820, A055821, A055822, A055823, A055824, A055825, A055827, A055828, A055829.

Programs

  • Maple
    T:= proc(i, j) option remember;
      if i=0 or j=0 then 1
    else add(add(T(h,m), m=0..j), h=0..i-1)
  fi; end:
seq(T(n+2, n), n=0..30); # G. C. Greubel, Jan 22 2020
  • Mathematica
    T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n+2, n], {n,0,25}] (* G. C. Greubel, Jan 22 2020 *)
  • Sage
    @CachedFunction
def T(i, j):
    if (i==0 or j==0): return 1
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[T(n+2, n) for n in (0..30)] # G. C. Greubel, Jan 22 2020

A055827 a(n) = T(2n+3,n), array T as in A055818.

Original entry on oeis.org

1, 23, 144, 848, 4880, 27816, 157920, 895264, 5074272, 28772280, 163262704, 927203184, 5270629104, 29988361032, 170780080320, 973422085184, 5552990609344, 31702646247768, 181128948471888, 1035584204252560
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055818, A055819, A055820, A055821, A055822, A055823, A055824, A055825, A055826, A055828, A055829.

Programs

  • Maple
    T:= proc(i, j) option remember;
      if i=0 or j=0 then 1
    else add(add(T(h,m), m=0..j), h=0..i-1)
  fi; end:
seq(T(n+3, n), n=0..30); # G. C. Greubel, Jan 22 2020
  • Mathematica
    T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n+3, n], {n,0,25}] (* G. C. Greubel, Jan 22 2020 *)
  • Sage
    @CachedFunction
def T(i, j):
    if (i==0 or j==0): return 1
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[T(n+3, n) for n in (0..30)] # G. C. Greubel, Jan 22 2020

A055828 a(n) = T(2n+4,n), array T as in A055818.

Original entry on oeis.org

1, 47, 336, 2128, 12848, 75808, 441824, 2557024, 14737312, 84726992, 486388912, 2789840688, 15995087184, 91689974592, 525608606912, 3013422222272, 17280193763392, 99117586397296, 568696489153808, 3263973649089808
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055818, A055819, A055820, A055821, A055822, A055823, A055824, A055825, A055826, A055827, A055829.

Programs

  • Maple
    T:= proc(i, j) option remember;
      if i=0 or j=0 then 1
    else add(add(T(h,m), m=0..j), h=0..i-1)
  fi; end:
seq(T(n+4, n), n=0..30); # G. C. Greubel, Jan 22 2020
  • Mathematica
    T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n+4, n], {n,0,25}] (* G. C. Greubel, Jan 22 2020 *)
  • Sage
    @CachedFunction
def T(i, j):
    if (i==0 or j==0): return 1
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[T(n+4, n) for n in (0..30)] # G. C. Greubel, Jan 22 2020
Showing 1-10 of 12 results. Next

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