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 1-6 of 6 results.

A055802 a(n) = T(n,n-2), array T as in A055801.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 7, 10, 11, 15, 16, 21, 22, 28, 29, 36, 37, 45, 46, 55, 56, 66, 67, 78, 79, 91, 92, 105, 106, 120, 121, 136, 137, 153, 154, 171, 172, 190, 191, 210, 211, 231, 232, 253, 254, 276, 277, 300, 301, 325, 326, 351, 352, 378, 379, 406, 407, 435
Offset: 2

Views

Author

Clark Kimberling, May 28 2000

Keywords

Comments

For n>2, a(n)+a(n+1) seems to be A002620(n+1)+1.

Links

Crossrefs

Cf. A002620, A134519.
Cf. A055801, A055803, A055804, A055805, A055806.

Programs

  • GAP
    Concatenation([1], List([3..65], n-> (2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16 )); # G. C. Greubel, Jan 23 2020
  • Magma
    [1] cat [(2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16: n in [3..65]]; // G. C. Greubel, Jan 23 2020
  • Maple
    seq( `if`(n==2, 1, (2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16), n=2..65); # G. C. Greubel, Jan 23 2020
  • Mathematica
    CoefficientList[Series[(1 -2*x^2 +x^3 +2*x^4 -x^5)/((1-x)^3*(1+x)^2), {x,0,65}], x] (* Wesley Ivan Hurt, Jan 20 2017 *)
Table[If[n==2,1, (2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16], {n,2,65}] (* G. C. Greubel, Jan 23 2020 *)
  • PARI
    Vec(x^2*(1-2*x^2+x^3+2*x^4-x^5)/((1-x)^3*(1+x)^2) + O(x^65)) \\ Charles R Greathouse IV, Feb 03 2013
  • PARI
    vector(65, n, my(m=n+1); if(m==2, 1, (2*m^2 -6*m +11 +(-1)^m*(2*m -11))/16)) \\ G. C. Greubel, Jan 23 2020
  • Sage
    [1]+[(2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16 for n in (3..65)] # G. C. Greubel, Jan 23 2020

Formula

G.f.: x^2*(1 -2*x^2 +x^3 +2*x^4 -x^5)/((1-x)^3*(1+x)^2).
a(n) = A114220(n-1), n>=3. - R. J. Mathar, Feb 03 2013
From Colin Barker, Jan 27 2016: (Start)
a(n) = (2*n^2 +2*(-1)^n*n -6*n -11*(-1)^n +11)/16 for n>2.
a(n) = (n^2 - 2*n)/8 for n>2 and even.
a(n) = (n^2 - 4*n + 11)/8 for n odd. (End)
E.g.f.: (4*x*(x-2) + x*(x-3)*cosh(x) + (x^2 -x +11)*sinh(x))/8. - G. C. Greubel, Jan 23 2020

A055803 a(n) = T(n,n-3), array T as in A055801.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 11, 14, 21, 25, 36, 41, 57, 63, 85, 92, 121, 129, 166, 175, 221, 231, 287, 298, 365, 377, 456, 469, 561, 575, 681, 696, 817, 833, 970, 987, 1141, 1159, 1331, 1350, 1541, 1561, 1772, 1793, 2025, 2047, 2301
Offset: 3

Views

Author

Clark Kimberling, May 28 2000

Keywords

Comments

Third differences seem to be A002620(n)+1.

Links

Crossrefs

Cf. A055801, A055802, A055804, A055805, A055806.

Programs

  • GAP
    Concatenation([1], List([4..60], n-> (-39 +55*n -15*n^2 +2*n^3 +(-1)^n*(135 -39*n +3*n^2))/96 )); # G. C. Greubel, Jan 23 2020
  • Magma
    [1] cat [(-39 +55*n -15*n^2 +2*n^3 +(-1)^n*(135 -39*n +3*n^2))/96: n in [4..60]]; // G. C. Greubel, Jan 23 2020
  • Maple
    seq( `if`(n=3, 1, (-39 +55*n -15*n^2 +2*n^3 +(-1)^n*(135 -39*n +3*n^2))/96), n=3..60); # G. C. Greubel, Jan 23 2020
  • Mathematica
    Table[If[n==3, 1, (-39 +55*n -15*n^2 +2*n^3 +(-1)^n*(135 -39*n +3*n^2))/96], {n,3,60}] (* G. C. Greubel, Jan 23 2020 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,1,1,2,3,5,7,11},50] (* Harvey P. Dale, Jan 28 2023 *)
  • PARI
    vector(60, n, my(m=n+2); if(m==3, 1, (-39 +55*m -15*m^2 +2*m^3 +(-1)^m*(135 -39*m +3*m^2))/96)) \\ G. C. Greubel, Jan 23 2020
  • Sage
    [1]+[(-39 +55*n -15*n^2 +2*n^3 +(-1)^n*(135 -39*n +3*n^2))/96 for n in (4..60)] # G. C. Greubel, Jan 23 2020

Formula

From Colin Barker, Nov 28 2014: (Start)
a(n) = (-39 +55*n -15*n^2 +2*n^3 +(-1)^n*(135 -39*n +3*n^2))/96 for n>3.
G.f.: x^3*(1 -3*x^2 +x^3 +4*x^4 -x^5 -2*x^6 +x^7)/((1-x)^4*(1+x)^3). (End)
E.g.f.: ( 8*(x^3 -3*x^2 +6*x -6) +(x^3 -3*x^2 +39*x +48)*cosh(x) +(x^3 -6*x^2 +3*x -87)*sinh(x) )/48. - G. C. Greubel, Jan 23 2020

A055804 a(n) = T(n,n-4), array T as in A055801.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 12, 19, 26, 40, 51, 76, 92, 133, 155, 218, 247, 339, 376, 505, 551, 726, 782, 1013, 1080, 1378, 1457, 1834, 1926, 2395, 2501, 3076, 3197, 3893, 4030, 4863, 5017, 6004, 6176, 7335, 7526, 8876, 9087
Offset: 4

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055801, A055802, A055803, A055805, A055806.

Programs

  • GAP
    Concatenation([1], List([5..50], n-> (2*n^4 -28*n^3 +178*n^2 -416*n +441 +(-1)^n*(4*n^3 -90*n^2 + 704*n -1977))/768 )); # G. C. Greubel, Jan 24 2020
  • Magma
    [1] cat [(2*n^4 -28*n^3 +178*n^2 -416*n +441 +(-1)^n*(4*n^3 -90*n^2 + 704*n -1977))/768: n in [5..50]]; // G. C. Greubel, Jan 24 2020
  • Maple
    seq( `if`(n=4, 1, (2*n^4 -28*n^3 +178*n^2 -416*n +441 +(-1)^n*(4*n^3 -90*n^2 + 704*n -1977))/768), n=4..50); # G. C. Greubel, Jan 24 2020
  • Mathematica
    Table[If[n==4, 1, (2*n^4 -28*n^3 +178*n^2 -416*n +441 +(-1)^n*(4*n^3 -90*n^2 + 704*n -1977))/768], {n,4,50}] (* G. C. Greubel, Jan 24 2020 *)
  • PARI
    vector(50, n, my(m=n+3); if(m==4, 1, (2*m^4 -28*m^3 +178*m^2 -416*m +441 +(-1)^m*(4*m^3 -90*m^2 + 704*m -1977))/768)) \\ G. C. Greubel, Jan 24 2020
  • Sage
    [1]+[(2*n^4 -28*n^3 +178*n^2 -416*n +441 +(-1)^n*(4*n^3 -90*n^2 + 704*n -1977))/768 for n in (5..50)] # G. C. Greubel, Jan 24 2020

Formula

G.f.: x^4*(-1 +4*x^2 -x^3 -7*x^4 +2*x^5 +5*x^6 -2*x^7 -2*x^8 +x^9)/((1-x)^5 (1+x)^4). - R. J. Mathar, Jul 10 2012
From G. C. Greubel, Jan 24 2020: (Start)
a(n) = (2*n^4 -28*n^3 +178*n^2 -416*n +441 +(-1)^n*(4*n^3 -90*n^2 + 704*n -1977))/768 for n>4, with a(4) = 1.
E.g.f.: ( (768 -768*x +192*x^2 -64*x^3 +16*x^4) +(-768 -441*x +15*x^2 -10*x^3 +x^4)*cosh(x) +(1209 +177*x +93*x^2 -6*x^3 +x^4)*sinh(x) )/384. (End)

A055805 a(n) = T(n,n-5), array T as in A055801.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 20, 32, 46, 72, 97, 148, 189, 281, 344, 499, 591, 838, 967, 1343, 1518, 2069, 2300, 3082, 3380, 4460, 4837, 6294, 6763, 8689, 9264, 11765, 12461, 15658, 16491, 20521, 21508, 26525, 27684, 33860
Offset: 5

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055801, A055802, A055803, A055804, A055806.

Programs

  • GAP
    Concatenation([1], List([6..50], n-> ((2*n^5 -45*n^4 +450*n^3 -2070*n^2 +4873*n -3585) +5*(-1)^n*(n^4 -34*n^3 +446*n^2 -2741*n +6861))/7680 )); # G. C. Greubel, Jan 24 2020
  • Magma
    [1] cat [((2*n^5 -45*n^4 +450*n^3 -2070*n^2 +4873*n -3585) +5*(-1)^n*(n^4 -34*n^3 +446*n^2 -2741*n +6861))/7680: n in [6..50]]; // G. C. Greubel, Jan 24 2020
  • Maple
    seq( `if`(n=5, 1, ((2*n^5 -45*n^4 +450*n^3 -2070*n^2 +4873*n -3585) +5*(-1)^n*(n^4 -34*n^3 +446*n^2 -2741*n +6861))/7680), n=5..50); # G. C. Greubel, Jan 24 2020
  • Mathematica
    Table[If[n==5, 1, ((2*n^5 -45*n^4 +450*n^3 -2070*n^2 +4873*n -3585) +5*(-1)^n*(n^4 -34*n^3 +446*n^2 -2741*n +6861))/7680], {n,5,50}] (* G. C. Greubel, Jan 24 2020 *)
LinearRecurrence[{1,5,-5,-10,10,10,-10,-5,5,1,-1},{1,1,1,2,3,5,8,13,20,32,46,72},50] (* Harvey P. Dale, Mar 08 2023 *)
  • PARI
    vector(50, n, my(m=n+4); if(m==5, 1, ((2*m^5 -45*m^4 +450*m^3 -2070*m^2 +4873*m -3585) +5*(-1)^m*(m^4 -34*m^3 +446*m^2 -2741*m +6861))/7680)) \\ G. C. Greubel, Jan 24 2020
  • Sage
    [1]+[((2*n^5 -45*n^4 +450*n^3 -2070*n^2 +4873*n -3585) +5*(-1)^n*(n^4 -34*n^3 +446*n^2 -2741*n +6861))/7680 for n in (6..50)] # G. C. Greubel, Jan 24 2020

Formula

From Colin Barker, Nov 28 2014: (Start)
a(n) = ((2*n^5 -45*n^4 +450*n^3 -2070*n^2 +4873*n -3585) +5*(-1)^n*(n^4 -34*n^3 +446*n^2 -2741*n +6861))/7680 for n>5.
G.f.: x^5*(1 -5*x^2 +x^3 +11*x^4 -3*x^5 -12*x^6 +5*x^7 +7*x^8 -3*x^9 -2*x^10 + x^11)/((1-x)^6*(1+x)^5). (End)

A055806 a(n) = T(n,n-6), array T as in A055801.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 21, 33, 53, 79, 125, 176, 273, 365, 554, 709, 1053, 1300, 1891, 2267, 3234, 3785, 5303, 6085, 8385, 9465, 12845, 14302, 19139, 21065, 27828, 30329, 39593, 42790, 55251, 59281, 75772, 80789, 102297, 108473, 136157, 143683, 178893
Offset: 6

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055801, A055802, A055803, A055804, A055805.

Programs

  • GAP
    Concatenation([1], List([7..50], n-> (48915 -58884*n +29723*n^2 -7200*n^3 +965*n^4 -66*n^5 +2*n^6 + 3*(-1)^n*(-231345 +98988*n -18505*n^2 +1840*n^3 -95*n^4 +2*n^5))/92160 )); # G. C. Greubel, Jan 24 2020
  • Magma
    [1] cat [(48915 -58884*n +29723*n^2 -7200*n^3 +965*n^4 -66*n^5 +2*n^6 + 3*(-1)^n*(-231345 +98988*n -18505*n^2 +1840*n^3 -95*n^4 +2*n^5))/92160: n in [7..50]]; // G. C. Greubel, Jan 24 2020
  • Maple
    seq( `if(n=6,1, (48915 -58884*n +29723*n^2 -7200*n^3 +965*n^4 -66*n^5 +2*n^6 + 3*(-1)^n*(-231345 +98988*n -18505*n^2 +1840*n^3 -95*n^4 +2*n^5))/92160), n=6..50); # G. C. Greubel, Jan 24 2020
  • Mathematica
    Table[If[n==6, 1, (48915 -58884*n +29723*n^2 -7200*n^3 +965*n^4 -66*n^5 +2*n^6 + 3*(-1)^n*(-231345 +98988*n -18505*n^2 +1840*n^3 -95*n^4 +2*n^5))/92160], {n, 6,50}] (* G. C. Greubel, Jan 24 2020 *)
  • PARI
    vector(50, n, my(m=n+5); if(m==6, 1, (48915 -58884*m +29723*m^2 -7200*m^3 +965*m^4 -66*m^5 +2*m^6 + 3*(-1)^m*(-231345 +98988*m -18505*m^2 +1840*m^3 -95*m^4 +2*m^5))/92160)) \\ G. C. Greubel, Jan 24 2020
  • Sage
    [1]+[(48915 -58884*n +29723*n^2 -7200*n^3 +965*n^4 -66*n^5 +2*n^6 + 3*(-1)^n*(-231345 +98988*n -18505*n^2 +1840*n^3 -95*n^4 +2*n^5))/92160 for n in (7..50)] # G. C. Greubel, Jan 24 2020

Formula

From G. C. Greubel, Jan 24 2020: (Start)
a(n) = (48915 -58884*n +29723*n^2 -7200*n^3 +965*n^4 -66*n^5 +2*n^6 + 3*(-1)^n*(-231345 +98988*n -18505*n^2 +1840*n^3 -95*n^4 +2*n^5))/92160, n > 6.
G.f.: x^6*(1 -6*x^2 +x^3 +16*x^4 -4*x^5 -23*x^6 +8*x^7 +20*x^8 -8*x^9 -9*x^10 + 4*x^11 +2*x^12 -x^13)/((1-x)^7*(1+x)^6). (End)

A011794 Triangle defined by T(n+1, k) = T(n, k-1) + T(n-1, k), T(n,1) = 1, T(1,k) = 1, T(2,k) = min(2,k).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 6, 7, 8, 1, 4, 7, 11, 12, 13, 1, 4, 10, 14, 19, 20, 21, 1, 5, 11, 21, 26, 32, 33, 34, 1, 5, 15, 25, 40, 46, 53, 54, 55, 1, 6, 16, 36, 51, 72, 79, 87, 88, 89, 1, 6, 21, 41, 76, 97, 125, 133, 142, 143, 144, 1, 7, 22, 57, 92, 148, 176, 212, 221, 231, 232, 233
Offset: 1

Views

Author

N. J. A. Sloane and David Broadhurst

Keywords

Examples

			matrix(10,10,n,k,a(n-1,k-1))
  [ 0 0 0 0 0 0 0 0 0 0 ]
  [ 0 1 1 1 1 1 1 1 1 1 ]
  [ 0 1 2 2 2 2 2 2 2 2 ]
  [ 0 1 2 3 3 3 3 3 3 3 ]
  [ 0 1 3 4 5 5 5 5 5 5 ]
  [ 0 1 3 6 7 8 8 8 8 8 ]
Triangle begins as:
  1;
  1, 2;
  1, 2,  3;
  1, 3,  4,  5;
  1, 3,  6,  7,  8;
  1, 4,  7, 11, 12, 13;
  1, 4, 10, 14, 19, 20, 21;
  1, 5, 11, 21, 26, 32, 33, 34;
  1, 5, 15, 25, 40, 46, 53, 54, 55;
  1, 6, 16, 36, 51, 72, 79, 87, 88, 89;

Links

Crossrefs

Columns include A008619 and (essentially) A055802, A055803, A055804, A055805, A055806.
Right-hand columns 1-14 are A000045, A000071, A001911, A001924, A001891, A014162, A053808, A014166, A053809, A053739, A054469, A053295, A054470, A053296.
Essentially a reflected version of A055801.
Sums include: A039834 (signed row), A131913 (row).
Cf. A008466, A074878, A103609.

Programs

  • Magma
    function T(n,k) // T = A011794(n,k)
  if k eq 1 or n eq 1 then return 1;
  elif n eq 2 then return Min(2, k);
  else return T(n-1,k-1) + T(n-2,k);
  end if;
end function;
[T(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Oct 21 2024
  • Mathematica
    T[n_, k_]:= T[n, k]= T[n-1, k-1] + T[n-2, k]; T[n_, 1] = 1; T[1, k_] = 1; T[2, k_] := Min[2, k]; Table[T[n, k], {n,15}, {k,n}]//Flatten (* Jean-François Alcover, Feb 26 2013 *)
  • PARI
    T(n,k)=if(n<=0 || k<=0,0, if(n<=2 || k==1, min(n,k), T(n-1,k-1)+T(n-2,k)))
  • SageMath
    def T(n, k): # T = A011794
    if (k==1 or n==1): return 1
    elif (n==2): return min(2,k)
    else: return T(n-1, k-1) + T(n-2, k)
flatten([[T(n, k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Oct 21 2024

Formula

T(n,n) = Fibonacci(n+1). - Jean-François Alcover, Feb 26 2013
From G. C. Greubel, Oct 21 2024: (Start)
Sum_{k=1..n} T(n, k) = A131913(n-1).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A039834(n).
Sum_{k=1..floor((n+1)/2)} T(n-k+1,k) = (1/2)*((1-(-1)^n)*A074878((n+3)/2) + (1+(-1)^n)*A008466((n+6)/2)) (diagonal row sums).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1,k) = (-1)^floor((n-1)/2)*A103609(n) + [n=1] (signed diagonal row sums). (End)

Extensions

Entry improved by comments from Michael Somos
More terms added by G. C. Greubel, Oct 21 2024
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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