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.

Previous Showing 81-88 of 88 results.

A213779 Principal diagonal of the convolution array A213778.

Original entry on oeis.org

1, 6, 15, 33, 58, 97, 146, 214, 295, 400, 521, 671, 840, 1043, 1268, 1532, 1821, 2154, 2515, 2925, 3366, 3861, 4390, 4978, 5603, 6292, 7021, 7819, 8660, 9575, 10536, 11576, 12665, 13838, 15063, 16377, 17746, 19209, 20730, 22350, 24031, 25816, 27665, 29623
Offset: 1

Views

Author

Clark Kimberling, Jun 21 2012

Keywords

Links

Crossrefs

Cf. A213778, A213500.

Programs

  • Mathematica
    (See A213778.)
LinearRecurrence[{2,1,-4,1,2,-1},{1,6,15,33,58,97},80] (* Harvey P. Dale, Dec 12 2016 *)
  • PARI
    Vec(x*(1+4*x+2*x^2+x^3)/((1-x)^4*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 31 2016

Formula

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
G.f.: x*(1+4*x+2*x^2+x^3) / ((1-x)^4*(1+x)^2).
From Colin Barker, Jan 31 2016: (Start)
a(n) = (16*n^3+30*n^2+2*(3*(-1)^n+7)*n+3*((-1)^n-1))/48.
a(n) = (8*n^3+15*n^2+10*n)/24 for n even.
a(n) = (8*n^3+15*n^2+4*n-3)/24 for n odd.
(End)

A213780 Antidiagonal sums of the convolution array A213778.

Original entry on oeis.org

1, 6, 17, 40, 78, 140, 230, 360, 535, 770, 1071, 1456, 1932, 2520, 3228, 4080, 5085, 6270, 7645, 9240, 11066, 13156, 15522, 18200, 21203, 24570, 28315, 32480, 37080, 42160, 47736, 53856, 60537, 67830, 75753, 84360, 93670, 103740, 114590
Offset: 1

Views

Author

Clark Kimberling, Jun 21 2012

Keywords

Links

Crossrefs

Cf. A213778, A213500.

Programs

Formula

a(n) = 3*a(n-1) - a(n-2) - 5*a(n-3) + 5*a(n-4) + a(n-5) - 3*a(n-6) + a(n-7).
G.f.: f(x)/g(x), where f(x) = x*(1 + 3*x) and g(x) = (1 + x)^2 *(1 - x)^5.

A213436 Principal diagonal of the convolution array A212891.

Original entry on oeis.org

1, 17, 84, 260, 625, 1281, 2352, 3984, 6345, 9625, 14036, 19812, 27209, 36505, 48000, 62016, 78897, 99009, 122740, 150500, 182721, 219857, 262384, 310800, 365625, 427401, 496692, 574084, 660185, 755625, 861056, 977152, 1104609
Offset: 1

Views

Author

Clark Kimberling, Jun 16 2012

Keywords

Links

Crossrefs

Cf. A212891, A213500.

Programs

Formula

a(n) = (11*n^4 + 6*n^3 - 5*n^2)/12.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1 + 12*x + 9*x^2)/(1 - x)^5.

A213503 Rectangular array: (row n) = b**c, where b(h) = h^2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 6, 2, 20, 11, 3, 50, 34, 16, 4, 105, 80, 48, 21, 5, 196, 160, 110, 62, 26, 6, 336, 287, 215, 140, 76, 31, 7, 540, 476, 378, 270, 170, 90, 36, 8, 825, 744, 616, 469, 325, 200, 104, 41, 9, 1210, 1110, 948, 756, 560, 380, 230, 118, 46, 10
Offset: 1

Views

Author

Clark Kimberling, Jun 16 2012

Keywords

Comments

Principal diagonal: A117066
Antidiagonal sums: A033455
For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....6....20....50....105....196...336
2....11...34....80....160....287...476
3....16...48....110...215....378...616
4....21...62....140...270....469...756
5....26...76....170...325....560...896
...
T(5,1) = (1)**(5) = 5
T(5,2) = (1,4)**(5,6) = 1*6+4*5 = 26
T(5,3) = (1,4,9)**(5,6,7) = 1*7+4*6+9*5 = 76

Links

Crossrefs

Cf. A213500.

Programs

  • GAP
    Flat(List([1..12], n-> List([1..n], k-> (n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12))); # G. C. Greubel, Jul 05 2019
  • Magma
    [[(n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12: k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 05 2019
  • Mathematica
    (* First program *)
b[n_]:= n^2; c[n_]:= n;
T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213503 *)
r[n_]:= Table[T[n, k], {k,40}]  (* columns of antidiagonal triangle *)
d = Table[T[n, n], {n, 1, 40}] (* A117066 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A033455 *)
(* Second program *)
Table[(n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 05 2019 *)
  • PARI
    t(n,k) = (n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12;
for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ G. C. Greubel, Jul 05 2019
  • Sage
    [[(n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12 for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 05 2019

Formula

T(n,k) = 5*T(n,k-1) - 10*T(n,k-2) + 10*T(n,k-3) - 5*T(n,k-4) + T(n,k-5).
G.f. for row n: f(x)/g(x), where f(x) = n + x - (n - 1)^2 x^2 and g(x) = (1 - x)^5.
T(n,k) = k*(k^3 + 4*k^2*n + 6*k*n - k + 2*n)/12. - G. C. Greubel, Jul 05 2019

A213853 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = binomial(2*n-4+2*h,n-2+h), n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 4, 2, 13, 10, 6, 42, 38, 32, 20, 141, 136, 128, 110, 70, 492, 486, 476, 452, 392, 252, 1767, 1760, 1748, 1718, 1638, 1428, 924, 6474, 6466, 6452, 6416, 6316, 6036, 5280, 3432, 24051, 24042, 24026, 23984, 23864, 23514, 22506, 19734, 12870
Offset: 1

Views

Author

Clark Kimberling, Jul 05 2012

Keywords

Comments

Row 1, (1,2,3,4,5,...)**(1,2,6,20,70,...):
Row 2, (1,2,3,4,5,...)**(2,6,20,70,252,...):
Row 3, (1,2,3,4,5,...)**(6,20,70,252,...):
For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
  1    4     13     42     141
  2    10    38     136    486
  6    32    128    476    1748
  20   110   452    1718   6416
  70   392   1638   6316   23864

Links

Crossrefs

Cf. A213500.

Programs

  • Mathematica
    b[n_]:=n;c[n_]:=Binomial[2n-2,n-1]
t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
r[n_]:=Table[t[n,k],{k,1,20}] (* A213853 *)

A213043 Convolution of (1,-1,2,-2,3,-3,...) and A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 0, 3, 1, 7, 5, 16, 17, 38, 50, 94, 138, 239, 370, 617, 979, 1605, 2575, 4190, 6755, 10956, 17700, 28668, 46356, 75037, 121380, 196431, 317797, 514243, 832025, 1346284, 2178293, 3524594, 5702870, 9227482, 14930334, 24157835, 39088150, 63246005, 102334135
Offset: 0

Views

Author

Clark Kimberling, Jun 10 2012

Keywords

Comments

(1,-1,2,-2,3,-3,...) = ((-1)^n)*(1+floor(n/2)), which results from A001057 by removing its initial 0.

Examples

			a(5) = (1,-1,2,-2,3,-3)**(1,1,2,3,5,8)=1*8-1*5+2*3-2*2+3*1-3*1 = 5.

Links

Crossrefs

Cf. A000045, A213500.

Programs

  • Mathematica
    f[x_] := (1 - x^2) (1 + x); g[x] := 1 - x - x^2;
s = Normal[Series[1/(f[x] g[x]), {x, 0, 60}]]
c = CoefficientList[s, x]  (* A213043 *)
LinearRecurrence[{0, 3, 1, -2, -1}, {1, 0, 3, 1, 7}, 60]
Table[Fibonacci[n+1] + ((-1)^n (2n+1) - 1)/4, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 29 2015 *)
  • PARI
    Vec(1/((1-x)*(1+x)^2*(1-x-x^2)) + O(x^50)) \\ Colin Barker, Mar 16 2016

Formula

a(n) = 3*a(n-2)+a(n-3)-2*a(n-4)-a(n-5).
G.f.: 1/((1 + x)^2 * (1 - 2*x + x^3)).
From Vladimir Reshetnikov, Oct 29 2015: (Start)
a(n) = Fibonacci(n+1) + ((-1)^n*(2*n+1)-1)/4, where Fibonacci(n) = A000045(n).
Recurrence (4-term): a(0) = 1, a(1) = 0, a(2) = 3, (2*n+1)*a(n) = n + 1 - 2*a(n-1) + 4*(n+1)*a(n-2) + (2*n+3)*a(n-3).
(End)
From Colin Barker, Mar 16 2016: (Start)
a(n) = (-5-5*(-1)^n+2^(1-n)*sqrt(5)*(-(1-sqrt(5))^(1+n)+(1+sqrt(5))^(1+n))+10*(-1)^n*(1+n))/20.
a(n) = (sqrt(5)*2^(1-n)*((1+sqrt(5))^(n+1)-(1-sqrt(5))^(n+1))+10*(n+1)-10)/20 for n even.
a(n) = (sqrt(5)*2^(1-n)*((1+sqrt(5))^(n+1)-(1-sqrt(5))^(n+1))-10*(n+1))/20 for n odd.
(End)

A213044 Convolution of Fibonacci numbers and positive integers repeated three times (A000045 and A008620).

Original entry on oeis.org

1, 1, 2, 5, 7, 12, 22, 34, 56, 94, 150, 244, 399, 643, 1042, 1691, 2733, 4424, 7164, 11588, 18752, 30348, 49100, 79448, 128557, 208005, 336562, 544577, 881139, 1425716, 2306866, 3732582, 6039448, 9772042, 15811490, 25583532, 41395035
Offset: 0

Views

Author

Clark Kimberling, Jun 10 2012

Keywords

Links

Crossrefs

Cf. A213500.

Programs

  • Mathematica
    f[x_] := (1 - x^3)^2; g[x] := 1 - x - x^2;
s = Normal[Series[1/(f[x] g[x]), {x, 0, 60}]]
c = CoefficientList[s, x]  (* A213044 *)

Formula

a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-2*a(n-4)-a(n-5)+a(n-6)+a(n-7).
G.f.: 1/((1 - x - x^2)*(1 - x^3)^2).

A213046 Convolution of Lucas numbers and positive integers repeated (A000032 and A008619).

Original entry on oeis.org

2, 3, 8, 13, 25, 41, 71, 116, 193, 314, 514, 834, 1356, 2197, 3562, 5767, 9339, 15115, 24465, 39590, 64067, 103668, 167748, 271428, 439190, 710631, 1149836, 1860481, 3010333, 4870829, 7881179, 12752024, 20633221, 33385262, 54018502, 87403782, 141422304
Offset: 0

Views

Author

Clark Kimberling, Jun 10 2012

Keywords

Links

Crossrefs

Cf. A213500.

Programs

  • Magma
    /* By definition */ A008619:=func; [&+[A008619(i)*Lucas(n-i): i in [0..n]]: n in [0..34]];
  • Mathematica
    f[x_] := (1 + x) (1 - x)^2; g[x] := 1 - x - x^2;
s = Normal[Series[(2 - x)/(f[x] g[x]), {x, 0, 60}]]
CoefficientList[s, x]  (* A213046 *)
LinearRecurrence[{2,1,-3,0,1},{2,3,8,13,25},40] (* Harvey P. Dale, Aug 31 2023 *)
  • PARI
    a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,0,-3,1,2]^n*[2;3;8;13;25])[1,1] \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    Vec((-2 + x)/((-1 + x)^2*(-1 + 2*x^2 + x^3)) + O(x^60)) \\ Colin Barker, Feb 09 2017

Formula

a(n) = 2*a(n-1) + a(n-2) - 3*a(n-3) + a(n-5).
G.f.: (-2 + x)/((-1 + x)^2*(-1 + 2*x^2 + x^3)).
a(n) = (-9/4 + (3*(-1)^n)/4 + (2^(-n)*((1-t)^n*(-5+2*t) + (1+t)^n*(5+2*t)))/t + (-1-n)/2) where t=sqrt(5). - Colin Barker, Feb 09 2017
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