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 11 results. Next

A067989 Row sums of triangle A067979; also of triangle A067990.

Original entry on oeis.org

1, 9, 34, 95, 240, 553, 1212, 2547, 5195, 10340, 20184, 38766, 73451, 137565, 255080, 468877, 855288, 1549583, 2790510, 4997895, 8907481, 15804634, 27928464, 49169100, 86268325, 150882993, 263124862
Offset: 0

Views

Author

Wolfdieter Lang, Feb 15 2002

Keywords

Formula

a(n)= (n+2)*((3*n+1)*F(n-1)/2+(2*n+1)*F(n)), with F(n) := A000045(n) (Fibonacci).
G.f.: (1+4*x-x^2)*(1+2*x)/(1-x-x^2)^3.

A067418 Triangle A067330 with rows read backwards.

Original entry on oeis.org

1, 2, 1, 5, 3, 2, 10, 7, 5, 3, 20, 15, 12, 8, 5, 38, 30, 25, 19, 13, 8, 71, 58, 50, 40, 31, 21, 13, 130, 109, 96, 80, 65, 50, 34, 21, 235, 201, 180, 154, 130, 105, 81, 55, 34, 420, 365, 331, 289, 250, 210, 170, 131, 89, 55, 744, 655, 600, 532, 469, 404, 340, 275, 212, 144, 89, 1308, 1164, 1075, 965
Offset: 0

Views

Author

Wolfdieter Lang, Feb 15 2002

Keywords

Comments

The column m (without leading 0's) gives the convolution of Fibonacci numbers F(n+1) := A000045(n+1), n>=0, with those with m-shifted index: a(n+m,m)=sum(F(k+1)*F(m+n+1-k),k=0..n), n>=0, m=0,1,...
The row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n) are generated by A(z)*(A(z)-x*A(x*z))/(1-x), with A(x) := 1/(1-x-x^2) (g.f. for Fibonacci F(n+1), n>=0).
The columns give A001629(n+2), A023610, A067331-4, A067430-1, A067977-8 for m= 0..9, respectively. Row sums give A067988.

Examples

			{1}; {2,1}; {5,3,2}; {10,7,5,3}; ...; p(2,n)=5+3*x+2*x^2.

Links

Programs

  • Mathematica
    Reverse /@ Table[Sum[Fibonacci[k + 1] Fibonacci[n - k + 1], {k, 0, m}], {n, 0, 11}, {m, 0, n}] // Flatten (* Michael De Vlieger, Apr 11 2016 *)

Formula

a(n, m)=A067330(n, n-m), n>=m>=0, else 0.
a(n, m)= (((3*(n-m)+5)*F(n-m+1)+(n-m+1)*F(n-m))*F(m+1)+((n-m)*F(n-m+1)+2*(n-m+1)*F(n-m))*F(m))/5.
G.f. for column m=0, 1, ...: (x^m)*(F(m+1)+F(m)*x)/(1-x-x^2)^2, with F(m) := A000045(m) (Fibonacci).
a(n, m) = ((-1)^m*F(n-2*m+1)-m*L(n+2)+n*L(n+2)+5*F(n)+4*F(n-1))/5, with F(-n) = (-1)^(n+1)*F(n), hence a(n, m) = (2*(n-m+1)*L(n+2)-A067990(n, m))/5, n>=m>=0. - Ehren Metcalfe, Apr 11 2016

A067979 Triangle read by rows of incomplete convolutions of Lucas numbers L(n+1) = A000204(n+1), n>=0.

Original entry on oeis.org

1, 3, 6, 4, 13, 17, 7, 19, 31, 38, 11, 32, 48, 69, 80, 18, 51, 79, 107, 140, 158, 29, 83, 127, 176, 220, 274, 303, 47, 134, 206, 283, 360, 432, 519, 566, 76, 217, 333, 459, 580, 706, 822, 963, 1039, 123, 351, 539, 742, 940, 1138, 1341, 1529, 1757, 1880, 199, 568, 872, 1201, 1520, 1844, 2163, 2492, 2796, 3165, 3364
Offset: 0

Views

Author

Wolfdieter Lang, Feb 15 2002

Keywords

Comments

The diagonals d>=0 (d=0: main diagonal) give convolutions of Lucas numbers L(n+1) := A000204(n+1), n>=0, with those with d-shifted index: a(d+n,d) = Sum_{k=0..n} L(k+1)*L(d+n+1-k).
The diagonals give A004799(n-1), A067980-7 for d=n-m= 0..8, respectively. Row sums give A067989.
The row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n) are generated by A(x*z)*(A(z)-x*A(x*z))/(1-x), with A(x) := (1+2*x)/(1-x-x^2) (g.f. Lucas L(n+1), n>=0).

Examples

			Triangle begins:
  {1};
  {3,6};
  {4,13,17};     p(2,x) = 4+13*x+17*x^2
  {7,19,31,38};
  ...

Links

Crossrefs

Cf. A067990 (triangle with rows read backwards).

Programs

  • Mathematica
    Table[Sum[LucasL[k + 1] LucasL[n - k + 1], {k, 0, m}], {n, 0, 10}, {m, 0, n}] // Flatten (* Michael De Vlieger, Apr 11 2016 *)
  • PARI
    for(n=0,10, for(k=0,n, print1(sum(m=0,k,(fibonacci(m+2) + fibonacci(m))*(fibonacci(n-m+2) + fibonacci(n-m))), ", "))) \\ G. C. Greubel, Dec 17 2017

Formula

a(n, m) = Sum_{k=0..m} L(k+1)*L(n-k+1), n>=m>=0, else 0.
a(n, m) = (m+1)*L(n-m+1)*F(m) + ((m+1)*L(n-m+1) + m*L(n-m))*F(m+1), n>=m>=0, with F(n) := A000045(n) (Fibonacci) and L(n) := A000032(n) (Lucas).
G.f. for diagonals d= n-m>=0: (x^d)*(L(d+1)+L(d)*x)*(1-2*x)/(1-x-x^2)^2.
a(n, m) = -(-1)^m*F(n-2*m-1) + m*L(n+2)+F(n+3), with F(-n) = (-1)^(n+1) * F(n), hence a(n, m) = -5*A067330(n, m)+2*(m+1)*L(n+2), n>=m>=0. - Ehren Metcalfe, Apr 11 2016

A067980 Convolution of L(n+1) := A000204(n+1) (Lucas), n>=0, with L(n+2), n>=0.

Original entry on oeis.org

3, 13, 31, 69, 140, 274, 519, 963, 1757, 3165, 5642, 9972, 17499, 30521, 52955, 91461, 157336, 269702, 460863, 785295, 1334713, 2263293, 3829846, 6468264, 10905075, 18355429, 30849559, 51776133, 86785892
Offset: 0

Views

Author

Wolfdieter Lang, Feb 15 2002

Keywords

Comments

Second diagonal of triangle A067979. Second column of triangle A067990.

Formula

a(n)= sum(L(k+1)*L(n+2-k), k=0..n) = (4*n+3)*F(n+1)+3*(n+1)*F(n), with F(n) := A000045(n) (Fibonacci).
G.f.: (3+x)*(1+2*x)/(1-x-x^2)^2.

A067981 Convolution of L(n+1) := A000204(n+1) (Lucas), n>=0, with L(n+3), n>=0.

Original entry on oeis.org

4, 19, 48, 107, 220, 432, 822, 1529, 2796, 5045, 9006, 15936, 27992, 48863, 84840, 146623, 252368, 432816, 739914, 1261285, 2144484, 3637609, 6157218, 10401792, 17541100, 29531947, 49644192, 83336339
Offset: 0

Views

Author

Wolfdieter Lang, Feb 15 2002

Keywords

Comments

Third diagonal of triangle A067979. Third column of triangle A067990.

Formula

a(n)= sum(L(k+1)*L(n+3-k), k=0..n) = (7*n+4)*F(n+1)+4*(n+1)*F(n), with F(n) := A000045(n) (Fibonacci).
G.f.: (4+3*x)*(1+2*x)/(1-x-x^2)^2.

A067982 Convolution of L(n+1) := A000204(n+1) (Lucas), n>=0, with L(n+4), n>=0.

Original entry on oeis.org

7, 32, 79, 176, 360, 706, 1341, 2492, 4553, 8210, 14648, 25908, 45491, 79384, 137795, 238084, 409704, 702518, 1200777, 2046580, 3479197, 5900902, 9987064, 16870056, 28446175, 47887376, 80493751, 135112472
Offset: 0

Views

Author

Wolfdieter Lang, Feb 15 2002

Keywords

Comments

Fourth diagonal of triangle A067979. Fourth column of triangle A067990.

Formula

a(n)= sum(L(k+1)*L(n+4-k), k=0..n) = (11*n+7)*F(n+1)+7*(n+1)*F(n), with F(n) := A000045(n) (Fibonacci).
G.f.: (7+4*x)*(1+2*x)/(1-x-x^2)^2.

A067983 Convolution of L(n+1) := A000204(n+1) (Lucas), n>=0, with L(n+5), n>=0.

Original entry on oeis.org

11, 51, 127, 283, 580, 1138, 2163, 4021, 7349, 13255, 23654, 41844, 73483, 128247, 222635, 384707, 662072, 1135334, 1940691, 3307865, 5623681, 9538511, 16144282, 27271848, 45987275, 77419323, 130137943
Offset: 0

Views

Author

Wolfdieter Lang, Feb 15 2002

Keywords

Comments

Fifth diagonal of triangle A067979. Fifth column of triangle A067990.

Formula

a(n)= sum(L(k+1)*L(n+5-k), k=0..n) = (18*n+11)*F(n+1)+11*(n+1)*F(n), with F(n) := A000045(n) (Fibonacci).
G.f.: (11+7*x)*(1+2*x)/(1-x-x^2)^2.

A067984 Convolution of L(n+1) := A000204(n+1) (Lucas), n>=0, with L(n+6), n>=0.

Original entry on oeis.org

18, 83, 206, 459, 940, 1844, 3504, 6513, 11902, 21465, 38302, 67752, 118974, 207631, 360430, 622791, 1071776, 1837852, 3141468, 5354445, 9102878, 15439413, 26131346, 44141904, 74433450, 125306699
Offset: 0

Views

Author

Wolfdieter Lang, Feb 15 2002

Keywords

Comments

Sixth diagonal of triangle A067979. Sixth column of triangle A067990.

Formula

a(n)= sum(L(k+1)*L(n+6-k), k=0..n) = (29*n+18)*F(n+1)+18*(n+1)*F(n), with F(n) := A000045(n) (Fibonacci).
G.f.: (18+11*x)*(1+2*x)/(1-x-x^2)^2.

A067985 Convolution of L(n+1) := A000204(n+1) (Lucas), n>=0, with L(n+7), n>=0.

Original entry on oeis.org

29, 134, 333, 742, 1520, 2982, 5667, 10534, 19251, 34720, 61956, 109596, 192457, 335878, 583065, 1007498, 1733848, 2973186, 5082159, 8662310, 14726559, 24977924, 42275628, 71413752, 120420725
Offset: 0

Views

Author

Wolfdieter Lang, Feb 15 2002

Keywords

Comments

Seventh diagonal of triangle A067979. Seventh column of triangle A067990.

Formula

a(n)= sum(L(k+1)*L(n+7-k), k=0..n) = (47*n+29)*F(n+1)+29*(n+1)*F(n), with F(n) := A000045(n) (Fibonacci).
G.f.: (29+18*x)*(1+2*x)/(1-x-x^2)^2.

A067986 Convolution of L(n+1) := A000204(n+1) (Lucas), n>=0, with L(n+8), n>=0.

Original entry on oeis.org

47, 217, 539, 1201, 2460, 4826, 9171, 17047, 31153, 56185, 100258, 177348, 311431, 543509, 943495, 1630289, 2805624, 4811038, 8223627, 14016755, 23829437, 40417337, 68406974, 115555656, 194854175
Offset: 0

Views

Author

Wolfdieter Lang, Feb 15 2002

Keywords

Comments

Eighth diagonal of triangle A067979. Eighth column of triangle A067990.

Formula

a(n)= sum(L(k+1)*L(n+8-k), k=0..n) = (76*n+47)*F(n+1)+47*(n+1)*F(n), with F(n) := A000045(n) (Fibonacci).
G.f.: (47+29*x)*(1+2*x)/(1-x-x^2)^2.
Showing 1-10 of 11 results. Next

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

Content is available under The OEIS End-User License Agreement.