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 1-3 of 3 results.

A006478 a(n) = a(n-1) + a(n-2) + F(n) - 1, a(0) = a(1) = 0, where F() = Fibonacci numbers A000045.

Original entry on oeis.org

0, 0, 0, 1, 3, 8, 18, 38, 76, 147, 277, 512, 932, 1676, 2984, 5269, 9239, 16104, 27926, 48210, 82900, 142055, 242665, 413376, 702408, 1190808, 2014608, 3401833, 5734251, 9650312, 16216602, 27213182, 45608092, 76345851, 127656829, 213230144, 355817324, 593205284
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Partial sums of A001629.
Number of edges in the Fibonacci hypercube FQ(n-2) (defined in the Rispoli and Cosares reference). - Emeric Deutsch, Oct 06 2014
Circuit rank (cyclomatic number) of the n-Fibonacci cube graph. - Eric W. Weisstein, Sep 05 2017

Examples

			G.f. = x^3 + 3*x^4 + 8*x^5 + 18*x^6 + 38*x^7 + 76*x^8 + 147*x^9 + 277*x^10 + ...

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000045, A000051, A006479, A175722.
Cf. A001629.

Programs

  • Haskell
    a006478 n = a006478_list !! (n-3)
a006478_list = scanl1 (+) $ drop 2 a001629_list
-- Reinhard Zumkeller, Sep 12 2015
  • Maple
    A006478 := proc(n)
  1 + ((n-5)*combinat[fibonacci](n-1)+(3*n-8)*combinat[fibonacci](n)) / 5;
end proc:
seq(A006478(n),n=0..20) ; # R. J. Mathar, Jun 12 2018
  • Mathematica
    CoefficientList[Series[x^3/((1 - x) (1 - x - x^2)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 13 2014 *)
LinearRecurrence[{3, -1, -3, 1, 1}, {0, 0, 0, 1, 3, 8}, 20] (* Eric W. Weisstein, Sep 05 2017 *)
Table[1 + (2 (n + 1) Fibonacci[n] + n Fibonacci[n + 1])/5 - Fibonacci[n + 2], {n, 0, 20}] (* Eric W. Weisstein, Sep 05 2017 *)
  • PARI
    {a(n) = if( n<0, polcoeff( x^2 / ((1 - x) * (1 + x - x^2)^2) + x * O(x^-n), -n), polcoeff( x^3 / ((1 - x) * (1 - x - x^2)^2) + x * O(x^n), n))}; /* Michael Somos, Mar 11 2014 */

Formula

a(n) - a(n-1) = A001629(n-1).
a(n) = 1 + ((n-5)*F(n-1) + (3*n-8)*F(n))/5.
G.f.: x^3/((1-x)*(1-x-x^2)^2). - Simon Plouffe in his 1992 dissertation
a(n) = Sum_{k=0..n-1} Sum_{i=0..k} F(i)*F(k-i). - Benoit Cloitre, Jan 26 2003
a(n) = A175722(-2-n). - Michael Somos, Mar 11 2014
a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + a(n-4) + a(n-5). - Eric W. Weisstein, Sep 05 2017
E.g.f.: exp(x) + exp(x/2)*(5*(3*x - 5)*cosh(sqrt(5)*x/2) + sqrt(5)*(5*x - 11)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Jul 24 2022

Extensions

a(0)-a(2) added and offset changed - N. J. A. Sloane, Jun 19 2021
Programs and b-file adapted by Georg Fischer, Jun 21 2021

A175721 Array T(n,m) read by antidiagonals: the coefficient of [x^m] of 1/(-x^n + 1 - x^(1+n) + x + 3*x^(2+n) - 2*x^2 - x^(3+n)) in row n, column 1 <= m.

Original entry on oeis.org

0, 3, -1, -3, 4, -1, 10, -6, 3, -1, -18, 14, -4, 3, -1, 42, -24, 10, -5, 3, -1, -87, 47, -19, 12, -5, 3, -1, 190, -83, 42, -22, 11, -5, 3, -1, -405, 152, -84, 45, -20, 11, -5, 3, -1
Offset: 1

Views

Author

Roger L. Bagula, Dec 04 2010

Keywords

Comments

Antidiagonal sums are: 0, 2, 0, 6, -6, 25, -50, 135, -304, ...
For large n, the rows approach A077925.

Examples

			The array starts in row n=1 with columns m >= 1 as
   0, 3, -3, 10, -18, 42, -87, 190, -405, 873, ... A077899
  -1, 4, -6, 14, -24, 47, -83, 152, -268, 476, ... A175722
  -1, 3, -4, 10, -19, 42, -84, 174, -353, 726, ...
  -1, 3, -5, 12, -22, 45, -87, 174, -340, 670, ...
  -1, 3, -5, 11, -20, 42, -83, 169, -339, 686, ...
  -1, 3, -5, 11, -21, 44, -86, 173, -343, 685, ...
  -1, 3, -5, 11, -21, 43, -84, 170, -339, 681, ...
  -1, 3, -5, 11, -21, 43, -85, 172, -342, 685, ...
  -1, 3, -5, 11, -21, 43, -85, 171, -340, 682, ...
  -1, 3, -5, 11, -21, 43, -85, 171, -341, 684, ...

Crossrefs

Cf. A000045, A077899, A077925, A175722.

Programs

  • Maple
    A175721 := proc(m,k) 1/(-x^m + 1 - x^(1 + m) + x + 3*x^(2 + m) - 2* x^2 - x^(3 + m)) ; coeftayl(%,x=0,k) ; end proc: # R. J. Mathar, Dec 22 2010
  • Mathematica
    f[x_, m_] = ExpandAll[(x - x^(m + 1))*(1 - x - x^2) - (1 - 2*x + x^(m + 1))];
g[x_, n_] = ExpandAll[x^(m + 3)*f[1/x, m]];
a = Table[Table[SeriesCoefficient[
      Series[1/g[x, m], {x, 0, 10}], n], {n, 0, 10}], {m, 1, 10}];
Table[Table[a[[m, n - m + 1]], {m, 1, n - 1}], {n, 1, 10}]
Flatten[%]

Formula

G.f.: 1/( - x^n + 1 - x^(1+n) + x + 3*x^(2+n) - 2*x^2 - x^(3+n) ).

A230449 T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, n) = A052952(n), n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 5, 4, 1, 4, 8, 9, 8, 1, 5, 12, 17, 17, 12, 1, 6, 17, 29, 34, 29, 21, 1, 7, 23, 46, 63, 63, 50, 33, 1, 8, 30, 69, 109, 126, 113, 83, 55, 1, 9, 38, 99, 178, 235, 239, 196, 138, 88, 1, 10, 47, 137, 277, 413, 474, 435, 334, 226, 144
Offset: 0

Views

Author

Johannes W. Meijer, Oct 19 2013

Keywords

Comments

The right hand columns of triangle T(n, k) represent the Kn2p sums of the ‘Races with Ties’ triangle A035317. See A180662 for the definitions of these sums.
The row sums lead to A094687, the convolution of Fibonacci and Jacobsthal numbers, and the alternating row sums lead to A008346.
The backwards antidiagonal sums equal Kn21(n) = (-1)^n*A175722(n).

Examples

			The first few rows of triangle T(n, k), n >= 0 and 0 <= k <= n.
n/k 0   1   2    3    4     5     6     7
------------------------------------------------
0|  1
1|  1,  1
2|  1,  2,  3
3|  1,  3,  5,   4
4|  1,  4,  8,   9,   8
5|  1,  5, 12,  17,  17,   12
6|  1,  6, 17,  29,  34,   29,   21
7|  1,  7, 23,  46,  63,   63,   50,   33
The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0.
n/k 0   1   2    3    4     5     6     7
------------------------------------------------
0|  1,  1,  3,   4,   8,   12,   21,   33
1|  1,  2,  5,   9,  17,   29,   50,   83
2|  1,  3,  8,  17,  34,   63,  113,  196
3|  1,  4, 12,  29,  63,  126,  239,  435
4|  1,  5, 17,  46, 109,  235,  474,  909
5|  1,  6, 23,  69, 178,  413,  887, 1796
6|  1,  7, 30,  99, 277,  690, 1577, 3373
7|  1,  8, 38, 137, 414, 1104, 2681, 6054

Crossrefs

Cf. (Triangle columns) A000012, A000027, A089071, A052952, A129696
Cf. A230447, A230448

Programs

  • Maple
    T:= proc(n, k) option remember: if k=0 then return(1) elif k=n then return(combinat[fibonacci](n+2) - (1-(-1)^n)/2) else procname(n-1,k-1)+procname(n-1,k) fi: end: seq(seq(T(n, k), k=0..n), n=0..10); # End first program.
T := proc(n, k): add(A035317(k-p+n-k, k-2*p), p=0..floor(k/2)) end: A035317 := proc(n, k): add((-1)^(i+k) * binomial(i+n-k+1, i), i=0..k) end: seq(seq(T(n, k), k=0..n), n=0..10); # End second program.

Formula

T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, n) = F(n+2) - (1-(-1)^n)/2 = A052952(n), with F(n) = A000045(n), the Fibonacci numbers, n >= 0 and 0 <= k <= n.
T(n+p-1, n) = sum(A035317(n-k+p-1, n-2*k), k=0..floor(n/2)), n >= 0 and p >= 1.
The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0.
Tsq(n, k) = sum(Tsq(n-1, i), i=0..k), n >= 1 and k >= 0, with Tsq(0, k) = A052952(k).
Tsq(n, k) = sum(A035317(n+k-i, k-2*i), i=0..floor(k/2)), n >= 0 and k >= 0.
Tsq(n, k) = A052952(2*n+k) - sum(A035317(n+k+i+1, k+2*i+2), i = 0..n-1)
The G.f. generates the terms in the n-th row of the square array Tsq(n, k).
G.f.: (-1)^(n)/((-1+x+x^2)*(x+1)*(x-1)^(n+1)), n >= 0.
Showing 1-3 of 3 results.

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

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