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.

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A117938 Triangle, columns generated from Lucas Polynomials.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 4, 1, 4, 11, 14, 7, 1, 5, 18, 36, 34, 11, 1, 6, 27, 76, 119, 82, 18, 1, 7, 38, 140, 322, 393, 198, 29, 1, 8, 51, 234, 727, 1364, 1298, 478, 47, 1, 9, 66, 364, 1442, 3775, 5778, 4287, 1154, 76, 1, 10, 83, 536, 2599, 8886, 19602, 24476, 14159, 2786, 123
Offset: 1

Views

Author

Gary W. Adamson, Apr 03 2006

Keywords

Comments

Companion triangle using Fibonacci polynomial generators = A073133. Inverse binomial transforms of the columns defines rows of A117937 (with some adjustments of offset).
A309220 is another version of the same triangle (except it omits the last diagonal), and perhaps has a clearer definition. - N. J. A. Sloane, Aug 13 2019

Examples

			First few rows of the triangle are:
  1;
  1, 1;
  1, 2,  3;
  1, 3,  6,   4;
  1, 4, 11,  14,   7;
  1, 5, 18,  36,  34,  11;
  1, 6, 27,  76, 119,  82,  18;
  1, 7, 38, 140, 322, 393, 198, 29;
  ...
For example, T(7,4) = 76 = f(4), x^3 + 3*x = 64 + 12 = 76.

Links

Crossrefs

Cf. A104509, A114525, A117936, A117937, A118980, A118981, A309220.
Cf. A000204 (diagonal), A059100 (column 3), A061989 (column 4).

Programs

  • Maple
    Lucas := proc(n,x) # see A114525
    option remember;
    if  n=0 then
        2;
    elif n =1 then
        x ;
    else
        x*procname(n-1,x)+procname(n-2,x) ;
    end if;
    expand(%) ;
end proc:
A117938 := proc(n::integer,k::integer)
    if k = 1 then
        1;
    else
        subs(x=n-k+1,Lucas(k-1,x)) ;
    end if;
end proc:
seq(seq(A117938(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Aug 16 2019
  • Mathematica
    T[n_, k_]:= LucasL[k-1, n-k+1] - Boole[k==1];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Oct 28 2021 *)
  • Sage
    def A117938(n,k): return 1 if (k==1) else round(2^(1-k)*( (n-k+1 + sqrt((n-k)*(n-k+2) + 5))^(k-1) + (n-k+1 - sqrt((n-k)*(n-k+2) + 5))^(k-1) ))
flatten([[A117938(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 28 2021

Formula

Columns are f(x), x = 1, 2, 3, ..., of the Lucas Polynomials: (1, defined different from A034807 and A114525); (x); (x^2 + 2); (x^3 + 3*x); (x^4 + 4*x^2 + 2); (x^5 + 5*x^3 + 5*x); (x^6 + 6*x^4 + 9*x^2 + 2); (x^7 + 7*x^5 + 14*x^3 + 7*x); ...

Extensions

Terms a(51) and a(52) corrected by G. C. Greubel, Oct 28 2021

A117937 Triangle, rows = inverse binomial transforms of A117938 columns.

Original entry on oeis.org

1, 1, 1, 3, 3, 2, 4, 10, 12, 6, 7, 27, 58, 60, 24, 11, 71, 240, 420, 360, 120, 18, 180, 920, 2460, 3504, 2520, 720, 29, 449, 3360, 13020, 27720, 32760, 20160, 5040, 47, 1107, 11898, 64620, 194184, 337680, 338400, 181440, 40320, 76, 2710, 41268, 307194, 1257120, 3029760, 4415040
Offset: 1

Views

Author

Gary W. Adamson, Apr 04 2006

Keywords

Comments

A117936 is the companion triangle using analogous Fibonacci polynomials. Left border of A117936 = the Lucas numbers; right border = factorials.
[Note that most of the comments here and in many related sequences by the same author refer to some unusual definition of binomial transforms for sequences starting at index 1. - R. J. Mathar, Jul 05 2012]

Examples

			First few rows of the triangle are:
1;
1, 1;
3, 3, 2;
4, 10, 12, 6;
7, 27, 58, 60, 24;
11, 71, 240, 420, 360, 120;
...
For example, row 4: (4, 10, 12, 6) = the inverse binomial transform of column 4 of A117938: (4, 14, 36, 76, 140...), being f(x), x =1,2,3...using the Lucas polynomial x^3 + 3x.

Crossrefs

Cf. A117936, A117938.

Programs

  • Maple
    A117937 := proc(n,k)
    add( A117938(n+i,n)*binomial(k-1,i)*(-1)^(1+i-k),i=0..k-1) ;
end proc:
seq(seq(A117937(n,k),k=1..n),n=1..13) ; # R. J. Mathar, Aug 16 2019

Formula

Rows of the triangle are inverse binomial transforms of A117938 columns. A117938 columns are generated from f(x), Lucas polynomials: (1); (x); (x^2 + 2); (x^3 + 3x); (x^4 + 4x + 2);...

A309717 Convolve Fibonacci, Pell and bronze Fibonacci numbers.

Original entry on oeis.org

0, 0, 0, 1, 6, 28, 114, 432, 1566, 5517, 19068, 65044, 219852, 738316, 2468028, 8222805, 27330858, 90685224, 300521622, 994991716, 3292117698, 10887332473, 35992718136, 118958691528, 393093822744, 1298783453112, 4290755845176, 14174217683209, 46821054068430, 154655837126740
Offset: 0

Views

Author

R. J. Mathar, Aug 16 2019

Keywords

Links

Crossrefs

Cf. A006684, A006190 (bronze Fibonacci numbers), A117936.

Programs

  • Maple
    -x^3/( (x^2+2*x-1)*(x^2+3*x-1)*(x^2+x-1) ) ;
taylor(%,x=0,30) ;
gfun[seriestolist](%) ;

Formula

G.f.: -x^3/( (x^2+2*x-1) * (x^2+3*x-1) * (x^2+x-1) ) = A006190(x) * A000045(x) * A000129(x).
Conjecture: 2*a(n) = A117936(n,3).
2*a(n) = A006190(n) + A000045(n) - 2*A000129(n). - R. J. Mathar, Mar 10 2023, typo corrected by Xiaoyuan Wang and Greg Dresden, May 08 2024
Showing 1-3 of 3 results.

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