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.

A048755 Partial sums of A048693.

Original entry on oeis.org

1, 7, 20, 52, 129, 315, 764, 1848, 4465, 10783, 26036, 62860, 151761, 366387, 884540, 2135472, 5155489, 12446455, 30048404, 72543268, 175134945, 422813163, 1020761276, 2464335720, 5949432721, 14363201167, 34675835060, 83714871292, 202105577649, 487926026595
Offset: 0

Views

Author

Barry E. Williams

Keywords

Links

Crossrefs

Cf. A048693, A048654, A048655.

Programs

  • Mathematica
    Accumulate[LinearRecurrence[{2,1},{1,6},30]] (* or *) LinearRecurrence[ {3,-1,-1},{1,7,20},40] (* Harvey P. Dale, Mar 29 2013 *)

Formula

a(n)=2*a(n-1)+a(n-2)+5; a(0)=1, a(1)=6.
a(n)=[ {(6+(7/2)*sqrt(2))(1+sqrt(2))^n - (6-(7/2)*sqrt(2))(1-sqrt(2))^n}/ 2*sqrt(2) ]-5/2.
G.f. ( 1+4*x ) / ( (x-1)*(x^2+2*x-1) ). - R. J. Mathar, Nov 08 2012
a(0)=1, a(1)=7, a(2)=20, a(n)=3*a(n-1)-a(n-2)-a(n-3). - Harvey P. Dale, Mar 29 2013

Extensions

More terms from Harvey P. Dale, Mar 29 2013

A117584 Generalized Pellian triangle.

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 1, 4, 7, 12, 1, 5, 9, 17, 29, 1, 6, 11, 22, 41, 70, 1, 7, 13, 27, 53, 99, 169, 1, 8, 15, 32, 65, 128, 239, 408, 1, 9, 17, 37, 77, 157, 309, 577, 985, 1, 10, 19, 42, 89, 186, 379, 746, 1393, 2378
Offset: 1

Views

Author

Gary W. Adamson, Mar 29 2006

Keywords

Examples

			First few rows of the triangle are:
  1;
  1, 2;
  1, 3,  5;
  1, 4,  7, 12;
  1, 5,  9, 17, 29;
  1, 6, 11, 22, 41, 70;
  1, 7, 13, 27, 53, 99, 169;
  ...
The triangle rows are antidiagonals of the generalized Pellian array:
  1, 2,  5, 12, 29, ...
  1, 3,  7, 17, 41, ...
  1, 4,  9, 22, 53, ...
  1, 5, 11, 27, 65, ...
  ...
For example, in the row (1, 5, 11, 27, 65, ...), 65 = 2*27 + 11.

Links

Crossrefs

Diagonals include A000129, A001333, A048654, A048655, A048693.
Cf. A117185.

Programs

  • Magma
    P:= func< n | Round( ((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2*Sqrt(2)) ) >;
T:= func< n,k | P(k) + (n-1)*P(k-1) >;
[T(n-k+1, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 05 2021
  • Mathematica
    T[n_, k_]:= Fibonacci[k, 2] + (n-1)*Fibonacci[k-1, 2];
Table[T[n-k+1, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jul 05 2021 *)
  • Sage
    def T(n,k): return lucas_number1(k,2,-1) + (n-1)*lucas_number1(k-1,2,-1)
flatten([[T(n-k+1, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jul 05 2021

Formula

Antidiagonals of the generalized Pellian array. First row of the array = A000129: (1, 2, 5, 12, ...). n-th row of the array starts (1, n+1, ...); as a Pellian sequence.
From G. C. Greubel, Jul 05 2021: (Start)
T(n, k) = P(k) + (n-1)*P(k-1), where P(n) = A000129(n) (square array).
Sum_{k=1..n} T(n-k+1, k) = A117185(n). (End)

A117895 Triangle T(n, k) = (k-n)*A000129(k+1) + (3*n-3*k+1)*A000129(k) with T(n,0) = 1, for 0 <= k <= n-1, read by rows.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 4, 8, 1, 5, 5, 11, 19, 1, 6, 6, 14, 26, 46, 1, 7, 7, 17, 33, 63, 111, 1, 8, 8, 20, 40, 80, 152, 268, 1, 9, 9, 23, 47, 97, 193, 367, 647, 1, 10, 10, 26, 54, 114, 234, 466, 886, 1562, 1, 11, 11, 29, 61, 131, 275, 565, 1125, 2139, 3771, 1, 12, 12, 32, 68, 148, 316, 664, 1364, 2716, 5164, 9104
Offset: 0

Views

Author

Gary W. Adamson, Mar 30 2006

Keywords

Comments

Successive deletions of the right borders of triangle A117894 produces triangles whose row sums = generalized Pell sequences starting (1, 2...), (1, 3...), (1, 4...); etc. Row sums of A117894 = A000129: (1, 2, 5...). Row sums of A117895 = A001333: (1, 3, 7...). Deletion of the border of A117895 would produce a triangle with row sums of the Pell sequence A048654 (1, 4, 9...); and so on.

Examples

			First few rows of the triangle are:
  1;
  1, 2;
  1, 3, 3;
  1, 4, 4,  8;
  1, 5, 5, 11, 19;
  1, 6, 6, 14, 26, 46;
  1, 7, 7, 17, 33, 63, 111;
  1, 8, 8, 20, 40, 80, 152, 268;
...
Row 4, (1, 4, 4, 8) is produced by adding (0, 1, 1, 3) to row 4 of A117894: (1, 3, 3, 5).

Links

Crossrefs

Cf. A000129, A001333, A048654, A048655, A048693, A117894.

Programs

  • Magma
    Pell:= func< n | Round(((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2*Sqrt(2))) >;
[k eq 0 select 1 else (k-n)*Pell(k+1) + (3*n-3*k+1)*Pell(k): k in [0..n-1], n in [0..12]]; // G. C. Greubel, Sep 27 2021
  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0, 1, (k-n)*Fibonacci[k+1, 2] + (3*n-3*k +1)*Fibonacci[k, 2]]; Table[T[n, k], {n,0,12}, {k,0,n-1}]//Flatten (* G. C. Greubel, Sep 27 2021 *)
  • Sage
    def P(n): return lucas_number1(n, 2, -1)
def A117895(n,k): return 1 if (k==0) else (k-n)*P(k+1) + (3*n-3*k+1)*P(k)
flatten([[A117895(n,k) for k in (0..n-1)] for n in (0..12)]) # G. C. Greubel, Sep 27 2021

Formula

Delete right border of triangle A117894. Alternatively, let row 1 = 1 and using the heading 0, 1, 1, 3, 7, 17, 41, 99, 239...(i.e. A001333 starting with 0, 1, 1, 3...); add the first n terms of the heading to n-th row of triangle A117894.
From G. C. Greubel, Sep 27 2021: (Start)
T(n, k) = (k-n)*A000129(k+1) + (3*n-3*k+1)*A000129(k) with T(n,0) = 1.
T(n, 1) = n+1 for n >= 1.
T(n, 2) = n+1 for n >= 2.
T(n, n) = 2*[n=0] + A078343(n). (End)

Extensions

New name and more terms added by G. C. Greubel, Sep 27 2021
Showing 1-3 of 3 results.

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