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.

A368518 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 + 3*x^2.

Original entry on oeis.org

1, 1, 2, 2, 4, 7, 3, 10, 18, 20, 5, 20, 51, 68, 61, 8, 40, 118, 220, 251, 182, 13, 76, 264, 584, 905, 888, 547, 21, 142, 558, 1452, 2678, 3540, 3076, 1640, 34, 260, 1145, 3380, 7279, 11536, 13418, 10456, 4921, 55, 470, 2286, 7548, 18391, 33990, 47600, 49552
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2024

Keywords

Comments

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

Examples

			First eight rows:
   1
   1    2
   2    4    7
   3   10   18    20
   5   20   51    68    61
   8   40  118   220   251   182
  13   76  264   584   905   888   547
  21  142  558  1452  2678  3540  3076  1640
Row 4 represents the polynomial p(4,x) = 3 + 10*x + 18*x^2 + 20*x^3, so (T(4,k)) = (3,10,18,20), k=0..3.

Links

Crossrefs

Cf. A000045 (column 1); A002605, (p(n,n-1)); A030195 (row sums), (p(n,1)); A182228 (alternating row sums), (p(n,-1)); A015545, (p(n,2)); A099012, (p(n,-2)); A087567, (p(n,3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152, A368153, A368154, A368155, A368156.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 + 3x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

Formula

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 2*x, u = p(2,x), and v = 1 + 32*x^2.
p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 4*x + 16*x^2), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

A225799 a(n) = Sum_{k=0..n} binomial(n,k) * 10^(n-k) * Fibonacci(n+k).

Original entry on oeis.org

0, 11, 143, 3058, 55341, 1052755, 19717984, 371084087, 6973353387, 131101759514, 2464418392865, 46327530894271, 870879506447808, 16371134451297043, 307750614069672631, 5785211638097121890, 108752568228856901349, 2044371455527726003547, 38430858858805840293152
Offset: 0

Views

Author

John Molokach, Jul 27 2013

Keywords

Comments

This sequence is part of a family of Fibonacci-like sequences, where:
Sum_{k=0..n} binomial(n,k)*m^(n-k)*Fibonacci(n+k) produces a sequence whose terms are divisible by (m+1); m>=1.
A recurrence relation for a(n) (m not equal to zero) is:
a(n) = (m+3)*a(n-1) + (m^2+m-1)*a(n-2); a(0)=0, a(1)=m+1.
Notable values of m include:
m = 1: Fibonacci(3n),
m = 0: Fibonacci(2n) (using recurrence relation only - the sum above is undefined for m=0),
m = -1: the zero sequence,
m = -2: (-1)*Fibonacci(n), or A152163(n+2).
For any value of m, the sequence gives a(n*k) divisible by a(n); n>=1, k>=1, m not equal to -1 (zero is not divisible by zero).
Equivalent sequences are given by: Sum_{k=0..n} binomial(n,k) * (m+1)^k * Fibonacci(k).
When these sequences are divided by m+1, we obtain the family of sequences A057088, A015553, A087567, A087579, A087584, A087603, and so on.
Another interesting value of m, m = -3, gives a(2n-1)= -2 * 5^(n-1); a(2n)=0.

Links

Crossrefs

Cf. A000045, A027941, A152163, A014445, A057088, A015553, A087567, A087579, A087584, A087603.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k]*10^(n - k)*Fibonacci[n + k], {k, 0, n}], {n, 0, 25}]
FullSimplify[Table[((13 + 11 Sqrt[5])^n - (13 - 11 Sqrt[5])^n)/(2^n Sqrt[5]), {n, 0, 25}]]
LinearRecurrence[{13,109},{0,11},30] (* Harvey P. Dale, Jul 31 2018 *)

Formula

a(n) = ((13 + 11*sqrt(5))^n - (13 - 11*sqrt(5))^n)/(2^n*sqrt(5)).
a(n) = 13*a(n-1) + 109*a(n-2); a(0)=0, a(1)=11.
G.f.: 11*x*/(1 - 13*x - 109*x^2). - Corrected by Georg Fischer, May 10 2019

A368157 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 + 2*x^2.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 3, 10, 16, 16, 5, 20, 46, 56, 44, 8, 40, 108, 184, 188, 120, 13, 76, 244, 496, 692, 608, 328, 21, 142, 520, 1248, 2088, 2480, 1920, 896, 34, 260, 1074, 2936, 5764, 8256, 8592, 5952, 2448, 55, 470, 2156, 6616, 14764, 24760, 31200, 28992
Offset: 1

Views

Author

Clark Kimberling, Jan 20 2024

Keywords

Comments

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

Examples

			First eight rows:
   1
   1    2
   2    4    6
   3   10   16    16
   5   20   46    56    44
   8   40  108   184   188   120
  13   76  244   496   692   608   328
  21  142  520  1248  2088  2480  1920  896
Row 4 represents the polynomial p(4,x) = 3 + 10*x + 16*x^2 + 16*x^3, so (T(4,k)) = (3,10,16,16), k=0..3.

Links

Crossrefs

Cf. A000045 (column 1); A002605, (p(n,n-1)); A030195 (row sums), (p(n,1)); A182228 (alternating row sums), (p(n,-1)); A015545, (p(n,2)); A099012, (p(n,-2)); A087567, (p(n,3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152, A368153, A368154, A368155, A368156.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 + 2x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

Formula

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 2*x, u = p(2,x), and v = 1 + 2*x^2.
p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 4*x + 13*x^2), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).
Showing 1-3 of 3 results.

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