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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A210792 Triangle of coefficients of polynomials v(n,x) jointly generated with A210791; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 5, 3, 1, 10, 11, 5, 1, 19, 28, 25, 8, 1, 36, 62, 81, 50, 13, 1, 69, 129, 218, 193, 98, 21, 1, 134, 261, 533, 597, 442, 185, 34, 1, 263, 522, 1235, 1631, 1559, 952, 343, 55, 1, 520, 1040, 2773, 4129, 4763, 3758, 1985, 625, 89, 1, 1033, 2071
Offset: 1

Views

Author

Clark Kimberling, Mar 26 2012

Keywords

Comments

Row n starts with 1 and ends with F(n+1), where F=A000045 (Fibonacci numbers).
Column 2: A052944.
Row sums: A000244 (powers of 3).
Alternating row sums: A001333.
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, 1/2, 3/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 29 2012

Examples

			First five rows:
  1;
  1,  2;
  1,  5,  3;
  1, 10, 11,  5;
  1, 19, 28, 25,  8;
First three polynomials v(n,x):
  1
  1 + 2x
  1 + 5x + 3x^2
From _Philippe Deléham_, Mar 29 2012: (Start)
(1, 0, 1/2, 3/2, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, ...) begins:
  1;
  1,   0;
  1,   2,   0;
  1,   5,   3,   0;
  1,  10,  11,   5,   0;
  1,  19,  28,  25,   8,   0;
  1,  36,  62,  81,  50,  13,   0;
  1,  69, 129, 218, 193,  98,  21,   0; (End)

Crossrefs

Cf. A210791, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 0; h = -1; p = 2; f = 0;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210791 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210792 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A007051 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A000244 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A001129 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A001333 *)

Formula

u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = (x-1)*u(n-1,x) + (x+2)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 29 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1 - 2*x - y*x + 3*y*x^2 - y^2*x^2)/(1 - 3*x - y*x + 2*x^2 + 2*y*x^2 - y^2*x^2).
T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k) - 2*T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(2,1) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)

A101760 Iccanobif prime indices: indices of prime numbers in A014258.

Original entry on oeis.org

3, 4, 5, 7, 9, 18, 19, 21, 22, 25, 27, 47, 97, 107, 154, 186, 205, 303, 363, 1385, 1515, 1546, 2642, 2795, 2825, 3193
Offset: 1

Views

Author

Jonathan Vos Post and Ray Chandler, Dec 15 2004

Keywords

Comments

No more terms through 10^4.

Crossrefs

Cf. A000040, A001129, A014258-A014260, A101759-A101762.

A072210 a(1)=a(2)=1; a(n)=reverse(reverse(a(n-1))+reverse(a(n-2))) for n > 2.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 31, 12, 43, 55, 98, 441, 332, 773, 16, 834, 994, 739, 6341, 3732, 9083, 2816, 1999, 37161, 46162, 73324, 10586, 838011, 933971, 771092, 615964, 396957, 9029221, 2098891, 1118123, 3107025, 4215248, 73123631, 16275022, 89398653, 95664775
Offset: 1

Views

Author

Joseph L. Pe, Jul 03 2002

Keywords

Comments

I call this sequence the Fibonacci mirror sequence for the following reason. For n>2, the expression "a(n)=a(n-1)+a(n-2)" is a valid equation if read backwards. For example, "a(9)=a(8)+a(7)" is "43=12+31", which read backwards is 13+21=34, a valid equation.
Reverse(a(n))=reverse(a(n-1))+reverse(a(n-2)). a(n) is the least natural number k such that reverse(k)=reverse(a(n-1))+reverse(a(n-2)).
(Added Jul 06 2002) Actually, the previous comments are true only if reverse(a(n-1))+reverse(a(n-2)) does not end in the digit 0. It ends in 0 for n = 15, but for no other n < 3 * 10^4. Mark Lewis claims that n = 15 is the only such value of n. He observes that the first fifteen terms of a(n) are the reverses of the first fifteen terms of the Fibonacci sequence. The later terms of a(n) are the reverses of the terms of the Fibonacci sequence starting with 377, 61 (excluding these initial two terms). Lewis' argument depends on his assertion that the (377,61)-sequence is, modulo 10, periodic with period 12 and with no zeros-one for which he, as yet, offers only empirical evidence.

Examples

			a(9)=reverse(reverse(a(8))+reverse(a(7)))=reverse(21+13)=43.

Links

Crossrefs

Cf. A001129, A014258.

Programs

  • Mathematica
    rev[n_] := FromDigits[Reverse[IntegerDigits[n]]]; r = {1, 1}; For[i = 1, i < 30, i++, l = Length[r]; r = Append[r, rev[rev[r[[l]]] + rev[r[[l - 1]]]]]]; r
rev[n_]:=FromDigits[Reverse[IntegerDigits[n]]]; nxt[{a_,b_}]:={b,rev[ rev[ a]+ rev[b]]}; Transpose[NestList[nxt,{1,1},50]][[1]] (* Harvey P. Dale, Apr 25 2014 *)

Extensions

More terms from Harvey P. Dale, Apr 25 2014

A101761 Iccanobif prime indices: Indices of prime numbers in A014259.

Original entry on oeis.org

3, 4, 5, 7, 21, 35, 97, 830, 947, 2627
Offset: 1

Views

Author

Jonathan Vos Post and Ray Chandler, Dec 15 2004

Keywords

Comments

No more terms through 10^4.

Crossrefs

Cf. A000040, A001129, A014258-A014260, A101759-A101762.

A102069 Analogous to the oblong (promic or heteromecic) sequence formed but with reversal digits of factors multiplied.

Original entry on oeis.org

2, 6, 12, 20, 30, 42, 56, 72, 9, 11, 231, 651, 1271, 2091, 3111, 4331, 5751, 7371, 182, 24, 264, 704, 1344, 2184, 3224, 4464, 5904, 7544, 276, 39, 299, 759, 1419, 2279, 3339, 4599, 6059, 7719, 372, 56, 336, 816, 1496, 2376, 3456, 4736, 6216, 7896, 470, 75, 375
Offset: 1

Views

Author

Michael Joseph Halm, Feb 12 2005

Keywords

Crossrefs

Cf. A001129, A002378.

Programs

  • Mathematica
    r[n_]:=FromDigits[Reverse[IntegerDigits[n]]];Table[r[n]*r[n+1],{n,51}] (* James C. McMahon, Jan 07 2024 *)

Formula

a(n) = R(n)*R(n+1), where R(n) is the digital reversal of n.
Previous Showing 31-35 of 35 results.

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