A227418 -id:A227418 - cp's OEIS frontend

A228879 a(n+2) = 3*a(n), starting 4,7.

4, 7, 12, 21, 36, 63, 108, 189, 324, 567, 972, 1701, 2916, 5103, 8748, 15309, 26244, 45927, 78732, 137781, 236196, 413343, 708588, 1240029, 2125764, 3720087, 6377292, 11160261, 19131876, 33480783, 57395628, 100442349, 172186884, 301327047, 516560652

Offset: 0

Richard R. Forberg, Sep 06 2013

Successive terms are the square roots of expressions of prior terms. The same recursive formula (see below) can be applied using any term of A001353 as the initializing value to produce the family of sequences, as given in the array A227418. This sequence uses A001353(2) = 4, and is the third row of that array.
a(4n) are the squares of A008776(n).
Binomial transform of a(n) is A021006.
First differences of a(n) = A083658 (without initial two terms).
2nd differences of a(n) = A068911 (with initial term).
a(n-1) is the number of n-digit base 10 numbers where all the digits are even numbers, and each digit differs by 2 from the previous and the next digit. - Graeme McRae, Jun 09 2014

Paolo Xausa, Table of n, a(n) for n = 0..1000
Twitter / MathQuizzes, Puzzle relating to this sequence
Index entries for linear recurrences with constant coefficients, signature (0,3).

Cf. A001353, A008776, A021006, A068911, A083658, A227418.

Row n = 5 of A377000.

LinearRecurrence[{0, 3}, {4, 7}, 50] (* Paolo Xausa, Oct 14 2024 *)

Vec(-(7*x+4)/(3*x^2-1) + O(x^100)) \\ Colin Barker, Jun 09 2014

a(n) = sqrt(3*a(n-1)^2 + (-3)^(n-1)), a(0) = 4.
This divisibility relation also applies: a(n+3) = a(n+2)*a(n+1)/a(n).
G.f.: -(7*x+4) / (3*x^2-1). - Colin Barker, Jun 09 2014
From Stefano Spezia, Mar 20 2022: (Start)
a(n) = 3^((n-1)/2)*(4*sqrt(3) + 7 + (-1)^n*(4*sqrt(3) - 7))/2.
E.g.f.: 4*cosh(sqrt(3)*x) + 7*sinh(sqrt(3)*x)/sqrt(3). (End)

More terms from Colin Barker, Jun 09 2014

A292466 Triangle read by rows: T(n,k) = 4T(n-1,k-1) + T(n,k-1) with T(2m,0) = 0 and T(2*m+1,0) = 5^m.

Original entry on oeis.org

0, 1, 1, 0, 4, 8, 5, 5, 21, 53, 0, 20, 40, 124, 336, 25, 25, 105, 265, 761, 2105, 0, 100, 200, 620, 1680, 4724, 13144, 125, 125, 525, 1325, 3805, 10525, 29421, 81997, 0, 500, 1000, 3100, 8400, 23620, 65720, 183404, 511392, 625, 625, 2625, 6625, 19025, 52625

Offset: 0

Seiichi Manyama, Sep 22 2017

			First few rows are:
    0;
    1,   1;
    0,   4,   8;
    5,   5,  21,   53;
    0,  20,  40,  124,  336;
   25,  25, 105,  265,  761,  2105;
    0, 100, 200,  620, 1680,  4724, 13144;
  125, 125, 525, 1325, 3805, 10525, 29421, 81997.
--------------------------------------------------------------
The diagonal is      {0, 1,  8,  53, 336, 2105, ...} and
the next diagonal is {1, 4, 21, 124, 761, 4724, ...}.
Two sequences have the following property:
     1^2 - 5*   0^2 = 1      (= 11^0),
     4^2 - 5*   1^2 = 11     (= 11^1),
    21^2 - 5*   8^2 = 121    (= 11^2),
   124^2 - 5*  53^2 = 1331   (= 11^3),
   761^2 - 5* 336^2 = 14641  (= 11^4),
  4724^2 - 5*2105^2 = 161051 (= 11^5),
  ...

Seiichi Manyama, Rows n = 0..139, flattened

The diagonal of the triangle is A091870.
The next diagonal of the triangle is A108404.
T(n,k) = b*T(n-1,k-1) + T(n,k-1): A292789 (b=-3), A292495 (b=-2), A117918 and A228405 (b=1), A227418 (b=2), this sequence (b=4).

T(n+1,n)^2 - 5*T(n,n)^2 = 11^n.

A292495 Triangle read by rows: T(n,k) = (-2)T(n-1,k-1) + T(n,k-1) with T(2m,0) = 0 and T(2*m+1,0) = (-1)^m.

Original entry on oeis.org

0, 1, 1, 0, -2, -4, -1, -1, 3, 11, 0, 2, 4, -2, -24, 1, 1, -3, -11, -7, 41, 0, -2, -4, 2, 24, 38, -44, -1, -1, 3, 11, 7, -41, -117, -29, 0, 2, 4, -2, -24, -38, 44, 278, 336, 1, 1, -3, -11, -7, 41, 117, 29, -527, -1199, 0, -2, -4, 2, 24, 38, -44, -278, -336, 718

Offset: 0

Seiichi Manyama, Sep 22 2017

			First few rows are:
   0;
   1,  1;
   0, -2, -4;
  -1, -1,  3,  11;
   0,  2,  4,  -2, -24;
   1,  1, -3, -11,  -7,  41;
   0, -2, -4,   2,  24,  38,  -44;
  -1, -1,  3,  11,   7, -41, -117, -29;
   0,  2,  4,  -2, -24, -38,   44, 278, 336.

Seiichi Manyama, Rows n = 0..139, flattened

The diagonal of the triangle is related to A099456.
The next diagonal of the triangle is related to A139011.
Cf. A006495, A006496.
T(n,k) = b*T(n-1,k-1) + T(n,k-1): A292789 (b=-3), this sequence (b=-2), A117918 and A228405 (b=1), A227418 (b=2), A292466 (b=4).

T(n+1,n)^2 + T(n,n)^2 = 5^n.

A292789 Triangle read by rows: T(n,k) = (-3)T(n-1,k-1) + T(n,k-1) with T(2m,0) = 0 and T(2*m+1,0) = (-2)^m.

Original entry on oeis.org

0, 1, 1, 0, -3, -6, -2, -2, 7, 25, 0, 6, 12, -9, -84, 4, 4, -14, -50, -23, 229, 0, -12, -24, 18, 168, 237, -450, -8, -8, 28, 100, 46, -458, -1169, 181, 0, 24, 48, -36, -336, -474, 900, 4407, 3864, 16, 16, -56, -200, -92, 916, 2338, -362, -13583, -25175, 0, -48

Offset: 0

Seiichi Manyama, Sep 23 2017

			First few rows are:
   0;
   1,   1;
   0,  -3,  -6;
  -2,  -2,   7,  25;
   0,   6,  12,  -9,  -84;
   4,   4, -14, -50,  -23,  229;
   0, -12, -24,  18,  168,  237,  -450;
  -8,  -8,  28, 100,   46, -458, -1169,  181;
   0,  24,  48, -36, -336, -474,   900, 4407, 3864.
--------------------------------------------------------------
The diagonal is      {0,  1, -6, 25, -84, ...} and
the next diagonal is {1, -3,  7, -9, -23, ...}.
Two sequences have the following property:
      1^2 + 2*    0^2 = 1      (= 11^0),
   (-3)^2 + 2*    1^2 = 11     (= 11^1),
      7^2 + 2* (-6)^2 = 121    (= 11^2),
   (-9)^2 + 2*   25^2 = 1331   (= 11^3),
  (-23)^2 + 2*(-84)^2 = 14641  (= 11^4),
  ...

Seiichi Manyama, Rows n = 0..139, flattened

T(n,k) = b*T(n-1,k-1) + T(n,k-1): this sequence (b=-3), A292495 (b=-2), A117918 and A228405 (b=1), A227418 (b=2), A292466 (b=4).

T(n+1,n)^2 + 2*T(n,n)^2 = 11^n.

cp's OEIS Frontend

A228879 a(n+2) = 3*a(n), starting 4,7.

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A292466 Triangle read by rows: T(n,k) = 4T(n-1,k-1) + T(n,k-1) with T(2m,0) = 0 and T(2*m+1,0) = 5^m.

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A292495 Triangle read by rows: T(n,k) = (-2)T(n-1,k-1) + T(n,k-1) with T(2m,0) = 0 and T(2*m+1,0) = (-1)^m.

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A292789 Triangle read by rows: T(n,k) = (-3)T(n-1,k-1) + T(n,k-1) with T(2m,0) = 0 and T(2*m+1,0) = (-2)^m.

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cp's OEIS Frontend

A228879 a(n+2) = 3*a(n), starting 4,7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A292466 Triangle read by rows: T(n,k) = 4*T(n-1,k-1) + T(n,k-1) with T(2*m,0) = 0 and T(2*m+1,0) = 5^m.

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Author

Keywords

Examples

Links

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Formula

A292495 Triangle read by rows: T(n,k) = (-2)*T(n-1,k-1) + T(n,k-1) with T(2*m,0) = 0 and T(2*m+1,0) = (-1)^m.

Views

Author

Keywords

Examples

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Formula

A292789 Triangle read by rows: T(n,k) = (-3)*T(n-1,k-1) + T(n,k-1) with T(2*m,0) = 0 and T(2*m+1,0) = (-2)^m.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A292466 Triangle read by rows: T(n,k) = 4T(n-1,k-1) + T(n,k-1) with T(2m,0) = 0 and T(2*m+1,0) = 5^m.

A292495 Triangle read by rows: T(n,k) = (-2)T(n-1,k-1) + T(n,k-1) with T(2m,0) = 0 and T(2*m+1,0) = (-1)^m.

A292789 Triangle read by rows: T(n,k) = (-3)T(n-1,k-1) + T(n,k-1) with T(2m,0) = 0 and T(2*m+1,0) = (-2)^m.