A083077 -id:A083077 - cp's OEIS frontend

A083075 Square array read by antidiagonals: T(n,k) = (k(2k+3)^n + 1)/(k+1).

1, 1, 1, 1, 3, 1, 1, 5, 13, 1, 1, 7, 33, 63, 1, 1, 9, 61, 229, 313, 1, 1, 11, 97, 547, 1601, 1563, 1, 1, 13, 141, 1065, 4921, 11205, 7813, 1, 1, 15, 193, 1831, 11713, 44287, 78433, 39063, 1, 1, 17, 253, 2893, 23801, 128841, 398581, 549029, 195313, 1, 1, 19, 321

Offset: 0

Paul Barry, Apr 23 2003

			Array begins:
  1     1     1     1     1 ...
  1     3    13    63   313 ...
  1     5    33   229  1601 ...
  1     7    61   547  4921 ...
  1     9    97  1065 11713 ...
  ...

Nathaniel Johnston, Antidiagonals 0..100, flattened

Rows include A034478, A083076, A066443, A083077, A083078.
Columns include odds, A082109, A083079.
Diagonals include A083079, A083080, A083081, A083082.

T := proc(n,k) return (k*(2*k+3)^n+1)/(k+1): end: seq(seq(T(k,n-k),k=0..n),n=0..10); # Nathaniel Johnston, Jun 26 2011

A083076 Third row of number array A083075.

Original entry on oeis.org

1, 5, 33, 229, 1601, 11205, 78433, 549029, 3843201, 26902405, 188316833, 1318217829, 9227524801, 64592673605, 452148715233, 3165041006629, 22155287046401, 155087009324805, 1085609065273633, 7599263456915429, 53194844198408001

Offset: 0

Paul Barry, Apr 23 2003

Binomial transform of A067411. Inverse binomial transform of A082412.

Trinomial transform of Jacobsthal numbers A001045. - Paul Barry, Sep 10 2007

Nathaniel Johnston, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A034478, A066443, A082761, A083077.

[(2*7^n+1)/3: n in [0..30]]; // Vincenzo Librandi, Nov 06 2011

seq((2*7^n+1)/3,n=0..20); # Nathaniel Johnston, Jun 26 2011

a(n) = (2*7^n + 1)/3.
G.f.: (1-3*x)/((1-x)*(1-7*x)).
E.g.f.: (2*exp(7*x) + exp(x))/3.
a(n) = Sum_{k=0..2*n} trinomial(n,k)*Fibonacci(k+1), where trinomial(n,k) are the trinomial coefficients (A027907). - Paul Barry, Sep 10 2007
a(n) = 7*a(n-1) - 2, a(n) = 8*a(n-1) - 7*a(n-2). - Vincenzo Librandi, Nov 06 2011

A083078 6th row of number array A083075.

Original entry on oeis.org

1, 11, 141, 1831, 23801, 309411, 4022341, 52290431, 679775601, 8837082811, 114882076541, 1493466995031, 19415070935401, 252395922160211, 3281146988082741, 42654910845075631, 554513840985983201

Offset: 0

Paul Barry, Apr 23 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (14,-13).

Cf. A083075, A083077.

LinearRecurrence[{14,-13},{1,11},30] (* Harvey P. Dale, Dec 27 2013 *)

a(n) = (5*13^n + 1)/6.
G.f.: (1-3*x)/((1-x)*(1-13*x)).
E.g.f.: (5*exp(13*x) + exp(x))/6.
a(n) = 13*a(n-1) - 2; a(n) = 14*a(n-1) - 13*a(n-2). - Vincenzo Librandi, Nov 11 2011

A199753 4*11^n+1.

Original entry on oeis.org

5, 45, 485, 5325, 58565, 644205, 7086245, 77948685, 857435525, 9431790765, 103749698405, 1141246682445, 12553713506885, 138090848575725, 1518999334332965, 16708992677662605, 183798919454288645, 2021788113997175085

Offset: 0

Vincenzo Librandi, Nov 10 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (12,-11).

Magma
```
[4*11^n+1: n in [0..30]];
```

4*11^Range[0,20]+1 (* or *) LinearRecurrence[{12,-11},{5,45},20] (* Harvey P. Dale, Mar 28 2012 *)

a(n) = 5*A083077
a(n) = 11*a(n-1)-10.
a(n) = 12*a(n-1)-11*a(n-2).
G.f.: 5*(1-3*x)/((1-x)*(1-11*x)).

cp's OEIS Frontend

A083075 Square array read by antidiagonals: T(n,k) = (k(2k+3)^n + 1)/(k+1).

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A083076 Third row of number array A083075.

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A083078 6th row of number array A083075.

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A199753 4*11^n+1.

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cp's OEIS Frontend

A083075 Square array read by antidiagonals: T(n,k) = (k*(2*k+3)^n + 1)/(k+1).

Views

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Maple

A083076 Third row of number array A083075.

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Magma

Maple

Formula

A083078 6th row of number array A083075.

Views

Author

Keywords

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Programs

Mathematica

Formula

A199753 4*11^n+1.

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Author

Keywords

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Programs

Magma

Mathematica

Formula

A083075 Square array read by antidiagonals: T(n,k) = (k(2k+3)^n + 1)/(k+1).