A257448 a(n) = 13*(2^n - 1) - 3*n^2 - 9*n.

1, 9, 37, 111, 283, 657, 1441, 3051, 6319, 12909, 26149, 52695, 105859, 212265, 425161, 851043, 1702903, 3406725, 6814477, 13630095, 27261451, 54524289, 109050097, 218101851, 436205503, 872412957, 1744828021, 3489658311, 6979319059, 13958640729

Offset: 1

Author

Luciano Ancora, Apr 23 2015

Keywords

• nonn
• easy

Comments

These numbers belong to a family of sequences obtained as follows:

. A000225: 1*(2^n-1);

. A050488: 3*(2^n-1) - 2*n;

. a(n): 13*(2^n-1) - 3*n^2 - 9*n;

. A257449: 75*(2^n-1) - 4*n^3 - 18*n^2 - 52*n;

. A257450: 541*(2^n-1) - 5*n^4 - 30*n^3 - 130*n^2 - 375*n,

where the sequence 1, 3, 13, 75, 541, ... is A000670 (after the first term), and A208744 gives the triangle of coefficients:

2;

3, 9;

4, 18, 52;

5, 30, 130, 375;

6, 45, 260, 1125, 3246;

7, 63, 455, 2625, 11361, 32781, etc.

Also, the antidiagonal sums in the array are given by the formula (6*n^2 + 6*k*n + (k-1)*k)*(k+n)!/((k+3)!*(n-1)!) for k = 0, 1, 2, 3, 4, ... (see Example field).

Examples

			By the second comment, the array begins (antidiagonals in A046902):
k=0: 1,  8, 27,  64,  125,  216, ...  A000578
k=1: 1,  9, 36, 100,  225,  441, ...  A000537
k=2: 1, 10, 46, 146,  371,  812, ...  A024166
k=3: 1, 11, 57, 203,  574, 1386, ...  A101094
k=4: 1, 12, 69, 272,  846, 2232, ...  A101097
k=5: 1, 13, 82, 354, 1200, 3432, ...  A101102
k=6: 1, 14, 96, 450, 1650, 5082, ...  A254469
...
See also A254469 (Example field).
		

Links

• Index entries for linear recurrences with constant coefficients, signature (5, -9, 7, -2).

Crossrefs

Cf. A000225, A000670, A050488, A208744, A257449, A257450.

Programs

Magma

[13*(2^n-1)-3*n^2-9*n: n in [1..30]]; // Vincenzo Librandi, Apr 24 2015

Mathematica

Table[13 (2^n - 1) - 3 n^2 - 9n, {n, 30}]
CoefficientList[Series[x (1 + 4 x + x^2)/((1 - x)^3*(1 - 2 x)), {x, 0, 30}], x] (* Michael De Vlieger, Nov 14 2016 *)

Formula

G.f.: x*(1+4*x+x^2)/((1-x)^3*(1-2*x)).

a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4) for n>4. - Ray Chandler, Jul 25 2015

Extensions

Edited by Bruno Berselli, Apr 28 2015