A227978 a(0)=1, a(1)=2; for n>1, a(n) = n*(2^n+4)/4.

1, 2, 4, 9, 20, 45, 102, 231, 520, 1161, 2570, 5643, 12300, 26637, 57358, 122895, 262160, 557073, 1179666, 2490387, 5242900, 11010069, 23068694, 48234519, 100663320, 209715225, 436207642, 905969691, 1879048220, 3892314141, 8053063710, 16642998303

Offset: 0

Author

Paul Curtz, Oct 07 2013

Keywords

• nonn
• easy

Comments

The inverse binomial transform of A176328/A176591 (see Comments field in A228827) begins: 1, -2, 25/6, -9, 599/30, -45, 4285/42, -231, 15599/30, -1161, 169625/66, ... Consider these values without sign and the fractions rounded to the nearest integer, the sequence lists the resulting numbers.

Differences table of a(n):

1, 2, 4, 9, 20, 45, 102, 231, 520, 1161, ...

1, 2, 5, 11, 25, 57, 129, 289, 641, 1409, ... After 2: 2^m*(m+4)+1.

1, 3, 6, 14, 32, 72, 160, 352, 768, 1664, ... A078836 (after 3).

2, 3, 8, 18, 40, 88, 192, 416, 896, 1920, ... A129955.

1, 5, 10, 22, 48, 104, 224, 480, 1024, 2176, ... A079861 (after 5).

4, 5, 12, 26, 56, 120, 256, 544, 1152, 2432, ... After 5: 2^m*(m+12).

1, 7, 14, 30, 64, 136, 288, 608, 1280, 2688, ... After 7: 2^m*(m+14).

6, 7, 16, 34, 72, 152, 320, 672, 1408, 2944, ..., etc.

(n-1)*a(n)-n*a(n-1) = A001788(n-1) for n>1. [Bruno Berselli, Oct 11 2013]

Links

• Bruno Berselli, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Crossrefs

Cf. A001788, A079862, A079863.

Programs

Magma

[1,2] cat [n*(2^n+4)/4: n in [2..40]]; // Bruno Berselli, Oct 11 2013

Mathematica

Join[{1, 2}, Table[n (2^n + 4)/4, {n, 2, 40}]] (* Bruno Berselli, Oct 11 2013 *)

PARI

a(n) = if (n == 0, 1, if (n == 1, 2, n*(2^n+4)/4)); \\ Michel Marcus, Oct 11 2013

Formula

a(2n+2) = A229135(n+1); a(2n-1) = -A228767(n) for n>0.

a(n) = 6*a(n-1) -13*a(n-2) +12*a(n-3) -4*a(n-4) for n>5.

G.f.: (1-4*x+5*x^2-x^3-2*x^4+2*x^5)/((1-x)^2*(1-2*x)^2). - Colin Barker, Oct 09 2013

Extensions

More terms from Colin Barker, Oct 09 2013