A168673 Binomial transform of A169609.

1, 4, 10, 20, 38, 74, 148, 298, 598, 1196, 2390, 4778, 9556, 19114, 38230, 76460, 152918, 305834, 611668, 1223338, 2446678, 4893356, 9786710, 19573418, 39146836, 78293674, 156587350, 313174700, 626349398, 1252698794, 2505397588, 5010795178, 10021590358

Offset: 0

Author

Paul Curtz, Dec 02 2009

Keywords

• nonn
• easy

Comments

Sequence and successive differences are identical to their third differences. See principal sequence A024495. Main diagonal of the array of successive differences is A083595 (1,6,8,20,36,...).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Crossrefs

Cf. A169609, A144437, A168615, A024495, A083595, A130772, A062092, A130151.

Programs

Magma

I:=[1,4,10]; [n le 3 select I[n] else 3*Self(n-1)- 3*Self(n-2)+2*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jul 30 2016

Mathematica

LinearRecurrence[{3,-3,2},{1,4,10},25] (* G. C. Greubel, Jul 29 2016 *)
RecurrenceTable[{a[0] == 1, a[1] == 4, a[2] == 10, a[n] == 3 a[n-1] - 3 a[n-2] + 2 a[n-3]}, a, {n, 40}] (* Vincenzo Librandi, Jul 30 2016 *)

PARI

a(n)=([0,1,0; 0,0,1; 2,-3,3]^n*[1;4;10])[1,1] \\ Charles R Greathouse IV, Jul 30 2016

Formula

a(n+1) - 2a(n) = A130772(n).

a(n) = A062092(n) - A130151(n+1).

a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) for n > 2; a(0) = 1, a(1) = 4, a(2) = 10.

G.f.: (1 + x + x^2)/(1 -3*x +3*x^2 -2*x^3). - Philippe Deléham, Dec 03 2009

Extensions

Edited and extended by Klaus Brockhaus, Dec 03 2009