A307688 a(n) = 2*a(n-1)-2*a(n-2)+a(n-3)+2*a(n-4) with a(0)=a(1)=0, a(2)=2, a(3)=3.

0, 0, 2, 3, 2, 0, 3, 14, 26, 27, 22, 44, 123, 234, 310, 363, 586, 1224, 2259, 3382, 4642, 7227, 13070, 23092, 36555, 54450, 85022, 143883, 245282, 396720, 616803, 973214, 1600106, 2664027, 4334662, 6887804, 10970523, 17828154, 29272390, 47634603, 76493626

Offset: 0

Author

Jean-François Alcover and Paul Curtz, Apr 22 2019

Keywords

• nonn
• easy

Comments

This is an autosequence of the second kind, the companion to A192395.

The array D(n, k) of successive differences begins:

0, 0, 2, 3, 2, 0, 3, 14, 26, 27, ...

0, 2, 1, -1, -2, 3, 11, 12, 1, -5, ...

2, -1, -2, -1, 5, 8, 1, -11, -6, 27, ...

-3, -1, 1, 6, 3, -7, -12, 5, 33, 30, ...

2, 2, 5, -3, -10, -5, 17, 28, -3, -55, ...

0, 3, -8, -7, 5, 22, 11, -31, -52, 13, ...

...

The main diagonal (0,2,-2,6,-10,22,...) is essentially the same as A151575.

It can be seen that abs(D(n, 1)) = D(1, n).

The diagonal starting from the third 0 is -(-1)^n*11*A001045(n), inverse binomial transform of 11*A001045(n).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• OEIS Wiki, Autosequence
• Index entries for linear recurrences with constant coefficients, signature (2,-2,1,2).

Crossrefs

Cf. A151575, A192395.

Cf. A001045 (first and fifth upper diagonals), A014551 (second upper diagonal), A115102 (third), A155980 (fourth).

Programs

Mathematica

a[0] = a[1] = 0; a[2] = 2; a[3] = 3; a[n_] := a[n] = 2*a[n-1] - 2*a[n-2] + a[n-3] + 2*a[n-4]; Table[a[n], {n, 0, 40}]
LinearRecurrence[{2,-2,1,2},{0,0,2,3},50] (* Harvey P. Dale, Oct 01 2021 *)

PARI

concat([0,0], Vec(x^2*(2 - x) / ((1 - x - x^2)*(1 - x + 2*x^2)) + O(x^40))) \\ Colin Barker, Apr 22 2019

Formula

G.f.: x^2*(2 - x) / ((1 - x - x^2)*(1 - x + 2*x^2)). - Colin Barker, Apr 22 2019