A064694 Add column entries of the table with rows (1,2,0,0...), (0,3,4,5,0,0...), (0,0,6,7,8,9,0,0...), (0,0,0,10,11,12,13,14,0,0...), ...

1, 5, 10, 22, 34, 58, 80, 120, 155, 215, 266, 350, 420, 532, 624, 768, 885, 1065, 1210, 1430, 1606, 1870, 2080, 2392, 2639, 3003, 3290, 3710, 4040, 4520, 4896, 5440, 5865, 6477, 6954, 7638, 8170, 8930, 9520, 10360, 11011, 11935, 12650, 13662, 14444

Offset: 1

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Oct 12 2001

Keywords

• nonn
• easy
• nice

Examples

			a(1)=1, a(2)=2+3=5, a(3)=4+6=10, a(4)=5+7+10=22.
		

Links

• Harry J. Smith, Table of n, a(n) for n=1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Crossrefs

Cf. A007584.

Programs

Mathematica

Table[ Sum[ Binomial[n-k+1, 2] + k, {k, 0, Floor[n/2]}], {n, 1, 45}] (* Jean-François Alcover, Sep 16 2013 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,5,10,22,34,58,80},50] (* Harvey P. Dale, Dec 11 2015 *)

PARI

{ for (n=1, 1000, a=sum(k=0, n\2, binomial(n - k + 1, 2) + k); write("b064694.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 22 2009

PARI

Vec(x*(2*x^2+4*x+1)/((x-1)^4*(x+1)^3) + O(x^100)) \\ Colin Barker, Feb 17 2015

Formula

a(2n-1) = n*(n+1)*(7*n-4)/6 (see A007584), a(2n) = n*(n+1)*(7*n+8)/6.

a(n) = sum{k=0..floor((n+1)/2), (n-k+1)ceiling((n-k+1)/2)+k+if(mod(n-k+1, 2) =0, ceiling((n-k+1)/2), 0)}. - Paul Barry, Aug 25 2004

a(n) = sum{k=0..floor(n/2), C(n-k+1,2)+k}; - Paul Barry, Jul 23 2008

a(n) = (2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(14*n+15+17*(-1)^n)/384. - Luce ETIENNE, Feb 17 2015

From Colin Barker, Feb 17 2015: (Start)

a(n) = (7*n^3+30*n^2+32*n)/48 for n even.

a(n) = (7*n^3+27*n^2+17*n-3)/48 for n odd.

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

(End)