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

1,2

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).

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)

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A007584.

Sequence in context: A271257 A343456 A087746 * A264147 A229440 A067622

Adjacent sequences: A064691 A064692 A064693 * A064695 A064696 A064697

Keyword

nonn,easy,nice

Author

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

Status

approved