A159693 Partial sums of A000463.

1, 2, 4, 8, 11, 20, 24, 40, 45, 70, 76, 112, 119, 168, 176, 240, 249, 330, 340, 440, 451, 572, 584, 728, 741, 910, 924, 1120, 1135, 1360, 1376, 1632, 1649, 1938, 1956, 2280, 2299, 2660, 2680, 3080, 3101, 3542, 3564, 4048, 4071, 4600, 4624, 5200, 5225, 5850

Offset

1,2

Comments

Sum of integers followed by squares.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Formula

a(n) = (n^3+3*n^2+8*n+r(n))/24, where r(n) = 3*n+9 if n is odd, 3*n^2 if n is even.

G.f.: x*(1+x-x^2+x^3)/((1+x)^3*(x-1)^4). - R. J. Mathar, Apr 20 2009

a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7). - R. J. Mathar, Apr 20 2009

a(n) = (2*n^3+9*n^2+19*n+9+3*(n^2-n-3)*(-1)^n)/48. - Luce ETIENNE, Dec 29 2014

E.g.f.: (2*x^3+15*x^2+30*x+9)*exp(x)/48 +(x^2-3)*exp(-x)/16. - Robert Israel, Dec 30 2014

Example

For n=9, a(n) = 1+1+2+4+3+9+4+16+5 = 45.

Maple

seq((2*n^3+9*n^2+19*n+9+3*(n^2-n-3)*(-1)^n)/48, n=1..100); # Robert Israel, Dec 30 2014

Mathematica

CoefficientList[Series[x*(1+x-x^2+x^3)/((1+x)^3*(x-1)^4), {x, 0, 50}], x] (* or *) Table[(2*n^3+9*n^2+19*n+9+3*(n^2-n-3)*(-1)^n)/48, {n, 0, 50}] (* G. C. Greubel, Jun 02 2018 *)
Accumulate[Flatten[{#, #^2}&/@Range[30]]] (* Harvey P. Dale, Nov 30 2019 *)

Prog

(Magma) S:=&cat[ [ n, n^2 ]: n in [1..25] ]; [ n eq 1 select S[1] else Self(n-1)+S[n]: n in [1..#S] ]; // Klaus Brockhaus, Apr 20 2009
(Haskell)
a159693 n = a159693_list !! (n-1)
a159693_list = scanl1 (+) a000463_list -- Reinhard Zumkeller, Nov 08 2015

Crossrefs

Cf. A000290, A000463.

Sequence in context: A282620 A320440 A288149 * A363735 A053437 A018277

Adjacent sequences: A159690 A159691 A159692 * A159694 A159695 A159696

Keyword

nonn

Author

Gerald Hillier, Apr 20 2009

Extensions

More terms from R. J. Mathar and Klaus Brockhaus, Apr 20 2009

Status

approved