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

Author

Gerald Hillier, Apr 20 2009

Keywords

• nonn

Comments

Sum of integers followed by squares.

Examples

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

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).

Crossrefs

Cf. A000290, A000463.

Programs

Haskell

a159693 n = a159693_list !! (n-1)
a159693_list = scanl1 (+) a000463_list -- Reinhard Zumkeller, Nov 08 2015

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

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 *)

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

Extensions

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