A114113 a(n) = sum{k=1 to n} (A114112(k)). (For n>=2, a(n) = sum{k=1 to n} (A014681(k)) =sum{k=1 to n} (A103889(k)).).

1, 3, 7, 10, 16, 21, 29, 36, 46, 55, 67, 78, 92, 105, 121, 136, 154, 171, 191, 210, 232, 253, 277, 300, 326, 351, 379, 406, 436, 465, 497, 528, 562, 595, 631, 666, 704, 741, 781, 820, 862, 903, 947, 990, 1036, 1081, 1129, 1176, 1226, 1275, 1327, 1378, 1432

Offset: 1

Author

Leroy Quet, Nov 13 2005

Keywords

• easy
• nonn

Comments

a(n) is not divisible by (A114112(n+1)).

Sequence is A130883 union A014105 - {0,2}. - Anthony Hernandez, Sep 08 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A014105, A014681, A026035, A084265, A103889, A114112.

Programs

Magma

I:=[1,3,7,10,16]; [n le 5 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..60]]; // Vincenzo Librandi, Mar 13 2018

Mathematica

Join[{1}, LinearRecurrence[{2, 0, -2, 1}, {3, 7, 10, 16}, 52]] (* Jean-François Alcover, Sep 22 2017 *)
CoefficientList[Series[(1 + x + x^2 -2 x^3 + x^4)/((1 + x) (1 - x)^3), {x, 0, 60}], x] (* Vincenzo Librandi, Mar 13 2018 *)

Python

def A114113(n): return 1 if n == 1 else (m:=n//2)*(n+1) + (n+1-m)*(n-2*m) # Chai Wah Wu, May 24 2022

Formula

a(1)=1. a(2n) = n*(2n+1). a(2n+1) = 2n^2 +3n +2.

From R. J. Mathar, Nov 04 2008: (Start)

a(n) = A026035(n+1) - A026035(n), n>1.

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

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4), n>5. (End)

This is (essentially) 1 + A084265, - N. J. A. Sloane, Mar 12 2018

Extensions

More terms from R. J. Mathar, Aug 31 2007