A171378 a(n) = (n+1)^2 - A006046(n+1).

0, 1, 4, 7, 14, 21, 30, 37, 52, 67, 84, 99, 120, 139, 160, 175, 206, 237, 270, 301, 338, 373, 410, 441, 486, 529, 574, 613, 662, 705, 750, 781, 844, 907, 972, 1035, 1104, 1171, 1240, 1303, 1380, 1455, 1532, 1603, 1684, 1759, 1836, 1899, 1992, 2083, 2176, 2263

Offset: 0

Author

Roger L. Bagula, Dec 07 2009

Keywords

• nonn
• easy

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000 (a(0..999) from Robert Price)
• Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 6, 30.

Crossrefs

Cf. A006046, A001316.

Programs

Magma

[(n+1)^2 - (&+[ (&+[ Binomial(m,k) mod 2: k in [0..m]]): m in [0..n]]): n in [0..60]]; // G. C. Greubel, Apr 11 2019

Mathematica

Table[(n+1)^2 -Sum[Sum[Mod[Binomial[m,k],2], {k,0,m}], {m,0,n}], {n,0, 60}]
a[0] = 0; a[1] = 1; a[n_] := a[n] = 2 a[Floor[#]] + a[Ceiling[#]] &[n/2]; Array[(# + 1)^2 - a[# + 1] &, 52, 0] (* Michael De Vlieger, Nov 01 2022 *)

PARI

{a(n) = (n+1)^2 - sum(m=0,n, sum(k=0,m, binomial(m,k)%2))};
for(n=0,60, print1(a(n), ", ")) \\ G. C. Greubel, Apr 11 2019

Sage

[(n+1)^2 - sum(sum(binomial(m,k)%2 for k in (0..m)) for m in (0..n)) for n in (0..60)] # G. C. Greubel, Apr 11 2019

Extensions

Edited by G. C. Greubel, Apr 11 2019

Definition corrected by Georg Fischer, Jun 21 2020