A268836 -id:A268836 - cp's OEIS frontend

A268837 Antidiagonal sums of array A268715: a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(n-k)).

0, 2, 7, 18, 17, 48, 56, 80, 67, 122, 136, 194, 204, 268, 281, 328, 291, 378, 396, 498, 510, 640, 675, 792, 790, 886, 965, 1098, 1093, 1208, 1248, 1344, 1227, 1378, 1356, 1530, 1538, 1792, 1815, 2016, 2008, 2218, 2339, 2602, 2619, 2892, 2970, 3208, 3150, 3294, 3385, 3586, 3691, 4012, 4174, 4440, 4367, 4554, 4644

Offset: 0

Antti Karttunen, Feb 15 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..4096

Cf. A003188, A006068.

Cf. also A268720, A268836.

(define (A268837 n) (add (lambda (k) (A003188 (+ (A006068 k) (A006068 (- n k))))) 0 n))
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(n-k)).

A342448 Partial sums of A066194.

Original entry on oeis.org

1, 3, 7, 10, 18, 25, 30, 36, 52, 67, 80, 94, 103, 113, 125, 136, 168, 199, 228, 258, 283, 309, 337, 364, 381, 399, 419, 438, 462, 485, 506, 528, 592, 655, 716, 778, 835, 893, 953, 1012, 1061, 1111, 1163, 1214, 1270, 1325, 1378, 1432, 1465, 1499, 1535, 1570

Offset: 1

John Erickson, Mar 12 2021

n^2/2 + n/2 <= a(n) <= (31/50)*n^2 + n/2. The lower and upper bounds are attained at n=2^k and n=5*2^k for k >= 0.

Alois P. Heinz, Table of n, a(n) for n = 1..16384

Cf. A066194, A268836, A007814, A000120, A010060, A000035.

b:= proc(n) option remember; `if`(n<2, n,
      Bits[Xor](n, b(iquo(n, 2))))
    end:
a:= proc(n) a(n):= 1+`if`(n<2, 0, a(n-1)+b(n-1)) end:
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 14 2021

a[1]=1;
a[n_/;EvenQ[n]]:= a[n] = 4a[n/2] - n/2;
a[n_/;OddQ[n]]:= a[n] = 2a[(n - 1)/2]+2a[(n + 1)/2]-(n-1)/2 - ThueMorse[n];
(* Second program: *)
b[n_] := If[n==0, 0, BitXor@@Table[Floor[n/2^m], {m, 0, Floor[Log[2, n]]}]];
A066194 = Table[b[n]+1, {n, 0, 60}];
A066194 // Accumulate (* Jean-François Alcover, Sep 10 2022 *)

a(n) = A268836(n)/2 + n. - Kevin Ryde, Mar 12 2021

a(1) = 1; a(n) = [n == 0 (mod 2)]*(4*a(n/2) - n/2) + [n == 1 (mod 2)]*(2*a((n - 1)/2)+2*a((n + 1)/2)-(n-1)/2 - A010060(n)) where [] is an Iverson bracket

cp's OEIS Frontend

A268837 Antidiagonal sums of array A268715: a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(n-k)).

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