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A239447 Partial sums of A030101.

%I A239447 #22 Jan 22 2025 14:18:35
%S A239447 0,1,2,5,6,11,14,21,22,31,36,49,52,63,70,85,86,103,112,137,142,163,
%T A239447 176,205,208,227,238,265,272,295,310,341,342,375,392,441,450,491,516,
%U A239447 573,578,615,636,689,702,747,776,837,840,875,894,945,956,999,1026,1085
%N A239447 Partial sums of A030101.
%H A239447 Alois P. Heinz, <a href="/A239447/b239447.txt">Table of n, a(n) for n = 0..10000</a>
%H A239447 Project Euler, <a href="https://projecteuler.net/problem=463">Problem 463: A weird recurrence relation</a>
%F A239447 a(4n) = 6 a(2n) - 5 a(n) - 3 a(n-1) - 1.
%F A239447 a(4n + 1) = 2 a(2n+1) + 4 a(2n) - 6 a(n) - 2 a(n-1) - 1.
%F A239447 a(4n + 2) = 3 a(2n+1) + 3 a(2n) - 6 a(n) - 2 a(n-1) - 1.
%F A239447 a(4n + 3) = 6 a(2n+1) - 8 a(n) - 1.
%t A239447 Accumulate[Table[FromDigits[Reverse[IntegerDigits[n,2]],2],{n,0,80}]] (* _Harvey P. Dale_, Jan 10 2023 *)
%o A239447 (Python)
%o A239447 A = {0: 0, 1: 1, 2: 2, 3: 5}
%o A239447 def a(n):
%o A239447     a_n = A.get(n)
%o A239447     if a_n is not None:
%o A239447         return a_n
%o A239447     q, r = divmod(n, 4)
%o A239447     if r == 0:
%o A239447         a_n = a(q*2)*6 - a(q)*5 - a(q - 1)*3 - 1
%o A239447     elif r == 1:
%o A239447         a_n = a(q*2 + 1)*2 + a(q*2)*4 - a(q)*6 - a(q - 1)*2 - 1
%o A239447     elif r == 2:
%o A239447         a_n = a(q*2 + 1)*3 + a(q*2)*3 - a(q)*6 - a(q - 1)*2 - 1
%o A239447     else:
%o A239447         a_n = a(q*2 + 1)*6 - a(q)*8 - 1
%o A239447     A[n] = a_n  # memoization
%o A239447     return a_n
%Y A239447 Cf. A030101.
%K A239447 nonn,base
%O A239447 0,3
%A A239447 _Olivier Pirson_, Mar 18 2014
%E A239447 More terms from _Alois P. Heinz_, May 19 2014