This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A102378 #15 Jun 09 2022 02:29:33
%S A102378 1,3,5,9,13,19,25,35,45,59,73,93,113,139,165,201,237,283,329,389,449,
%T A102378 523,597,691,785,899,1013,1153,1293,1459,1625,1827,2029,2267,2505,
%U A102378 2789,3073,3403,3733,4123,4513,4963,5413,5937,6461,7059,7657,8349
%N A102378 a(n) = a(n-1) + a([n/2]) + 1, a(1) = 1.
%C A102378 From _Gus Wiseman_, Mar 23 2019: (Start)
%C A102378 The offset could safely be changed to zero by setting the boundary condition to a(0) = 0.
%C A102378 Also the number of integer partitions of 2n into powers of 2 with at least one part > 1. The Heinz numbers of these partitions are given by A324927. For example, the a(1) = 1 through a(5) = 13 integer partitions are:
%C A102378 (2) (4) (42) (8) (82)
%C A102378 (22) (222) (44) (442)
%C A102378 (211) (411) (422) (811)
%C A102378 (2211) (2222) (4222)
%C A102378 (21111) (4211) (4411)
%C A102378 (22211) (22222)
%C A102378 (41111) (42211)
%C A102378 (221111) (222211)
%C A102378 (2111111) (421111)
%C A102378 (2221111)
%C A102378 (4111111)
%C A102378 (22111111)
%C A102378 (211111111)
%C A102378 (End)
%F A102378 a(n) - a(n-1) = A018819(n+1)
%F A102378 G.f. A(x) satisfies (1-x)*A(x) = 2(1 + x)*B(x^2), where B(x) is the gf of A033485
%F A102378 a(n) = A000123(n) - 1. - _Gus Wiseman_, Mar 23 2019
%F A102378 G.f. A(x) satisfies: A(x) = (x + (1 - x^2) * A(x^2)) / (1 - x)^2. - _Ilya Gutkovskiy_, Aug 11 2021
%t A102378 Table[Length[Select[IntegerPartitions[n],And[Max@@#>1,And@@IntegerQ/@Log[2,#]]&]],{n,0,30,2}] (* _Gus Wiseman_, Mar 23 2019 *)
%o A102378 (Python)
%o A102378 from itertools import islice
%o A102378 from collections import deque
%o A102378 def A102378_gen(): # generator of terms
%o A102378 aqueue, f, b, a = deque([2]), True, 1, 2
%o A102378 yield from (1, 3)
%o A102378 while True:
%o A102378 a += b
%o A102378 yield 2*a - 1
%o A102378 aqueue.append(a)
%o A102378 if f: b = aqueue.popleft()
%o A102378 f = not f
%o A102378 A102378_list = list(islice(A102378_gen(),40)) # _Chai Wah Wu_, Jun 08 2022
%Y A102378 Cf. A000123, A033485.
%Y A102378 Cf. A018819, A318400, A324927, A324928.
%K A102378 nonn
%O A102378 1,2
%A A102378 _Mitch Harris_, Jan 05 2005