cp's OEIS Frontend

A090864 Complement of generalized pentagonal numbers (A001318).

%I A090864 #37 Sep 15 2024 06:50:17
%S A090864 3,4,6,8,9,10,11,13,14,16,17,18,19,20,21,23,24,25,27,28,29,30,31,32,
%T A090864 33,34,36,37,38,39,41,42,43,44,45,46,47,48,49,50,52,53,54,55,56,58,59,
%U A090864 60,61,62,63,64,65,66,67,68,69,71,72,73,74,75,76,78,79,80,81,82,83,84
%N A090864 Complement of generalized pentagonal numbers (A001318).
%C A090864 Also n for which A006906(n) is even, or equivalently n for which A000009(n) is even (since A006906 and A000009 have the same parity).
%C A090864 The number of partitions of a(n) into distinct parts with an even number of parts equals the number of such partitions with an odd number of parts: A067661(a(n)) = A067659(a(n)). See, e.g., the Freitag-Busam reference, p. 410 given in A036499. - _Wolfdieter Lang_, Jan 19 2016
%H A090864 G. C. Greubel, <a href="/A090864/b090864.txt">Table of n, a(n) for n = 1..1000</a>
%F A090864 A080995(a(n)) = 0; A000009(a(n)) = A118303(n). - _Reinhard Zumkeller_, Apr 22 2006
%F A090864 A010815(a(n)) = A067661(a(n)) - A067659(a(n)) = 0, n >= 1. See a comment above. - _Wolfdieter Lang_, Jan 19 2016
%F A090864 a(n) = n+1 + A085141(n-1) + A111651(n). - _Kevin Ryde_, Sep 15 2024
%t A090864 Complement[Range[200], Select[Accumulate[Range[0,200]]/3, IntegerQ]] (* _G. C. Greubel_, Jun 06 2017 *)
%o A090864 (Python)
%o A090864 from math import isqrt
%o A090864 def A090864(n):
%o A090864     def f(x): return n+(m:=isqrt(24*x+1)+1)//6+(m-2)//6
%o A090864     kmin, kmax = 0,1
%o A090864     while f(kmax) > kmax:
%o A090864         kmax <<= 1
%o A090864     while kmax-kmin > 1:
%o A090864         kmid = kmax+kmin>>1
%o A090864         if f(kmid) <= kmid:
%o A090864             kmax = kmid
%o A090864         else:
%o A090864             kmin = kmid
%o A090864     return kmax # _Chai Wah Wu_, Aug 29 2024
%o A090864 (PARI) a(n) = my(q,r); [q,r]=divrem(sqrtint(24*n),3); n + q + (r >= bitnegimply(1,q)); \\ _Kevin Ryde_, Sep 15 2024
%Y A090864 Cf. A000009, A001318, A067659, A067661, A080995, A006906, A010815, A118303.
%Y A090864 Cf. A085141, A111651.
%K A090864 nonn,easy
%O A090864 1,1
%A A090864 _Jon Perry_, Feb 12 2004
%E A090864 More terms from _Reinhard Zumkeller_, Apr 22 2006
%E A090864 Edited by _Ray Chandler_, Dec 14 2011
%E A090864 Edited by _Jon E. Schoenfield_, Nov 25 2016