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%I #5 Mar 30 2012 18:40:44
%S 2,5,6,9,10,13,14,16,17,21,23,26,28,30,31,36,37,38,39,41,44,47,48,50,
%T 51,52,54,55,58,60,61,65,67,69,71,74,76,79,82,84,86,87,93,95,96,99,
%U 100,101,103,105,106,108,112,115,116,117,118,119,121,122,126,128,132,133,134
%N Numbers that are the sum of a nonzero pentagonal number and a nonzero square in at least one way.
%C These pentagonal numbers P(k) that can be represented as the sum of P(i)+j^2, i,j>0, are at k= 2, 6, 9, 10, 13, 17, 21, 22, 24, 26, 29, 34, 35, 38, 41, 45, 46, 53. Are almost all positive integers in this sequence and, if so, what is the largest value in the complement? The largest square in the complement? The largest pentagonal number in the complement?
%F {A000326(i) + A000290(j) for i, j > 0}. {i(3*i-1)/2 + j^2 for i, j > 0}.
%e Let P(n) = n-th pentagonal number:
%e a(1) = P(1) + 1^2 = 1 + 1 = 2.
%e a(2) = P(1) + 2^2 = 1 + 4 = 5 = P(2).
%e a(3) = P(2) + 1^2 = 5 + 1 = 6.
%e a(4) = P(2) + 2^2 = 5 + 4 = 9 = 3^2.
%e a(5) = P(1) + 3^2 = 1 + 9 = 10 = a(P(2)).
%e a(8) = P(3) + 2^2 = 12 + 4 = 16 = 4^2.
%e a(10) = P(2) + 4^2 = 5 + 16 = P(3) + 3^2 = 12 + 9 = 21.
%e a(12) = P(1) + 5^2 = 1 + 25 = P(4) + 2^2 = 22 + 4 = 26 = a(P(3)).
%e a(16) = P(5) + 1^2 = 35 + 1 = 36 = 6^2.
%e a(17) = P(1) + 6^2 = 1 + 36 = P(3) + 5^2 = 12 + 25 = 37.
%e a(25) = P(5) + 4^2 = 35 + 16 = 51 = P(6).
%e a(30) = P(6) + 3^2 = 51 + 9 = P(5) + 5^2 = 35 + 25 = 60.
%e a(35) = P(7) + 1^2 = 70 + 1 = P(5) + 6^2 = 35 + 36 = P(4) + 7^2 = 22 + 49 = 71 = a(P(5)).
%e a(37) = P(6) + 5^2 = 51 + 25 = P(3) + 8^2 = 12 + 64 = 76.
%e a(41) = P(7) + 4^2 = 70 + 16 = P(4) + 8^2 = 22 + 64 = 86.
%Y Cf. A000290, A000326, A134935-A134938.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Jan 21 2008
%E Corrected and extended by _R. J. Mathar_, Jan 21 2008
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