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%I #67 Apr 09 2021 02:26:58
%S 0,4,16,81,471,2031,1381,11781,6906,17956,34531,123256,40056,305256,
%T 863281,448906,200281,1957231,520731,10563906,1001406,11222656,
%U 7631406,3454506,1482081,75865156,7172606106,8852431,25035156,334020781,13018281,38531031,7410406,7014160156
%N a(n) is the smallest nonnegative integer that can be written as a sum of two distinct nonzero triangular numbers in exactly n ways or -1 if no such integer exists.
%C Conjecture: This sequence has a positive a(n) for every positive integer n, and each sequence in the infinite indexed family, of which this sequence offers the initial terms, is infinite, as well.
%C From _David A. Corneth_, Mar 08 2021: (Start)
%C a(40) = 37052031, a(45) = 221310781, a(48) = 60765331, a(39) <= 2782318906, a(42) <= 325457031, a(47) <= 927577056, a(50) <= 2200089531, a(54) <= 327539956, a(56) <= 926300781, a(60) <= 481676406, a(63) <= 4598740656, a(64) <= 303826656, a(71) <= 4579579956, a(72) <= 789949306, a(80) <= 1519133281, a(96) <= 3220562556. Terms for n <= 96 not listed here and terms for which only upper bounds are known are >= 3*10^8.
%C Is a(n) == 6 (mod 25) for n >= 5? It holds for all terms known to date.
%C The triangular numbers mod 25 are periodic with period 25. Constructing all 25*25 = 625 sums of two distinct triangular numbers mod 25 gives 65 cases for 6 (mod 25). The second largest occurs 40 times. (End)
%C a(47) = 550240551, a(59) = 7629645156, a(67) = 6418012656, a(81) = 9498658731, a(90) = 8188498906. All upper bounds listed in the above comments for n other than 47 are the exact values of a(n). For all n for which no value is listed here or above, a(n) > 10^10 (or a(n) = -1). - _Jon E. Schoenfield_, Mar 09 2021
%C From _Martin Ehrenstein_, Mar 09 2021: (Start)
%C a(44) = 15646972656. For n<=51, all terms not mentioned here or above, a(n) >= 6.5*10^10 (or a(n) = -1).
%C a(47) == 1 (mod 25) and a(95) = 47652012541 == 16 (mod 25). Thus the answer to Corneth's question is 'No'. (End)
%H Martin Ehrenstein, <a href="/A342326/b342326.txt">Table of n, a(n) for n = 0..37</a>
%F a(n) = min { m >= 0 : A307597(m) = n }. - _Alois P. Heinz_, Mar 08 2021
%e a(1) = 4 = 1 + 3;
%e a(2) = 16 = 1 + 15 = 6 + 10;
%e a(3) = 81 = 3 + 78 = 15 + 66 = 36 + 45.
%t r = 125000; (* generates the first 12 terms of the sequence *)
%t lst = Table[0, {r}];
%t lim = Floor[Sqrt[2r]];
%t Do[ num = (i^2 + i)/2 + (j^2 + j)/2;
%t If[num <= r, lst[[num]]++], {i, lim}, {j, i - 1}];
%t First /@ (Flatten@Position[lst, #] & /@ Range[Max[lst]])
%o (PARI) upto(n) = {my(v = vector(n)); res = vector(10); for(i = 1, (sqrtint(8*n + 1)-1)\2, bi = binomial(i + 1, 2); for(j = i+1, (sqrtint(8*(n - bi))-1)\2, v[bi + binomial(j+1, 2)]++ ) ); for(i = 1, #v, if(v[i] > 0, if(v[i] > #res, res = concat(res, vector(v[i] - #res)); ); if(res[v[i]] == 0, res[v[i]] = i ) ) ); concat(0, res) } \\ _David A. Corneth_, Mar 08 2021
%Y Cf. A000217, A260647, A051533, A262749, A265140, A265134, A265136, A265138, A307597.
%K nonn
%O 0,2
%A _Robert G. Root_, Mar 08 2021
%E a(13)-a(18) from _Alois P. Heinz_, Mar 08 2021
%E a(19)-a(25) from _David A. Corneth_, Mar 08 2021
%E a(26)-a(33) from _Jon E. Schoenfield_, Mar 09 2021 (some terms first found by _David A. Corneth_)
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