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A306122
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Numbers that are product of a second hexagonal number (A014105) and a square pyramidal numbers (A000330) in at least two ways.
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1
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0, 105, 300, 855, 1155, 2940, 13860, 14700, 17850, 20790, 22230, 27300, 33930, 70125, 73920, 87780, 114400, 116025, 135135, 145530, 157080, 195000, 213150, 235290, 304590, 347655, 381150, 431340, 451044, 471975, 566580, 632700, 764400, 796950, 942480, 950040
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OFFSET
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1,2
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COMMENTS
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We have A000330(n) = 1 + 2^2 + ... + n^2 and A014105(m) = 0^2 - 1^2 + 2^2 -+ ... + (2m)^2, so the terms of this sequence are the numbers that are a product, in at least two ways, of a partial sum of squares times a (positive) partial sum of squares with alternating signs (with + for even terms; cf. A306121 for the opposite convention).
The initial a(1) = 0 is added for completeness.
Below 10^8, the number 17850 is the only one to have four representations of the given form, and 6347250 is the only one to have exactly three.
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LINKS
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Table of n, a(n) for n=1..36.
Geoffrey Campbell, Integer solutions of (1²-2²+3²-...+(2a-1)²) × (1²+2²+3²+...+b²) = (1²-2²+3²-...+(2c-1)²) × (1²+2²+3²+...+d²) where a ≠ c and b ≠ d, Number Theory group on LinkedIn, June 2018.
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PROG
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(PARI) {my(L=10^6, A14105(a)=a*(2*a+1), A330(b)=(b+1)*b*(2*b+1)/6, A=S=[]); for(b=1, sqrtnint(L\A14105(1)\3, 3), for(a=1, oo, if( setsearch(S, t=A14105(a)*A330(b)), A=setunion(A, [t]), t>L&&next(2); S=setunion(S, [t])))); A}
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CROSSREFS
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Cf. A000330, A014105, A306121.
Sequence in context: A075764 A046299 A010090 * A174830 A325312 A147576
Adjacent sequences: A306119 A306120 A306121 * A306123 A306124 A306125
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KEYWORD
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nonn
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AUTHOR
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Geoffrey B. Campbell and M. F. Hasler, Jul 03 2018
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STATUS
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approved
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