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A274978
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Integers of the form m*(m + 6)/7.
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31
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0, 1, 13, 16, 40, 45, 81, 88, 136, 145, 205, 216, 288, 301, 385, 400, 496, 513, 621, 640, 760, 781, 913, 936, 1080, 1105, 1261, 1288, 1456, 1485, 1665, 1696, 1888, 1921, 2125, 2160, 2376, 2413, 2641, 2680, 2920, 2961, 3213, 3256, 3520, 3565, 3841, 3888, 4176, 4225, 4525, 4576
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OFFSET
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1,3
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COMMENTS
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Nonnegative values of m are listed in A047274.
Also, numbers h such that 7*h + 9 is a square.
Equivalently, numbers of the form i*(7*i - 6) with i = 0, 1, -1, 2, -2, 3, -3, ...
Infinitely many squares belong to this sequence.
Generalized 16-gonal (or hexadecagonal) numbers. See the third comment. - Omar E. Pol, Jun 06 2018
Exponents in expansion of Product_{n >= 1} (1 + x^(14*n-13))*(1 + x^(14*n-1))*(1 - x^(14*n)) = 1 + x + x^13 + x^16+ x^40 + .... - Peter Bala, Dec 10 2020
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LINKS
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FORMULA
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O.g.f.: x^2*(1 + 12*x + x^2)/((1 + x)^2*(1 - x)^3).
E.g.f.: (5*(2*x + 1)*exp(-x) + (14*x^2 - 5)*exp(x))/8.
a(n) = (14*(n-1)*n - 5*(2*n-1)*(-1)^n - 5)/8.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n >= 6. - Wesley Ivan Hurt, Dec 18 2020
Sum_{n>=2} 1/a(n) = (7 + 6*Pi*cot(Pi/7))/36. - Amiram Eldar, Feb 28 2022
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EXAMPLE
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88 is in the sequence because 88 = 22*(22+6)/7 or also 88 = 4*(7*4-6).
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MATHEMATICA
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Select[Table[(n(n+6))/7, {n, 0, 200}], IntegerQ] (* Harvey P. Dale, Sep 20 2022 *)
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PROG
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(Sage)
h = lambda m: m*(m+6)/7
return [h(m) for m in (0..len) if h(m) in ZZ]
(Magma) [t: m in [0..200] | IsIntegral(t) where t is m*(m+6)/7];
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CROSSREFS
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Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), this sequence (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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