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%I A274978 #62 Sep 20 2022 16:22:36
%S A274978 0,1,13,16,40,45,81,88,136,145,205,216,288,301,385,400,496,513,621,
%T A274978 640,760,781,913,936,1080,1105,1261,1288,1456,1485,1665,1696,1888,
%U A274978 1921,2125,2160,2376,2413,2641,2680,2920,2961,3213,3256,3520,3565,3841,3888,4176,4225,4525,4576
%N A274978 Integers of the form m*(m + 6)/7.
%C A274978 Nonnegative values of m are listed in A047274.
%C A274978 Also, numbers h such that 7*h + 9 is a square.
%C A274978 Equivalently, numbers of the form i*(7*i - 6) with i = 0, 1, -1, 2, -2, 3, -3, ...
%C A274978 Infinitely many squares belong to this sequence.
%C A274978 Generalized 16-gonal (or hexadecagonal) numbers. See the third comment. - _Omar E. Pol_, Jun 06 2018
%C A274978 Partial sums of A317312. - _Omar E. Pol_, Jul 28 2018
%C A274978 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
%H A274978 Bruno Berselli, <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL0EyNzQ5NzgvYjI3NDk3OC50eHQ">Table of n, a(n) for n = 1..1000</a>
%H A274978 <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL2luZGV4L1JlYyNvcmRlcl8wNQ">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A274978 O.g.f.: x^2*(1 + 12*x + x^2)/((1 + x)^2*(1 - x)^3).
%F A274978 E.g.f.: (5*(2*x + 1)*exp(-x) + (14*x^2 - 5)*exp(x))/8.
%F A274978 a(n) = (14*(n-1)*n - 5*(2*n-1)*(-1)^n - 5)/8.
%F A274978 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
%F A274978 Sum_{n>=2} 1/a(n) = (7 + 6*Pi*cot(Pi/7))/36. - _Amiram Eldar_, Feb 28 2022
%e A274978 88 is in the sequence because 88 = 22*(22+6)/7 or also 88 = 4*(7*4-6).
%t A274978 Select[m = Range[0, 200]; m (m + 6)/7, IntegerQ] (* _Jean-François Alcover_, Jul 21 2016 *)
%t A274978 Select[Table[(n(n+6))/7,{n,0,200}],IntegerQ] (* _Harvey P. Dale_, Sep 20 2022 *)
%o A274978 (Sage)
%o A274978 def A274978_list(len):
%o A274978     h = lambda m: m*(m+6)/7
%o A274978     return [h(m) for m in (0..len) if h(m) in ZZ]
%o A274978 print(A274978_list(179)) # _Peter Luschny_, Jul 18 2016
%o A274978 (Magma) [t: m in [0..200] | IsIntegral(t) where t is m*(m+6)/7];
%Y A274978 Supersequence of A051868.
%Y A274978 Cf. A317312.
%Y A274978 Cf. sequences of the form m*(m+k)/(k+1): A000290 (k=0), A000217 (k=1), A001082 (k=2), A074377 (k=3), A195162 (k=4), A144065 (k=5), A274978 (k=6), A274979 (k=7), A218864 (k=8).
%Y A274978 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).
%K A274978 nonn,easy
%O A274978 1,3
%A A274978 _Bruno Berselli_, Jul 15 2016

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