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A274978 Integers of the form m*(m + 6)/7. 31

%I #62 Sep 20 2022 16:22:36

%S 0,1,13,16,40,45,81,88,136,145,205,216,288,301,385,400,496,513,621,

%T 640,760,781,913,936,1080,1105,1261,1288,1456,1485,1665,1696,1888,

%U 1921,2125,2160,2376,2413,2641,2680,2920,2961,3213,3256,3520,3565,3841,3888,4176,4225,4525,4576

%N Integers of the form m*(m + 6)/7.

%C Nonnegative values of m are listed in A047274.

%C Also, numbers h such that 7*h + 9 is a square.

%C Equivalently, numbers of the form i*(7*i - 6) with i = 0, 1, -1, 2, -2, 3, -3, ...

%C Infinitely many squares belong to this sequence.

%C Generalized 16-gonal (or hexadecagonal) numbers. See the third comment. - _Omar E. Pol_, Jun 06 2018

%C Partial sums of A317312. - _Omar E. Pol_, Jul 28 2018

%C 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 Bruno Berselli, <a href="/A274978/b274978.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F O.g.f.: x^2*(1 + 12*x + x^2)/((1 + x)^2*(1 - x)^3).

%F E.g.f.: (5*(2*x + 1)*exp(-x) + (14*x^2 - 5)*exp(x))/8.

%F a(n) = (14*(n-1)*n - 5*(2*n-1)*(-1)^n - 5)/8.

%F 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 Sum_{n>=2} 1/a(n) = (7 + 6*Pi*cot(Pi/7))/36. - _Amiram Eldar_, Feb 28 2022

%e 88 is in the sequence because 88 = 22*(22+6)/7 or also 88 = 4*(7*4-6).

%t Select[m = Range[0, 200]; m (m + 6)/7, IntegerQ] (* _Jean-François Alcover_, Jul 21 2016 *)

%t Select[Table[(n(n+6))/7,{n,0,200}],IntegerQ] (* _Harvey P. Dale_, Sep 20 2022 *)

%o (Sage)

%o def A274978_list(len):

%o h = lambda m: m*(m+6)/7

%o return [h(m) for m in (0..len) if h(m) in ZZ]

%o print(A274978_list(179)) # _Peter Luschny_, Jul 18 2016

%o (Magma) [t: m in [0..200] | IsIntegral(t) where t is m*(m+6)/7];

%Y Supersequence of A051868.

%Y Cf. A317312.

%Y 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 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 nonn,easy

%O 1,3

%A _Bruno Berselli_, Jul 15 2016

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Last modified March 28 14:02 EDT 2024. Contains 371254 sequences. (Running on oeis4.)