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

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

Offset: 1

Author

Bruno Berselli, Jul 15 2016

Keywords

• nonn
• easy

Comments

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

Partial sums of A317312. - Omar E. Pol, Jul 28 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

Examples

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

Links

• Bruno Berselli, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Supersequence of A051868.

Cf. A317312.

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).

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).

Programs

Magma

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

Mathematica

Select[m = Range[0, 200]; m (m + 6)/7, IntegerQ] (* Jean-François Alcover, Jul 21 2016 *)
Select[Table[(n(n+6))/7,{n,0,200}],IntegerQ] (* Harvey P. Dale, Sep 20 2022 *)

Sage

def A274978_list(len):
    h = lambda m: m*(m+6)/7
    return [h(m) for m in (0..len) if h(m) in ZZ]
print(A274978_list(179)) # Peter Luschny, Jul 18 2016

Formula

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