A274979 Integers of the form m*(m + 7)/8.

0, 1, 15, 18, 46, 51, 93, 100, 156, 165, 235, 246, 330, 343, 441, 456, 568, 585, 711, 730, 870, 891, 1045, 1068, 1236, 1261, 1443, 1470, 1666, 1695, 1905, 1936, 2160, 2193, 2431, 2466, 2718, 2755, 3021, 3060, 3340, 3381, 3675, 3718, 4026, 4071, 4393, 4440, 4776, 4825

Offset: 1

Author

Bruno Berselli, Jul 15 2016

Keywords

• nonn
• easy

Comments

Nonnegative values of m are listed in A047393.

Also, numbers h such that 32*h + 49 is a square.

Equivalently, numbers of the form i*(8*i + 7) with i = 0, -1, 1, -2, 2, -3, 3, ...

Infinitely many squares belong to this sequence.

The first bisection is A139278, and 0 followed by the second bisection gives A051870.

Generalized 18-gonal (or octadecagonal) numbers (see the third comment). - Omar E. Pol, Jun 06 2018

Partial sums of A317314. - Omar E. Pol, Jul 28 2018

Exponents in expansion of Product_{n >= 1} (1 + x^(16*n-15))*(1 + x^(16*n-1))*(1 - x^(16*n)) = 1 + x + x^15 + x^18 + x^46 + .... - Peter Bala, Dec 10 2020

Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, k >= 5. They are also the partial sums of the sequence formed by the multiples of (k - 4) and the odd numbers (A005408) interleaved, k >= 5. In this case k = 18. - Omar E. Pol, Apr 25 2021

Examples

			100 is in the sequence because 100 = 25*(25+7)/8 or also 100 = 4*(8*4-7).
From _Omar E. Pol_, Apr 24 2021: (Start)
Illustration of initial terms as vertices of a rectangular spiral:
        46_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _18
         |                                                       |
         |                           0                           |
         |                           |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
         |                           1                           15
         |
        51
More generally, all generalized k-gonal numbers can be represented with this kind of spirals, k >= 5. In this case  k = 18. (End)
		

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

Cf. sequences of the form m*(m+k)/(k+1) listed in A274978.

Cf. similar sequences listed in A299645.

Cf. A317314.

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), A274978 (k=16), A303305 (k=17), this sequence (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+7)/8];

Mathematica

Select[m = Range[0, 200]; m (m + 7)/8, IntegerQ] (* Jean-François Alcover, Jul 21 2016 *)
Select[Table[(m(m+7))/8,{m,0,200}],IntegerQ] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{0,1,15,18,46},50] (* Harvey P. Dale, May 07 2019 *)

Python

def A274979(n): return (n>>1)*((n<<2)+(3 if n&1 else -7)) # Chai Wah Wu, Mar 11 2025

Sage

def A274979_list(len):
    h = lambda m: m*(m+7)/8
    return [h(m) for m in (0..len) if h(m) in ZZ]
print(A274979_list(199)) # Peter Luschny, Jul 18 2016

Formula

O.g.f.: x^2*(1 + 14*x + x^2)/((1 + x)^2*(1 - x)^3).

E.g.f.: (3*(2*x + 1)*exp(-x) + (8*x^2 - 3)*exp(x))/4.

a(n) = (8*(n-1)*n - 3*(2*n-1)*(-1)^n - 3)/4.

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

From Amiram Eldar, Feb 28 2022: (Start)

Sum_{n>=2} 1/a(n) = (8 + 7*(sqrt(2)+1)*Pi)/49.

Sum_{n>=2} (-1)^n/a(n) = 8*log(2)/7 + 2*sqrt(2)*log(sqrt(2)+1)/7 - 8/49. (End)

a(n) = (n-1)*(4*n+3)/2 if n is odd and a(n) = n*(4*n-7)/2 if n is even. - Chai Wah Wu, Mar 11 2025