A218864 Numbers of the form 9*k^2 + 8*k, k an integer.

0, 1, 17, 20, 52, 57, 105, 112, 176, 185, 265, 276, 372, 385, 497, 512, 640, 657, 801, 820, 980, 1001, 1177, 1200, 1392, 1417, 1625, 1652, 1876, 1905, 2145, 2176, 2432, 2465, 2737, 2772, 3060, 3097, 3401, 3440, 3760, 3801, 4137, 4180, 4532, 4577, 4945, 4992

Offset: 1

Author

Jason Kimberley, Nov 08 2012

Keywords

• nonn
• easy

Comments

Numbers m such that 9*m + 16 is a square. - Vincenzo Librandi, Apr 07 2013

Equivalently, integers of the form h*(h + 8)/9 (nonnegative values of h are listed in A090570). - Bruno Berselli, Jul 15 2016

Generalized 20-gonal (or icosagonal) numbers: r*(9*r - 8) with r = 0, +1, -1, +2, -2, +3, -3, ... - Omar E. Pol, Jun 06 2018

Partial sums of A317316. - Omar E. Pol, Jul 28 2018

Exponents in expansion of Product_{n >= 1} (1 + x^(18*n-17))*(1 + x^(18*n-1))*(1 - x^(18*n)) = 1 + x + x^17 + x^20 + x^52 + .... - Peter Bala, Dec 10 2020

Links

• Jason Kimberley, Table of n, a(n) for n = 1..2000
• S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math., Vol. 274, No. 1-3 (2004), pp. 9-24. See C(q).
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Characteristic function is A205987.

Numbers of the form 9*m^2+k*m, for integer n: A016766 (k=0), A132355 (k=2), A185039 (k=4), A057780 (k=6), this sequence (k=8).

Cf. A074377 (numbers m such that 16*m+9 is a square).

Cf. A317316.

For similar sequences of numbers m such that 9*m+i is a square, see list in A266956.

Cf. sequences of the form m*(m+i)/(i+1) listed in A274978. [Bruno Berselli, Jul 25 2016]

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), A274979 (k=18), A303813 (k=19), this sequence (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

a:=func; [0]cat[a(n*m): m in [-1,1], n in [1..20]];

Mathematica

Array[(18 # (# - 1) - 7 (-1)^#*(2 # - 1) - 7)/8 &, 48] (* or *)
CoefficientList[Series[x (1 + 16 x + x^2)/((1 + x)^2*(1 - x)^3), {x, 0, 47}], x] (* Michael De Vlieger, Jun 06 2018 *)

Formula

a(n) = (18*n*(n - 1) - 7*(-1)^n*(2*n - 1) - 7)/8. - Bruno Berselli, Nov 13 2012

G.f.: x*(1 + 16*x + x^2)/((1 + x)^2*(1 - x)^3). - Bruno Berselli, Nov 14 2012

Sum_{n>=2} 1/a(n) = (9 + 8*Pi*cot(Pi/9))/64. - Amiram Eldar, Feb 28 2022