A266956 -id:A266956 - cp's OEIS frontend

A218864 Numbers of the form 9k^2 + 8k, 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

Jason Kimberley, Nov 08 2012

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

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).

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).

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

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 *)

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

A132355 Numbers of the form 9h^2 + 2h, for h an integer.

Original entry on oeis.org

0, 7, 11, 32, 40, 75, 87, 136, 152, 215, 235, 312, 336, 427, 455, 560, 592, 711, 747, 880, 920, 1067, 1111, 1272, 1320, 1495, 1547, 1736, 1792, 1995, 2055, 2272, 2336, 2567, 2635, 2880, 2952, 3211, 3287, 3560, 3640, 3927, 4011, 4312, 4400, 4715, 4807

Offset: 1

Mohamed Bouhamida, Nov 08 2007

X values of solutions to the equation 9*X^3 + X^2 = Y^2.
The set of all m such that 9*m + 1 is a perfect square. - Gary Detlefs, Feb 22 2010
The concatenation of any term with 11..11 (1 repeated an even number of times, see A099814) belongs to the list. Example: 87 is a term, so also 8711, 871111, 87111111, 871111111111, ... are terms of this sequence. - Bruno Berselli, May 15 2017

Jason Kimberley, Table of n, a(n) for n = 1..2108
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 A(q).
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

A205808 is the characteristic function.
Cf. A000217, A001082, A002378, A005563, A028347, A036666, A046092, A054000, A056220, A062717, A087475, A132209, A010701, A056020.
Numbers of the form 9*n^2+k*n, for integer n: A016766 (k=0), this sequence (k=2), A185039 (k=4), A057780 (k=6), A218864 (k=8). - Jason Kimberley, Nov 09 2012
For similar sequences of numbers m such that 9*m+k is a square, see list in A266956.

a:=func; [0] cat [a(n*m): m in [-1,1], n in [1..25]]; // Jason Kimberley, Nov 08 2012

readlib(issqr); for n from 0 to 3560 do if(issqr(9*n+1)) then print(n) fi od; # Gary Detlefs, Feb 22 2010
seq(n^2+n+5*ceil(n/2)^2,n=0..39); # Gary Detlefs, Feb 23 2010

f[n_]:=IntegerQ[Sqrt[1+9*n]]; Select[Range[0,8! ],f[ # ]&] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
Sort[Table[9n^2+2n,{n,-30,30}]] (* Harvey P. Dale, Dec 06 2013 *)

a(n)=n^2-n+5*(n\2)^2 \\ Charles R Greathouse IV, Sep 28 2015

a(2*k) = k*(9*k-2), a(2*k+1) = k*(9*k+2).
a(n) = n^2 - n + 5*floor(n/2)^2. - Gary Detlefs, Feb 23 2010
From R. J. Mathar, Mar 17 2010: (Start)
a(n) = +a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).
G.f.: x^2*(7 + 4*x + 7*x^2)/((1 + x)^2*(1 - x)^3). (End)
a(n) = (2*n - 1 + (-1)^n)*(9*(2*n - 1) + (-1)^n)/16. - Luce ETIENNE, Sep 13 2014
Sum_{n>=2} 1/a(n) = 9/4 - cot(2*Pi/9)*Pi/2. - Amiram Eldar, Mar 15 2022

Simpler definition and minor edits from N. J. A. Sloane, Feb 03 2012

Since this is a list, offset changed to 1 and formulas translated by Jason Kimberley, Nov 18 2012

A185039 Numbers of the form 9m^2 + 4m, m an integer.

Original entry on oeis.org

0, 5, 13, 28, 44, 69, 93, 128, 160, 205, 245, 300, 348, 413, 469, 544, 608, 693, 765, 860, 940, 1045, 1133, 1248, 1344, 1469, 1573, 1708, 1820, 1965, 2085, 2240, 2368, 2533, 2669, 2844, 2988, 3173, 3325, 3520, 3680, 3885, 4053, 4268, 4444, 4669, 4853, 5088

Offset: 1

N. J. A. Sloane, Feb 04 2012

Also, numbers m such that 9*m+4 is a square. After 0, therefore, there are no squares in this sequence. - Bruno Berselli, Jan 07 2016

Bruno Berselli, Table of n, a(n) for n = 1..1000
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math. 274 (2004), no. 1-3, 9-24. See B(q).
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Characteristic function is A205809.
Numbers of the form 9*n^2+k*n, for integer n: A016766 (k=0), A132355 (k=2), this sequence (k=4), A057780 (k=6), A218864 (k=8). [Jason Kimberley, Nov 08 2012]
For similar sequences of numbers m such that 9*m+k is a square, see list in A266956.

[0] cat &cat[[9*n^2-4*n,9*n^2+4*n]: n in [1..32]]; // Bruno Berselli, Feb 04 2011

CoefficientList[Series[x*(5+8*x+5*x^2)/((x+1)^2*(1-x)^3), {x,0,50}], x] (* G. C. Greubel, Jun 20 2017 *)
LinearRecurrence[{1,2,-2,-1,1},{0,5,13,28,44},50] (* Harvey P. Dale, Jan 23 2018 *)

x='x+O('x^50); Vec(x*(5+8*x+5*x^2)/((x+1)^2*(1-x)^3)) \\ G. C. Greubel, Jun 20 2017

From Bruno Berselli, Feb 04 2012: (Start)
G.f.: x*(5+8*x+5*x^2)/((x+1)^2*(1-x)^3).
a(n) = a(-n+1) = (18*n*(n-1)+(2*n-1)*(-1)^n+1)/8 = A004526(n)*A156638(n). (End).

A266957 Numbers m such that 9*m+10 is a square.

Original entry on oeis.org

-1, 6, 10, 31, 39, 74, 86, 135, 151, 214, 234, 311, 335, 426, 454, 559, 591, 710, 746, 879, 919, 1066, 1110, 1271, 1319, 1494, 1546, 1735, 1791, 1994, 2054, 2271, 2335, 2566, 2634, 2879, 2951, 3210, 3286, 3559, 3639, 3926, 4010, 4311, 4399, 4714, 4806, 5135, 5231, 5574

Offset: 1

Bruno Berselli, Jan 07 2016

Equivalently, numbers of the form h*(9*h+2)-1, where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ...

Also, integer values of k*(k+2)/9 minus 1.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A132355.
Cf. similar sequences listed in A266956.
Cf. A056020: square roots of 9*a(n)+10.

[n: n in [-1..6000] | IsSquare(9*n+10)];

[(18*(n-1)*n+5*(2*n-1)*(-1)^n-3)/8: n in [1..50]];

Select[Range[-1, 6000], IntegerQ[Sqrt[9 # + 10]] &]
Table[(18 (n - 1) n + 5 (2 n - 1) (-1)^n - 3)/8, {n, 1, 50}]

for(n=-1, 6000, if(issquare(9*n+10), print1(n, ", ")))

vector(50, n, n; (18*(n-1)*n+5*(2*n-1)*(-1)^n-3)/8)

from gmpy2 import is_square
[n for n in range(-1,6000) if is_square(9*n+10)]

[(18*(n-1)*n+5*(2*n-1)*(-1)**n-3)/8 for n in range(1, 60)]

[n for n in (-1..6000) if is_square(9*n+10)]

[(18*(n-1)*n+5*(2*n-1)*(-1)^n-3)/8 for n in (1..50)]

G.f.: x*(-1 + 7*x + 6*x^2 + 7*x^3 - x^4)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n+1) = (18*(n-1)*n + 5*(2*n-1)*(-1)^n - 3)/8.
a(n) = A132355(n) + 1.

A266958 Numbers m such that 9*m+13 is a square.

Original entry on oeis.org

-1, 4, 12, 27, 43, 68, 92, 127, 159, 204, 244, 299, 347, 412, 468, 543, 607, 692, 764, 859, 939, 1044, 1132, 1247, 1343, 1468, 1572, 1707, 1819, 1964, 2084, 2239, 2367, 2532, 2668, 2843, 2987, 3172, 3324, 3519, 3679, 3884, 4052, 4267, 4443, 4668, 4852, 5087, 5279, 5524

Offset: 1

Bruno Berselli, Jan 07 2016

Equivalently, numbers of the form h*(9*h+4)-1, where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ...
Also, integer values of k*(k+4)/9 minus 1.
Is A063289 (after -1) the list of the square roots of 9*a(n)+13?

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A185039.

Cf. similar sequences listed in A266956.

[n: n in [-1..6000] | IsSquare(9*n+13)];

[(18*(n-1)*n+(2*n-1)*(-1)^n-7)/8: n in [1..50]];

Select[Range[-1, 6000], IntegerQ[Sqrt[9 # + 13]] &]
Table[(18 (n-1) n + (2 n - 1) (-1)^n - 7)/8, {n, 1, 50}]
LinearRecurrence[{1,2,-2,-1,1},{-1,4,12,27,43},50] (* Harvey P. Dale, Jan 20 2020 *)

for(n=-1, 6000, if(issquare(9*n+13), print1(n, ", ")))

vector(50, n, n; (18*(n-1)*n+(2*n-1)*(-1)^n-7)/8)

from gmpy2 import is_square
[n for n in range(-1,6000) if is_square(9*n+13)]

[(18*(n-1)*n+(2*n-1)*(-1)**n-7)/8 for n in range(1,60)]

[n for n in range(-1,6000) if is_square(9*n+13)]

[(18*(n-1)*n+(2*n-1)*(-1)^n-7)/8 for n in range(1,50)]

G.f.: x*(-1 + 5*x + 10*x^2 + 5*x^3 - x^4)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n+1) = (18*(n-1)*n + (2*n-1)*(-1)^n - 7)/8.
a(n) = A185039(n) + 1.

A375352 Numbers k such that 14*k + 2 is a square.

Original entry on oeis.org

1, 7, 23, 41, 73, 103, 151, 193, 257, 311, 391, 457, 553, 631, 743, 833, 961, 1063, 1207, 1321, 1481, 1607, 1783, 1921, 2113, 2263, 2471, 2633, 2857, 3031, 3271, 3457, 3713, 3911, 4183, 4393, 4681, 4903, 5207, 5441, 5761, 6007, 6343, 6601, 6953, 7223, 7591, 7873

Offset: 1

Juri-Stepan Gerasimov, Aug 12 2024

a(11) = 391 is first composite number in this sequence.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Numbers k such that (m + (16-m)*k) is a square: A204221 (m = 1), this sequence (m = 2), A001082 (m = 4), A181433 (m = 5), A273367 (m = 6), A266956 (m = 7), A056220 (m = 8), A274978 (m = 9), A028872 (m = 12), A161532 (m = 14).

Cf. A113804, A212965.

[k: k in [0..8000] | IsSquare(14*k + 2)];

((Table[14*n + {4, 10}, {n, 0, 23}] // Flatten)^2 - 2)/14 (* Amiram Eldar, Aug 13 2024 *)

a(n) = (A113804(n)^2 - 2)/14. - Amiram Eldar, Aug 13 2024
a(n) = 2*A212965(n-1) - 1. - Hugo Pfoertner, Aug 13 2024
E.g.f.: ((2 + x + 7*x^2)*cosh(x) + (1 - x + 7*x^2)*sinh(x) - 2)/2. - Stefano Spezia, Aug 13 2024

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

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A132355 Numbers of the form 9*h^2 + 2*h, for h an integer.

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A185039 Numbers of the form 9*m^2 + 4*m, m an integer.

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A266957 Numbers m such that 9*m+10 is a square.

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A266958 Numbers m such that 9*m+13 is a square.

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A375352 Numbers k such that 14*k + 2 is a square.

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

A132355 Numbers of the form 9h^2 + 2h, for h an integer.

A185039 Numbers of the form 9m^2 + 4m, m an integer.