A179088 -id:A179088 - cp's OEIS frontend

A113338 Positive integers of the form (18*m^2+1)/11.

41, 59, 419, 473, 1193, 1283, 2363, 2489, 3929, 4091, 5891, 6089, 8249, 8483, 11003, 11273, 14153, 14459, 17699, 18041, 21641, 22019, 25979, 26393, 30713, 31163, 35843, 36329, 41369, 41891, 47291, 47849, 53609, 54203, 60323, 60953

Offset: 1

Zak Seidov, Jan 08 2006

Here m=(11*(2*n-1)-9*(-1)^n)/4 for n>0.

(18*m^2 + 1)/11 is an integer for m=5+11k and m=6+11k, k=0,1,2,.... [ Corrected by Bruno Berselli, Jun 28 2010]

Bruno Berselli, Table of n, a(n) for n = 1..10000.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A179088, A179337, A179338, A179339, A179370.

a(n)=(198*n*(n-1)-81*(2*n-1)*(-1)^n+83)/4 \\ Charles R Greathouse IV, Oct 07 2015

From Bruno Berselli, Jun 29 2010: (Start)
G.f.: x*(41+18*x+278*x^2+18*x^3+41*x^4)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
a(n) = (198*n*(n-1)-81*(2*n-1)*(-1)^n+83)/4. (End)

Definition corrected, comment and cross-reference added by Bruno Berselli, Jul 13 2010

A179337 Positive integers of the form (6*m^2 + 1)/11.

Original entry on oeis.org

5, 35, 107, 197, 341, 491, 707, 917, 1205, 1475, 1835, 2165, 2597, 2987, 3491, 3941, 4517, 5027, 5675, 6245, 6965, 7595, 8387, 9077, 9941, 10691, 11627, 12437, 13445, 14315, 15395, 16325, 17477, 18467, 19691, 20741, 22037, 23147, 24515

Offset: 1

Bruno Berselli, Jul 11 2010

Here m = (11*(2*n-1) - (-1)^n)/4 for n > 0.

B. Berselli, Table of n, a(n) for n = 1..10000.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A113338, A179088, A179338, A179339, A179370.

[(66*n*(n-1)-3*(2*n-1)*(-1)^n+17)/4: n in [1..40]]; // Vincenzo Librandi, Nov 16 2011

LinearRecurrence[{1,2,-2,-1,1},{5,35,107,197,341},40] (* Vincenzo Librandi, Nov 16 2011 *)

a(n) = (66*n*(n-1) - 3*(2*n-1)*(-1)^n + 17)/4.
G.f.: x*(5 + 30*x + 62*x^2 + 30*x^3 + 5*x^4)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).

A179370 Positive integers of the form (7*m^2+1)/11.

Original entry on oeis.org

16, 23, 163, 184, 464, 499, 919, 968, 1528, 1591, 2291, 2368, 3208, 3299, 4279, 4384, 5504, 5623, 6883, 7016, 8416, 8563, 10103, 10264, 11944, 12119, 13939, 14128, 16088, 16291, 18391, 18608, 20848, 21079, 23459, 23704, 26224, 26483

Offset: 1

Bruno Berselli, Jul 12 2010

Here m=(11*(2*n-1)-9*(-1)^n)/4 for n>0.

B. Berselli, Table of n, a(n) for n = 1..10000.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A113338, A179088, A179337, A179338, A179339.

a(n) = (154*n*(n-1)-63*(2*n-1)*(-1)^n+65)/8.
G.f.: x*(16+7*x+108*x^2+7*x^3+16*x^4)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).

A179338 Positive integers of the form (10*m^2+1)/11.

Original entry on oeis.org

1, 91, 131, 401, 481, 931, 1051, 1681, 1841, 2651, 2851, 3841, 4081, 5251, 5531, 6881, 7201, 8731, 9091, 10801, 11201, 13091, 13531, 15601, 16081, 18331, 18851, 21281, 21841, 24451, 25051, 27841, 28481, 31451, 32131, 35281, 36001, 39331

Offset: 1

Bruno Berselli, Jul 11 2010

Here m=(11*(2*n-1)+7*(-1)^n)/4 for n>0.

B. Berselli, Table of n, a(n) for n = 1..10000.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A113338, A179088, A179337, A179339, A179370.

[(110*n*(n-1)+35*(2*n-1)*(-1)^n+39)/4: n in [1..50]]; // Vincenzo Librandi, Jul 10 2014

Select[(10Range[300]^2+1)/11,IntegerQ] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{1,91,131,401,481},50] (* Harvey P. Dale, Jul 10 2014 *)

isok(n) = (((11*n-1) % 10 == 0) && issquare((11*n-1)/10)) \\ Michel Marcus, Jun 07 2013

a(n) = (110*n*(n-1)+35*(2*n-1)*(-1)^n+39)/4.
G.f.: x*(1+90*x+38*x^2+90*x^3+1*x^4)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).

A179339 Positive integers of the form (30*m^2+1)/11.

Original entry on oeis.org

11, 221, 461, 1091, 1571, 2621, 3341, 4811, 5771, 7661, 8861, 11171, 12611, 15341, 17021, 20171, 22091, 25661, 27821, 31811, 34211, 38621, 41261, 46091, 48971, 54221, 57341, 63011, 66371, 72461, 76061, 82571, 86411, 93341, 97421

Offset: 1

Bruno Berselli, Jul 11 2010 - Dec 10 2010

Here m = (11*(2*n-1)+3*(-1)^n)/4 for n>0.

More generally, (t*((11*(2*n-1)+k*(-1)^n)/4)^2 +1)/11 = ( 22*t*n*(n-1) +t*k*(2*n-1)*(-1)^n+(t*(k^2+121)+16)/22 )/8 for any natural number t == 2, 6, 7, 8, 10 (mod 11) and k = -5, -1, -9, 3, 7, respectively.

B. Berselli, Table of n, a(n) for n = 1..10000.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A113338, A179088, A179337, A179338, A179370.

LinearRecurrence[{1,2,-2,-1,1},{11,221,461,1091,1571},40] (* Harvey P. Dale, Mar 03 2023 *)

a(n) = (330*n*(n-1)+45*(2*n-1)*(-1)^n+89)/4.
G.f.: x*(11+210*x+218*x^2+210*x^3+11*x^4)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).

A281445 Nonnegative k for which (2*k^2 + 1)/11 is an integer.

Original entry on oeis.org

4, 7, 15, 18, 26, 29, 37, 40, 48, 51, 59, 62, 70, 73, 81, 84, 92, 95, 103, 106, 114, 117, 125, 128, 136, 139, 147, 150, 158, 161, 169, 172, 180, 183, 191, 194, 202, 205, 213, 216, 224, 227, 235, 238, 246, 249, 257, 260, 268, 271, 279, 282, 290, 293, 301, 304, 312, 315

Offset: 1

Bruno Berselli, Apr 13 2017

For prime d < 11, (2*k^2 + 1)/d can provide integers when d = 3 (A186424).
Corresponding values of (2*k^2 + 1)/11 are listed in A179088.
All k == 4 or 7 (mod 11). - Robert Israel, Apr 25 2017

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

Cf. A179088.

Cf. A001651 (nonnegative k for which (2*k^2 + 1)/3 is an integer).

Magma
```
&cat [[11*n+4, 11*n+7]: n in [0..30]];
```

seq(seq(11*i+j,j=[4,7]),i=0..50); # Robert Israel, Apr 25 2017

Select[Range[400], IntegerQ[(2*#^2 + 1)/11] &]

[k for k in range(400) if ((2*k^2+1)/11).is_integer()]

O.g.f.: x*(4 + 3*x + 4*x^2)/((1 + x)*(1 - x)^2).
E.g.f.: 4 - 5*exp(-x)/4 - 11*(1 - 2*x)*exp(x)/4.
a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
a(n) = (22*n - 5*(-1)^n - 11)/4. Therefore: a(2*h) = 11*h - 4, a(2*h+1) = 11*h + 4.
If h>0,
h*a(n) + (6*h - 5*(-1)^h - 11)/4 = a(h*n) for odd n; otherwise:
h*a(n) + 4*(h - 1) = a(h*n). Some special cases:
h=2: 2*a(n) - 1 = a(2*n) for odd n, 2*a(n) +  4 = a(2*n) for even n;
h=3: 3*a(n) + 3 = a(3*n) for odd n, 3*a(n) +  8 = a(3*n) for even n;
h=4: 4*a(n) + 2 = a(4*n) for odd n, 4*a(n) + 12 = a(4*n) for even n;
h=5: 5*a(n) + 6 = a(5*n) for odd n, 5*a(n) + 16 = a(5*n) for even n, and so on.
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(3*Pi/22)*Pi/11. - Amiram Eldar, Feb 27 2023
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = cos(Pi/22)*sec(3*Pi/22).
Product_{n>=1} (1 + (-1)^n/a(n)) = 2*sin(3*Pi/22). (End)

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A113338 Positive integers of the form (18*m^2+1)/11.

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A179337 Positive integers of the form (6*m^2 + 1)/11.

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A179370 Positive integers of the form (7*m^2+1)/11.

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A179338 Positive integers of the form (10*m^2+1)/11.

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A179339 Positive integers of the form (30*m^2+1)/11.

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A281445 Nonnegative k for which (2*k^2 + 1)/11 is an integer.

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