A113804 -id:A113804 - cp's OEIS frontend

A113801 Numbers that are congruent to {1, 13} mod 14.

1, 13, 15, 27, 29, 41, 43, 55, 57, 69, 71, 83, 85, 97, 99, 111, 113, 125, 127, 139, 141, 153, 155, 167, 169, 181, 183, 195, 197, 209, 211, 223, 225, 237, 239, 251, 253, 265, 267, 279, 281, 293, 295, 307, 309, 321, 323, 335, 337, 349, 351, 363, 365, 377, 379

Offset: 1

Giovanni Teofilatto, Jan 22 2006

If 14k+1 is a perfect square..(0,12,16,52,60,120..) then the square root of 14k+1 = a(n) - Gary Detlefs, Feb 22 2010

More generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1==0 (mod h); in our case, a(n)^2-1==0 (mod 14). Also a(n)^2-1==0 (mod 28). - Bruno Berselli, Oct 26 2010 - Nov 17 2010

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

Cf. A000217, A113802, A113803, A113804, A113805, A113806, A113807, A008589, A045472 (primes), A195145 (partial sums), A005408, A047209, A007310, A047336, A047522, A056020, A090771, A175885, A091998, A175886, A175887.

a113801 n = a113801_list !! (n-1)
a113801_list = 1 : 13 : map (+ 14) a113801_list
-- Reinhard Zumkeller, Jan 07 2012

LinearRecurrence[{1,1,-1},{1,13,15},60] (* or *) Select[Range[500], MemberQ[{1,13},Mod[#,14]]&] (* Harvey P. Dale, May 11 2011 *)

a(n)=n\2*14-(-1)^n \\ Charles R Greathouse IV, Sep 15 2015

a(n) = 14*(n-1)-a(n-1), n>1. - R. J. Mathar, Jan 30 2010
From Bruno Berselli, Oct 26 2010: (Start)
a(n) = -a(-n+1) = (14*n+5*(-1)^n-7)/2.
G.f.: x*(1+12*x+x^2)/((1+x)*(1-x)^2).
a(n) = a(n-2)+14 for n>2.
a(n) = 14*A000217(n-1)+1 - 2*sum[i=1..n-1] a(i) for n>1. (End)
a(0)=1, a(1)=13, a(2)=15, a(n)=a(n-1)+a(n-2)-a(n-3). - Harvey P. Dale, May 11 2011
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/14)*cot(Pi/14). - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((14*x - 7)*exp(x) + 5*exp(-x))/2. - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/14).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/14)*cosec(Pi/14). (End)

Corrected and extended by Giovanni Teofilatto, Nov 14 2008

Replaced the various formulas by a correct one - R. J. Mathar, Jan 30 2010

A113807 Permutation of natural numbers generated by 7-rowed array shown below.

Original entry on oeis.org

1, 13, 2, 15, 12, 3, 27, 16, 11, 4, 29, 26, 17, 10, 5, 41, 30, 25, 18, 9, 6, 43, 40, 31, 24, 19, 8, 7, 55, 44, 39, 32, 23, 20, 14, 57, 54, 45, 38, 33, 22, 21, 69, 58, 53, 46, 37, 34, 28, 71, 68, 59, 52, 47, 36, 35, 83, 72, 67, 60, 51, 48, 42, 85, 82, 73, 66, 61, 50, 49

Offset: 1

Giovanni Teofilatto, Jan 22 2006

nonn

For such arrays A_k see the k-family of 2k-periodic sequences P_k defined in a comment on A203571. There the k residue classes mod n have been defined. The present array is A_7 if the last class, starting with 7, is taken as first class [0] after adding a 0 in front. Then one obtains a permutation of the nonnegative integers. However, each complete residue class also includes its negative members. - Wolfdieter Lang, Feb 02 2012

			1 13 15 27 29 41 43 55 57 69 71 83 85 ... (A113801)
2 12 16 26 30 40 44 54 58 68 72 82 86 ... (A113802)
3 11 17 25 31 39 45 53 59 67 73 81 87 ... (A113803)
4 10 18 24 32 38 46 52 60 66 74 80 88 ... (A113804)
5  9 19 23 33 37 47 51 61 65 75 70 89 ... (A113805)
6  8 20 22 34 36 48 50 62 64 76 78 90 ... (A113806)
7 14 21 28 35 42 49 56 63 70 77 84 91 ... (A008589)

Cf. A088520, A090243, A090298, A092260, A204453.

A-numbers added for array rows by Wolfdieter Lang, Dec 15 2011

More terms from Ray Chandler, Dec 15 2011

A193053 a(n) = (14n(n+3) + (2n-5)(-1)^n + 21)/16.

Original entry on oeis.org

1, 5, 10, 17, 26, 36, 49, 62, 79, 95, 116, 135, 160, 182, 211, 236, 269, 297, 334, 365, 406, 440, 485, 522, 571, 611, 664, 707, 764, 810, 871, 920, 985, 1037, 1106, 1161, 1234, 1292, 1369, 1430, 1511, 1575, 1660, 1727, 1816, 1886, 1979, 2052, 2149, 2225, 2326

Offset: 0

Bruno Berselli, Oct 20 2011 - based on remarks and sequences by Omar E. Pol

For an origin of this sequence, see the numerical spiral illustrated in the Links section.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A195020 (vertices of the numerical spiral in link).

Cf. A001106, A022264, A033572, A144555, A152760, A158482, A158485, A185019, A195021, A195023-A195030, A195320, A198017 [incomplete list].

[(14*n*(n+3)+(2*n-5)*(-1)^n+21)/16: n in [0..50]];

Table[(14*n*(n + 3) + (2*n - 5)*(-1)^n + 21)/16, {n, 0, 50}] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{1,5,10,17,26},60] (* Harvey P. Dale, Jun 19 2020 *)

for(n=0, 50, print1((14*n*(n+3)+(2*n-5)*(-1)^n+21)/16", "));

O.g.f.: (1 + 4*x + 3*x^2 - x^3)/((1 + x)^2*(1 - x)^3).
E.g.f.: (1/16)*((21 + 56*x + 14*x^2)*exp(x) - (5 + 2*x)*exp(-x)). - G. C. Greubel, Aug 19 2017
a(n) = A195020(n) + n + 1.
a(n)   - a(-n-1)  = A047336(n+1).
a(n+1) - a(-n)    = A113804(n+1).
a(n+2) - a(n)     = A047385(n+3).
a(n+4) - a(n)     = A113803(n+4).
a(2*n) + a(2*n-1) = A069127(n+1).
a(2*n) - a(2*n-1) = A016813(n).
a(2*n+1) - a(2*n) = A016777(n+1).
a(n+2) + 2*a(n+1) + a(n) = A024966(n+2).

A047385 Numbers that are congruent to {2, 5} mod 7.

Original entry on oeis.org

2, 5, 9, 12, 16, 19, 23, 26, 30, 33, 37, 40, 44, 47, 51, 54, 58, 61, 65, 68, 72, 75, 79, 82, 86, 89, 93, 96, 100, 103, 107, 110, 114, 117, 121, 124, 128, 131, 135, 138, 142, 145, 149, 152, 156, 159, 163, 166, 170

Offset: 1

N. J. A. Sloane

Also, numbers n such that (n^2+3)/7 is a nonnegative integer. - Bruno Berselli, Jan 16 2016

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

Cf. A113804, A255240, A255249.

A047385:=n->4*n-2-floor(n/2); seq(A047385(k),k=1..100); # Wesley Ivan Hurt, Oct 16 2013

Table[4 n - 2 - Floor[n/2], {n,100}] (* Wesley Ivan Hurt, Oct 16 2013 *)
#+{2,5}&/@(7*Range[0,30])//Flatten (* Harvey P. Dale, Jul 15 2017 *)

a(n)=(14*n-6)>>2 \\ Charles R Greathouse IV, Dec 05 2011

G.f.: x*(2 + 3*x + 2*x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Dec 05 2011
a(n) = (14*n - (-1)^n - 7)/4. - Bruno Berselli, Dec 05 2011
a(n) = 4*n - 2 - floor(n/2). - Wesley Ivan Hurt, Oct 16 2013
E.g.f.: 2 + ((14*x - 7)*exp(x) - exp(-x))/4. - David Lovler, Sep 01 2022
From Amiram Eldar, Sep 26 2022: (Start)
a(n) = A113804(n)/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(3*Pi/14)*Pi/7. (End)
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*sin(3*Pi/14) (A255249).
Product_{n>=1} (1 + (-1)^n/a(n)) = 1/(2*cos(Pi/7)) (A255240). (End)

A113803 Numbers that are congruent to {3, 11} mod 14.

Original entry on oeis.org

3, 11, 17, 25, 31, 39, 45, 53, 59, 67, 73, 81, 87, 95, 101, 109, 115, 123, 129, 137, 143, 151, 157, 165, 171, 179, 185, 193, 199, 207, 213, 221, 227, 235, 241, 249, 255, 263, 269, 277, 283, 291, 297, 305, 311, 319, 325, 333, 339, 347, 353, 361, 367

Offset: 1

Giovanni Teofilatto, Jan 22 2006

Third row A113807.

Cf. A008589, A113801, A113802, A113804, A113805, A113806.

{3+#,11+#}&/@(14*Range[0,30])//Flatten (* Harvey P. Dale, Jun 28 2020 *)

a(n) = 14*n - a(n-1) - 14 (with a(1) = 3). - Vincenzo Librandi, Nov 13 2010
From Wolfdieter Lang, Dec 15 2011: (Start)
a(n) = 7*n-(7-(-1)^n)/2.
O.g.f.: x*(3+8*x+3*x^2)/((1+x)*(1-x)^2).
See the Bruno Berselli contribution under A113801. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = cot(3*Pi/14)*Pi/14. - Amiram Eldar, Dec 30 2021
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = cot(3*Pi/14).
Product_{n>=1} (1 + (-1)^n/a(n)) = sin(Pi/7)*cosec(3*Pi/14). (End)

A113802 Numbers that are congruent to {2, 12} mod 14.

Original entry on oeis.org

2, 12, 16, 26, 30, 40, 44, 54, 58, 68, 72, 82, 86, 96, 100, 110, 114, 124, 128, 138, 142, 152, 156, 166, 170, 180, 184, 194, 198, 208, 212, 222, 226, 236, 240, 250, 254, 264, 268, 278, 282, 292, 296, 306, 310, 320, 324, 334, 338, 348, 352, 362, 366, 376, 380

Offset: 1

Giovanni Teofilatto, Jan 22 2006

Second row A113807.

Cf. A008589, A113801, A113803, A113804, A113805, A113806.

Select[Range[400],MemberQ[{2,12},Mod[#,14]]&] (* Harvey P. Dale, Oct 30 2011 *)

a(n) = 14*n - a(n-1) - 14 (with a(1) = 2). - Vincenzo Librandi, Nov 13 2010
From Wolfdieter Lang, Dec 15 2011: (Start)
a(n) = 7*n-(7-3*(-1)^n)/2.
O.g.f.: 2*x*(1+5*x+x^2)/((1+x)*(1-x)^2).
See the contribution of Bruno Berselli under A113801. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = cot(Pi/7)*Pi/14. - Amiram Eldar, Dec 30 2021
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = cosec(Pi/7)*sin(3*Pi/14).
Product_{n>=1} (1 + (-1)^n/a(n)) = cosec(Pi/7)*sin(Pi/14). (End)

A113805 Numbers that are congruent to {5, 9} mod 14.

Original entry on oeis.org

5, 9, 19, 23, 33, 37, 47, 51, 61, 65, 75, 79, 89, 93, 103, 107, 117, 121, 131, 135, 145, 149, 159, 163, 173, 177, 187, 191, 201, 205, 215, 219, 229, 233, 243, 247, 257, 261, 271, 275, 285, 289, 299, 303, 313, 317, 327, 331, 341, 345, 355, 359, 369

Offset: 1

Giovanni Teofilatto, Jan 22 2006

Cf. A113801, A113802, A113803, A113804, A113806, A008589, A113807.

Flatten[Table[14n + {5, 9}, {n, 0, 28}]] (* Alonso del Arte, Dec 15 2011 *)

a(n) = 14*n - a(n-1) - 14 (with a(1) = 5). - Vincenzo Librandi, Nov 13 2010
From Wolfdieter Lang, Dec 15 2011: (Start)
a(n) = 7*n-(7+3*(-1)^n)/2.
O.g.f.: x*(5+4*x+5*x^2)/((1+x)*(1-x)^2).
See the Bruno Berselli contribution under A113801. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(Pi/7)*Pi/14. - Amiram Eldar, Dec 30 2021
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 4*sin(Pi/7)*sin(3*Pi/14).
Product_{n>=1} (1 + (-1)^n/a(n)) = 2*sin(Pi/7). (End)

A113806 Numbers that are congruent to {6, 8} mod 14.

Original entry on oeis.org

6, 8, 20, 22, 34, 36, 48, 50, 62, 64, 76, 78, 90, 92, 104, 106, 118, 120, 132, 134, 146, 148, 160, 162, 174, 176, 188, 190, 202, 204, 216, 218, 230, 232, 244, 246, 258, 260, 272, 274, 286, 288, 300, 302, 314, 316, 328, 330, 342, 344, 356, 358, 370

Offset: 1

Giovanni Teofilatto, Jan 22 2006

Cf. A008589, A113801, A113802, A113803, A113804, A113805, A113807.

Flatten[# + {6, 8} &/@ (14 Range[0, 30])] (* Harvey P. Dale, Jan 11 2011 *)

a(n) = 14*n - a(n-1) - 14 (with a(1) = 6). - Vincenzo Librandi, Nov 13 2010
From Wolfdieter Lang, Dec 15 2011: (Start)
a(n) = 7*n-(7+5*(-1)^n)/2.
O.g.f.: 2*x*(3+x+3*x^2)/((1+x)*(1-x)^2).
See the Bruno Berselli contribution under A113801. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(Pi/14)*Pi/14. - Amiram Eldar, Dec 30 2021
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sec(Pi/14).
Product_{n>=1} (1 + (-1)^n/a(n)) = cosec(Pi/7)*cosec(3*Pi/14)/4. (End)

A164131 Numbers k such that k^2 == 2 (mod 31).

Original entry on oeis.org

8, 23, 39, 54, 70, 85, 101, 116, 132, 147, 163, 178, 194, 209, 225, 240, 256, 271, 287, 302, 318, 333, 349, 364, 380, 395, 411, 426, 442, 457, 473, 488, 504, 519, 535, 550, 566, 581, 597, 612, 628, 643, 659, 674, 690, 705, 721, 736, 752, 767, 783, 798, 814

Offset: 1

Vincenzo Librandi, Aug 11 2009

Sequences of the type n^2 == 2 (mod m) are basically defined for each m of A057126. See A047341 (m=7), A113804 (m=14), A155449 (m=17), A155450 (m=23), A158803 (m=41) etc. - R. J. Mathar, Aug 26 2009

			At n= 4, a(4)=(31-1+186)/4=54. At n=5, a(5)=(31+1+248)/4=70.

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

Cf. A057126, A047341, A113804, A155449, A155450, A158803.

Select[Range[850],Mod[#^2,31]==2&]  (* Harvey P. Dale, Feb 04 2011 *)

isok(k) = Mod(k, 31)^2 == 2; \\ Michel Marcus, Nov 22 2022

a(n) = a(n-1)+a(n-2)-a(n-3).
a(n) = (31+(-1)^(n-1)+62(n-1))/4.
G.f.: x*(8+15*x+8*x^2)/((1+x)*(x-1)^2). - R. J. Mathar, Aug 26 2009
a(n) = 31*(n-1)-a(n-1) with n>1, a(1)=8. - Vincenzo Librandi, Nov 30 2010
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(15*Pi/62)*Pi/31. - Amiram Eldar, Feb 28 2023

Entries checked by R. J. Mathar, Aug 26 2009

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

cp's OEIS Frontend

A113801 Numbers that are congruent to {1, 13} mod 14.

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A113807 Permutation of natural numbers generated by 7-rowed array shown below.

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A193053 a(n) = (14*n*(n+3) + (2*n-5)*(-1)^n + 21)/16.

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Magma

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A047385 Numbers that are congruent to {2, 5} mod 7.

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Maple

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A113803 Numbers that are congruent to {3, 11} mod 14.

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A113802 Numbers that are congruent to {2, 12} mod 14.

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A113805 Numbers that are congruent to {5, 9} mod 14.

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A113806 Numbers that are congruent to {6, 8} mod 14.

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A164131 Numbers k such that k^2 == 2 (mod 31).

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A193053 a(n) = (14n(n+3) + (2n-5)(-1)^n + 21)/16.