A092242 -id:A092242 - cp's OEIS frontend

A195143 a(n) = n-th concentric 12-gonal number.

0, 1, 12, 25, 48, 73, 108, 145, 192, 241, 300, 361, 432, 505, 588, 673, 768, 865, 972, 1081, 1200, 1321, 1452, 1585, 1728, 1873, 2028, 2185, 2352, 2521, 2700, 2881, 3072, 3265, 3468, 3673, 3888, 4105, 4332, 4561, 4800, 5041, 5292, 5545, 5808, 6073, 6348

Offset: 0

Omar E. Pol, Sep 17 2011

Concentric dodecagonal numbers. [corrected by Ivan Panchenko, Nov 09 2013]
Sequence found by reading the line from 0, in the direction 0, 12,..., and the same line from 1, in the direction 1, 25,..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Main axis, perpendicular to A028896 in the same spiral.
Partial sums of A091998. - Reinhard Zumkeller, Jan 07 2012
Column 12 of A195040. - Omar E. Pol, Sep 28 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

A135453 and A069190 interleaved.
Cf. A001082, A032527, A032528, A077221, A195043, A195045, A195040, A195142, A195145, A195146, A195147, A195148, A195149.
Cf. A016921 (6n+1), A016969 (6n+5), A091998 (positive integers of the form 12*k +- 1), A092242 (positive integers of the form 12*k +- 5).

a195143 n = a195143_list !! n
a195143_list = scanl (+) 0 a091998_list
-- Reinhard Zumkeller, Jan 07 2012

[(3*n^2+(-1)^n-1): n in [0..50]]; // Vincenzo Librandi, Sep 27 2011

Table[Sum[2*(-1)^(n - k + 1) + 6*k - 3, {k, n}], {n, 0, 47}] (* L. Edson Jeffery, Sep 14 2014 *)

From Vincenzo Librandi, Sep 27 2011: (Start)
a(n) = 3*n^2+(-1)^n-1.
a(n) = -a(n-1) + 6*n^2 - 6*n + 1. (End)
G.f.: -x*(1+10*x+x^2) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Sep 18 2011
a(n) = Sum_{k=1..n} (2*(-1)^(n-k+1) + 3*(2*k-1)), n>0, a(0) = 0. - L. Edson Jeffery, Sep 14 2014
Sum_{n>=1} 1/a(n) = Pi^2/72 + tan(Pi/sqrt(6))*Pi/(4*sqrt(6)). - Amiram Eldar, Jan 16 2023

A092260 Permutation of natural numbers generated by 6-rowed array shown below.

Original entry on oeis.org

1, 11, 2, 13, 10, 3, 23, 14, 9, 4, 25, 22, 15, 8, 5, 35, 26, 21, 16, 7, 6, 37, 34, 27, 20, 17, 12, 47, 38, 33, 28, 19, 18, 49, 46, 39, 32, 29, 24, 59, 50, 45, 40, 31, 30, 61, 58, 51, 44, 41, 36, 71, 62, 57, 52, 43, 42, 73, 70, 63, 56, 53, 48, 83, 74, 69, 64, 55, 54, 85, 82, 75

Offset: 1

Giovanni Teofilatto, Feb 19 2004

nonn

11 13 23 25 35 37 47 49 59... (A091998)
10 14 22 26 34 38 46 50 58... (A091999)
9 15 21 27 33 39 45 51 57... (A016945)
8 16 20 28 32 40 44 52 56... (A092259)
7 17 19 29 31 41 43 53 55... (A092242)
12 18 24 30 36 42 48 54 60... (A008588, excluding initial term)
For such arrays A_k, here A_6, see a W. Lang comment on A113807, the A_7 case. However, to get the array A_6 one should take the last line as the first one and add a 0 in front (thus obtaining a permutation of the nonnegative integers). - Wolfdieter Lang, Feb 02 2012

Cf. A088520, A090243, A090298, A113807.

Edited and extended by Ray Chandler, Feb 21 2004

A120683 Decimal expansion of secant of 15 degrees (cosecant of 75 degrees).

Original entry on oeis.org

1, 0, 3, 5, 2, 7, 6, 1, 8, 0, 4, 1, 0, 0, 8, 3, 0, 4, 9, 3, 9, 5, 5, 9, 5, 3, 5, 0, 4, 9, 6, 1, 9, 3, 3, 1, 3, 3, 9, 6, 2, 7, 5, 6, 0, 5, 2, 7, 9, 7, 2, 2, 0, 5, 5, 2, 5, 6, 0, 1, 2, 8, 2, 9, 2, 6, 0, 2, 2, 7, 8, 9, 8, 9, 9, 5, 2, 0, 7, 9, 8, 7, 6, 8, 9, 4, 7, 1, 8, 9, 8, 7, 7, 6, 9, 9, 8, 6, 6, 2, 0, 8, 3, 5, 8

Offset: 1

Rick L. Shepherd, Jun 24 2006

Side length of the largest equilateral triangle that can be inscribed in a unit square (as stated in MathWorld/Weisstein link).

A quartic integer. - Charles R Greathouse IV, Aug 27 2017

			1.03527618041008304939559535049619331339627560527972...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

Eric Weisstein's World of Mathematics, Equilateral Triangle.
Index entries for algebraic numbers, degree 4.

Cf. A002193, A019679, A019691, A019884, A010464, A092242, A101263.

RealDigits[Sec[15 Degree],10,120][[1]] (* Harvey P. Dale, Jun 03 2015 *)

sqrt(6) - sqrt(2) \\ Charles R Greathouse IV, Aug 27 2017

Equals sec(Pi/12) = sec(A019679) = sqrt(6) - sqrt(2) = A010464 - A002193 = csc(5*Pi/12) = 1/sin(5*Pi/12) = 1/sin(10*A019691) = 1/A019884.
Equals Product_{k >= 1} 1/(1 - 1/(36*(2*k - 1)^2)). - Antonio Graciá Llorente, Mar 20 2024
From Amiram Eldar, Nov 24 2024: (Start)
Equals 2*A101263.
Equals Product_{k>=1} (1 - (-1)^k/A092242(k)). (End)
Smallest positive of the 4 real-valued roots of x^4-16*x^2+16=0. - R. J. Mathar, Aug 31 2025

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A195143 a(n) = n-th concentric 12-gonal number.

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A092260 Permutation of natural numbers generated by 6-rowed array shown below.

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A120683 Decimal expansion of secant of 15 degrees (cosecant of 75 degrees).

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