A106833 -id:A106833 - cp's OEIS frontend

A158552 a(n) = A144433(n) - A106833(n).

5, -1, 7, -3, 9, -5, 11, -7, 13, -9, 15, -11, 17, -13, 19, -15, 21, -17, 23, -19, 25, -21, 27, -23, 29, -25, 31, -27, 33, -29, 35, -31, 37, -33, 39, -35, 41, -37, 43, -39, 45, -41, 47, -43, 49, -45, 51, -47, 53, -49, 55, -51, 57, -53, 59, -55, 61, -57, 63, -59, 65, -61, 67

Offset: 1

Paul Curtz, Mar 21 2009

sign

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

Cf. A118402, A106833, A144433.

a(n) = A118402(n+3).
From R. J. Mathar, Apr 08 2009: (Start)
G.f.: x*(5+4*x+x^2)/((1-x)*(1+x)^2).
a(n) = -a(n-1) + a(n-2) + a(n-3).
(End)

Edited and extended by R. J. Mathar, Apr 08 2009

A158674 Period 18: repeat 3, 4, 0, 8, 6, 3, 3, 7, 0, 2, 6, 6, 3, 1, 0, 5, 6, 0.

Original entry on oeis.org

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

Offset: 1

Paul Curtz, Mar 24 2009

Also the decimal expansion of 378737078073678400/111111111111111111 .

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

a(n)=[0, 3, 4, 0, 8, 6, 3, 3, 7, 0, 2, 6, 6, 3, 1, 0, 5, 6][n%18+1] \\ Charles R Greathouse IV, Jul 17 2016

a(n) = A106833(n) mod 9.

G.f.: -(3+4*x+8*x^3+6*x^4+3*x^5+3*x^6+7*x^7+2*x^9+6*x^10+6*x^11+3*x^12+x^13+5*x^15+6*x^16) / ((x-1) * (1+x+x^2) * (1+x^3+x^6) * (1+x) * (1-x+x^2) * (1-x^3+x^6)).

References to apparently unrelated sequences removed by R. J. Mathar, Mar 02 2010

A168068 Array T(n,k) read by antidiagonals: T(n,2k+1) = 2k+1. T(n,2k) = 2^n*k.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 4, 3, 2, 0, 1, 8, 3, 4, 5, 0, 1, 16, 3, 8, 5, 3, 0, 1, 32, 3, 16, 5, 6, 7, 0, 1, 64, 3, 32, 5, 12, 7, 4, 0, 1, 128, 3, 64, 5, 24, 7, 8, 9, 0, 1, 256, 3, 128, 5, 48, 7, 16, 9, 5, 0, 1, 512, 3, 256, 5, 96, 7, 32, 9, 10, 11, 0, 1, 1024, 3, 512, 5, 192, 7, 64, 9, 20, 11, 6, 0, 1, 2048, 3, 1024, 5

Offset: 0

Paul Curtz, Nov 18 2009

The array is constructed multiplying the even-indexed A026741(k) by 2^n, and keeping the odd-indexed A026471(k) as they are.

Connections to the hydrogen spectrum: The squares of the second row are T(1,k)^2 = A001477(k)^2 = A000290(k) which are the denominators of the Lyman lines (see A171522). The squares of the row T(2,k) are in A154615, denominators of the Balmer series. Row T(3,k) is related to A106833 and A061038.

			The array starts in row n=0 with columns k>=0 as:
0,1,1,3,2,5,3,7,4, A026741
0,1,2,3,4,5,6,7,8, A001477
0,1,4,3,8,5,12,7,16, A022998
0,1,8,3,16,5,24,7,32, A144433
0,1,16,3,32,5,48,7,64,
0,1,32,3,64,5,96,7,128,

A168068 := proc(n,k) if type(k,'odd') then k; else 2^(n-1)*k ; end if; end proc: # R. J. Mathar, Jan 22 2011

A298950 Numbers k such that 5*k - 4 is a square.

Original entry on oeis.org

1, 4, 8, 17, 25, 40, 52, 73, 89, 116, 136, 169, 193, 232, 260, 305, 337, 388, 424, 481, 521, 584, 628, 697, 745, 820, 872, 953, 1009, 1096, 1156, 1249, 1313, 1412, 1480, 1585, 1657, 1768, 1844, 1961, 2041, 2164, 2248, 2377, 2465, 2600, 2692, 2833, 2929, 3076, 3176, 3329, 3433

Offset: 1

Bruno Berselli, Jan 30 2018

a(n) is a member of A140612. Proof: a(n) = n^2 + (n/2-1)^2 for even n, otherwise a(n) = (n-1)^2 + ((n+1)/2)^2; also, a(n) + 1 = (n-1)^2 + (n/2+1)^2 for even n, otherwise a(n) + 1 = n^2 + ((n-3)/2)^2. Therefore, both a(n) and a(n) + 1 belong to A001481.
Primes in sequence are listed in A245042.
Squares in sequence are listed in A081068.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A195162: numbers k such that 5*k + 4 is a square.
Subsequence of A001481, A020668, A036404, A140612.
Cf. A036666, A081068, A106833 (first differences), A245042.

List([1..60], n -> (10*n*(n-1)+(2*n-1)*(-1)^n+9)/8);

[(10*n*(n-1)+(2*n-1)*(-1)^n+9)/8: n in [1..60]];

Table[(10 n (n - 1) + (2 n - 1) (-1)^n + 9)/8, {n, 1, 60}]
LinearRecurrence[{1,2,-2,-1,1},{1,4,8,17,25},60] (* Harvey P. Dale, Sep 16 2022 *)

makelist((10*n*(n-1)+(2*n-1)*(-1)^n+9)/8, n, 1, 60);

Vec((1+x^2)*(1+3*x+x^2)/((1-x)^3*(1+x)^2)+O(x^60))

vector(60, n, nn; (10*n*(n-1)+(2*n-1)*(-1)^n+9)/8)

[(10*n*(n-1)+(2*n-1)*(-1)**n+9)/8 for n in range(1, 60)]

[(10*n*(n-1)+(2*n-1)*(-1)^n+9)/8 for n in (1..60)]

G.f.: x*(1 + x^2)*(1 + 3*x + x^2)/((1 - x)^3*(1 + x)^2).
a(n) = a(1-n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (10*n*(n-1) + (2*n-1)*(-1)^n + 9)/8.
a(n) = A036666(n) + 1.

A152053 a(n) = A144433(3n+1) + A144433(3n+2) + A144433(3n+3).

Original entry on oeis.org

27, 36, 81, 72, 135, 108, 189, 144, 243, 180, 297, 216, 351, 252, 405, 288, 459, 324, 513, 360, 567, 396, 621, 432, 675, 468, 729, 504, 783, 540, 837, 576, 891, 612, 945, 648, 999, 684, 1053, 720, 1107

Offset: 0

Paul Curtz, Nov 22 2008

All terms are multiples of 9.

Index entries for linear recurrences with constant coefficients, signature (0, 2, 0, -1).

Cf. A106933, A144433.

Table[(9/2)(5 + (-1)^n)(n + 1), {n, 0, 40}] (* Jean-François Alcover, Feb 02 2019 *)
LinearRecurrence[{0,2,0,-1},{27,36,81,72},50] (* Harvey P. Dale, Nov 07 2019 *)

a(n) = 45*(n+1)/2 + 9*(-1)^n*(n+1)/2 \\ Jianing Song, Feb 04 2019

From R. J. Mathar, May 21 2009: (Start)
G.f.: 9*(3+4*x+3*x^2)/((x-1)^2*(1+x)^2).
a(n) = 45*(n+1)/2 + 9*(-1)^n*(n+1)/2. (End)
a(n) = 9*A106833(n+1). - Jean-François Alcover, Feb 02 2019, after Paul Curtz
a(n+4) = 2*a(n+2) - a(n). - Jianing Song, Feb 04 2019

Edited by R. J. Mathar, May 21 2009

A340519 Smallest order of a non-abelian group with a center of order n.

Original entry on oeis.org

6, 8, 18, 16, 30, 24, 42, 32, 54, 40, 66, 48, 78, 56, 90, 64, 102, 72, 114, 80, 126, 88, 138, 96, 150, 104, 162, 112, 174, 120, 186, 128, 198, 136, 210, 144, 222, 152, 234, 160, 246, 168, 258, 176, 270, 184, 282, 192, 294, 200, 306, 208, 318, 216, 330, 224, 342, 232, 354, 240, 366, 248

Offset: 1

Bob Heffernan and Des MacHale, Jan 24 2021; corrected Feb 14 2021

nonn

a(n) is 6n if n is odd and 4n if n is even. This is because the groups involved are C(n) X S3 if n is odd, where S3 is the symmetric group of order 6, and C(n/2) X D8 if n is even, where D8 is the dihedral group of order 8 and C(m) is the cyclic group of order m.

By Lagrange's Theorem a(n) is a multiple of n.

Equals 2*A106833.

Cf. A059806, A060702, A340518.

Table[If[OddQ[n],6n,4n],{n,100}] (* Harvey P. Dale, Mar 03 2023 *)

cp's OEIS Frontend

A158552 a(n) = A144433(n) - A106833(n).

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A158674 Period 18: repeat 3, 4, 0, 8, 6, 3, 3, 7, 0, 2, 6, 6, 3, 1, 0, 5, 6, 0.

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A168068 Array T(n,k) read by antidiagonals: T(n,2k+1) = 2k+1. T(n,2k) = 2^n*k.

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A298950 Numbers k such that 5*k - 4 is a square.

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A152053 a(n) = A144433(3n+1) + A144433(3n+2) + A144433(3n+3).

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A340519 Smallest order of a non-abelian group with a center of order n.

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