A014695 -id:A014695 - cp's OEIS frontend

A177841 Decimal expansion of (5+sqrt(221))/14.

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

Offset: 1

Klaus Brockhaus, May 14 2010

Continued fraction expansion of (5+sqrt(221))/14 is A014695.

			(5+sqrt(221))/14 = 1.41900491052275039447...

Cf. A177157 (decimal expansion of sqrt(221)), A014695 (repeat 1, 2, 2, 1).

RealDigits[(5+Sqrt[221])/14,10,120][[1]] (* Harvey P. Dale, Nov 12 2011 *)

A134430 Period 4: repeat [1, -2, -2, 1].

Original entry on oeis.org

1, -2, -2, 1, 1, -2, -2, 1, 1, -2, -2, 1, 1, -2, -2, 1, 1, -2, -2, 1, 1, -2, -2, 1, 1, -2, -2, 1, 1, -2, -2, 1, 1, -2, -2, 1, 1, -2, -2, 1, 1, -2, -2, 1, 1, -2, -2, 1, 1, -2, -2, 1, 1, -2, -2, 1, 1, -2, -2, 1, 1, -2, -2, 1, 1, -2, -2, 1, 1, -2, -2, 1, 1, -2

Offset: 0

Paul Curtz, Jan 31 2008

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

Cf. A014695, A134813.

&cat [[1, -2, -2, 1]^^30]; // Wesley Ivan Hurt, Jul 09 2016

seq(op([1, -2, -2, 1]), n=0..40); # Wesley Ivan Hurt, Jul 09 2016

PadRight[{}, 100, {1, -2, -2, 1}] (* Wesley Ivan Hurt, Jul 09 2016 *)

Binomial transform of A134813.
|a(n)| = A014695(n).
G.f.: (1-3*x+x^2) / ((1-x)*(1+x^2)). - R. J. Mathar, Jan 18 2011
From Wesley Ivan Hurt, Jul 09 2016: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) for n>2, a(n) = a(n-4) for n>3.
a(n) = (3*cos(n*Pi/2) - 3*sin(n*Pi/2) - 1)/2. (End)

A227168 a(n) = gcd(2n, n(n+1)/2)^2.

Original entry on oeis.org

1, 1, 36, 4, 25, 9, 196, 16, 81, 25, 484, 36, 169, 49, 900, 64, 289, 81, 1444, 100, 441, 121, 2116, 144, 625, 169, 2916, 196, 841, 225, 3844, 256, 1089, 289, 4900, 324, 1369, 361, 6084, 400

Offset: 1

Paul Curtz, Jul 03 2013

a(n) is defined as A062828(n)^2 for n >= 1. If we extend the sequence to n=0 and negative n by use of the recurrence that relates a(n) to a(n+12), a(n+8) and a(n+4), we obtain a(0)=0, a(-1)=4 and a(-n) = A176743(n-2)^2 for n >= 2.
Define c(n) = a(n+2) - a(n-2) for c >= 0. Because a(n) is a shuffle of three interleaved 2nd-order polynomials, c(n) is a shuffle of three interleaved 1st-order polynomials: c(n) = 4* A062828(n)*(periodically repeated 1, 8, 1, 1).
The sequence a(n) is case p=0 of the family A062828(n)*A062828(n+p):
0, 1, 1, 36,  4, 25,  9, 196, ... = a(n).
0, 1, 6, 12, 10, 15, 42,  56, ... = A130658(n)*A000217(n) = A177002(n-1)*A064038(n+1).
0, 6, 2, 30,  6, 70, 12, 126, ... = 2*A198148(n)
0, 2, 5, 18, 28, 20, 27,  70, ... = A177002(n+2)*A160050(n+1) = A014695(n+2)*A000096(n).

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Cf. A062828, A016814, A017138, A005565, A006752, A061037, A061038.

[GCD(2*n, n*(n+1)/2)^2: n in [1..50]]; // G. C. Greubel, Sep 20 2018

A227168 := proc(n)
    A062828(n)^2 ;
end proc: # R. J. Mathar, Jul 25 2013

a[n_] := GCD[2*n, n*(n + 1)/2]^2; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jul 03 2013 *)

a(n)=if(n%2, n*if(n%4>2, 2, 1), n/2)^2 \\ Charles R Greathouse IV, Jul 07 2013

a(n) = A062828(n)^2.
a(4n) = (4*n+1)^2; a(2n+1) = (n+1)^2; a(4n+2) = 4*(4*n+3)^2.
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
a(n) * (period 4: repeat 4, 1, 1, 4) = A061038(n).
A005565(n-3) = a(n+1) * A061037(n). - Corrected by R. J. Mathar, Jul 25 2013
a(n) = A130658(n-1)^2 * A181318(n). - Corrected by R. J. Mathar, Aug 01 2013
G.f.: -x*(1 + x + 36*x^2 + 4*x^3 + 22*x^4 + 6*x^5 + 88*x^6 + 4*x^7 + 9*x^8 + x^9 + 4*x^10) / ( (x-1)^3*(1+x)^3*(x^2+1)^3 ). - R. J. Mathar, Jul 20 2013
Sum_{n>=1} 1/a(n) = 47*Pi^2/192 + 3*G/8, where G is Catalan's constant (A006752). - Amiram Eldar, Aug 21 2022

A244840 Denominators of the triangle T(n,k) = (n*(n+1)/2+k+1)/(k+1) for n >= k >= 0.

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Jul 07 2014

Numerators: A244734(n,k).

See A244734 for the first entries of the rational triangle T(n,k).

			T(0,0) = 1/1, T(1,0) = 2/1, T(1,1) = 3/2,... .
The triangle a(n,k) begins:
n/k  0 1 2 3 4 5 6 7 8  9 10 11 12 13 14 15 16 17 18 19 20 ...
0:   1
1:   1 2
2:   1 2 1
3:   1 1 1 2
4:   1 1 3 2 1
5:   1 2 1 4 1 2
6:   1 2 1 4 5 2 1
7:   1 1 3 1 5 3 1 2
8:   1 1 1 1 5 1 7 2 1
9:   1 2 1 4 1 2 7 8 1  2
10:  1 2 3 4 1 6 7 8 9  2  1
11:  1 1 1 2 5 1 7 4 3  5  1  2
12:  1 1 1 2 5 1 7 4 3  5 11  2 1
13:  1 2 3 4 5 6 1 8 9 10 11 12 1   2
14:  1 2 1 4 1 2 1 8 3  2 11  4 13  2  1
15:  1 1 1 1 1 1 7 1 3  1 11  1 13  7  1  2
16:  1 1 3 1 5 3 7 1 9  5 11  3 13  7 15  2  1
17:  1 2 1 4 5 2 7 8 1 10 11  4 13 14  5 16  1  2
18:  1 2 1 4 5 2 7 8 1 10 11  4 13 14  5 16 17  2  1
19:  1 1 3 2 1 3 7 4 9  1 11  6 13  7  3  8 17  9  1  2
20:  1 1 1 2 1 1 1 4 3  1 11  2 13  1  1  8 17  3 19  2  1
n/k  0 1 2 3 4 5 6 7 8  9 10 11 12 13 14 15 16 17 18 19 20 ...
.. reformatted - _Wolfdieter Lang_, Jul 28 2014 .
The second column is of period 4: repeat 2, 2, 1, 1. From A014695 or A130658.
The third column is of period 3: repeat 1, 1, 3. From A109007.
The fourth column is of period 8: repeat 2, 2, 4, 4, 1, 1, 4, 4.
The fifth column is of period 5: repeat 1, 1, 5, 5, 5.
The sixth column is of period 12: repeat 2, 2, 3, 1, 2, 6, 1, 1, 6, 2, 1, 3 .
The seventh column is of period 7: repeat 1, 1, 7, 7, 7, 7, 7.
Hence the positive terms of A022998.
Main diagonal: A000034(n).
Alternate main and second diagonal: A130658(n).
Common denominator by row: 1, 2, 2, 2, 6, 4, 20, 30, 70, ... .

Cf. A014695, A130658, A109007, A002260, A022998, A244734, A000034.

Table[(n*(n+1)/2+k+1)/(k+1) // Denominator, {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 08 2014 *)

a(n,k) = denominator((n*(n+1)/2 + k + 1)/(k+1)) for n >= k >= 0.

Editse: Name reformulated, comment with T(n,k) reference added. - Wolfdieter Lang, Jul 28 2014

A251091 a(n) = n^2 / gcd(n+2, 4).

Original entry on oeis.org

0, 1, 1, 9, 8, 25, 9, 49, 32, 81, 25, 121, 72, 169, 49, 225, 128, 289, 81, 361, 200, 441, 121, 529, 288, 625, 169, 729, 392, 841, 225, 961, 512, 1089, 289, 1225, 648, 1369, 361, 1521, 800, 1681, 441, 1849, 968, 2025, 529, 2209, 1152, 2401, 625, 2601, 1352

Offset: 0

Paul Curtz, May 08 2015

A061038(n), which appears in 4*a(n) formula, is a permutation of n^2.
Origin. In December 2010, I wrote in my 192-page Exercise Book no. 5, page 41, the array (difference table of the first row):
   1     0,   1/3,     1,   9/5,    8/3,   25/7,    9/2,   49/9, ...
  -1,  1/3,   2/3,   4/5, 13/15,  19/21,  13/14,  17/18,  43/45, ...
Numerators are listed in A176126, denominators are in A064038, and denominator - numerator = 2, 2, 1, 1,... (A014695).
  4/3, 1/3,  2/15,  1/15, 4/105,   1/42,   1/63,   1/90,  4/495, ...
  -1, -1/5, -1/15, -1/35, -1/70, -1/126, -1/210, -1/330, -1/495, ...
where the denominators of the second row are listed in A000332.
Also for those of the inverse binomial transform
  1, -1, 4/3, -1, 4/5, -2/3, 4/7, -1/2, 4/9, -2/5, 4/11, -1/3, ... ?
a(n) is the (n+1)-th term of the numerators of the first row.

			a(0) = 0/2, a(1) = 1/1, a(2) = 4/4, a(3) = 9/1.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Cf. A000290, A000332, A016754, A026741, A060819, A061038, A109008, A139098, A168077, A176895, A181900.

[(1-(1/16)*(1+(-1)^n)*(5-(-1)^(n div 2)) )*n^2: n in [0..60]]; // Vincenzo Librandi, Jun 12 2015

seq(seq((4*i+j-1)^2/[2,1,4,1][j],j=1..4),i=0..30); # Robert Israel, May 14 2015

f[n_] := Switch[ Mod[n, 4], 0, n^2/2, 1, n^2, 2, n^2/4, 3, n^2]; Array[f, 50, 0] (* or *) Table[(4 i + j - 1)^2/{2, 1, 4, 1}[[j]], {i, 0, 12}, {j, 4}] // Flatten (* after Robert Israel *) (* or *) LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {0, 1, 1, 9, 8, 25, 9, 49, 32, 81, 25, 121}, 53] (* or *) CoefficientList[ Series[-((x (1 + x (1 + x (9 + x (8 + x (22 + x (6 + x (22 + x (8 + x (9 + x + x^2))))))))))/(-1 + x^4)^3), {x, 0, 52}], x] (* Robert G. Wilson v, May 19 2015 *)

concat(0, Vec(-x*(x^10 + x^9 + 9*x^8 + 8*x^7 + 22*x^6 + 6*x^5 + 22*x^4 + 8*x^3 + 9*x^2 + x + 1) / ((x-1)^3*(x+1)^3*(x^2+1)^3) + O(x^100))) \\ Colin Barker, May 14 2015

a(n) = n^2/(period 4: repeat 2, 1, 4, 1).
a(4n) = 8*n^2, a(2n+1) = a(4n+2) = (2*n+1)^2.
a(n+4) = a(n) + 8*A060819(n).
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12), n>11.
4*a(n) = (period 4: repeat 2, 1, 4, 1) * A061038(n).
G.f.: -x*(x^10+x^9+9*x^8+8*x^7+22*x^6+6*x^5+22*x^4+8*x^3+9*x^2+x+1) / ((x-1)^3*(x+1)^3*(x^2+1)^3). - Colin Barker, May 14 2015
a(2n) = A181900(n), a(2n+1) = A016754(n). [Bruno Berselli, May 14 2015]
a(n) = ( 1 - (1/16)*(1+(-1)^n)*(5-(-1)^(n/2)) )*n^2. - Bruno Berselli, May 14 2015
Sum_{n>=1} 1/a(n) = 13*Pi^2/48. - Amiram Eldar, Aug 12 2022

Missing term (1521) inserted in the sequence by Colin Barker, May 14 2015
Definition uses a formula by Jean-François Alcover, Jul 01 2015
Keyword:mult added by Andrew Howroyd, Aug 06 2018

A288994 a(n) = n(n+3) when n is congruent to 0 or 3 (mod 4), and n(n+3)/2 otherwise.

Original entry on oeis.org

0, 2, 5, 18, 28, 20, 27, 70, 88, 54, 65, 154, 180, 104, 119, 270, 304, 170, 189, 418, 460, 252, 275, 598, 648, 350, 377, 810, 868, 464, 495, 1054, 1120, 594, 629, 1330, 1404, 740, 779, 1638, 1720, 902, 945, 1978, 2068, 1080, 1127, 2350, 2448, 1274, 1325

Offset: 0

Jean-François Alcover and Paul Curtz, Jun 21 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Cf. A000096, A014695, A028552, A060819, A130658, A145979, A227316.

a[n_] := n (n+3) Switch[Mod[n, 4], 0|3, 1, _, 1/2]; Table[a[n], {n, 0, 50}]
Table[If[MemberQ[{0,3},Mod[n,4]],n(n+3),(n(n+3))/2],{n,0,50}] (* or *) LinearRecurrence[{3,-6,10,-12,12,-10,6,-3,1},{0,2,5,18,28,20,27,70,88},60] (* Harvey P. Dale, Jun 05 2021 *)

concat(0, Vec(x*(2 - x + 15*x^2 - 16*x^3 + 18*x^4 - 9*x^5 + 5*x^6 - 2*x^7) / ((1 - x)^3*(1 + x^2)^3) + O(x^60))) \\ Colin Barker, Jun 21 2017

i=I; a(n) = (1/8 + i/8)*(((3 - 3*i) - i*(-i)^n + i^n)*n*(3 + n)) \\ Colin Barker, Jun 21 2017

a(n) = n*(n+3)/2 * (2 - floor((n+1)/2) mod 2), where n*(n+3)/2 is A000096(n).
a(n) = A060819(n+3)*A145979(n-2).
a(n) = (2*n*(n+3))/(GCD(4, n+2)*GCD(4, n+3)).
a(n) = A227316(n+1) - (period 4 repeat 2,1,1,2).
From Colin Barker, Jun 21 2017: (Start)
G.f.: x*(2 - x + 15*x^2 - 16*x^3 + 18*x^4 - 9*x^5 + 5*x^6 - 2*x^7) / ((1 - x)^3*(1 + x^2)^3).
a(n) = (1/8 + i/8)*(((3 - 3*i) - i*(-i)^n + i^n)*n*(3 + n)), where i=sqrt(-1). (End)
Sum_{n>=1} 1/a(n) = 17/18 + log(2)/6. - Amiram Eldar, Aug 12 2022

A070937 Number of times maximal coefficient (A025591) appears in Product_{k<=n} (x^k + 1), i.e., number of times highest value appears in n-th row of A053632 or n-th column of A070936.

Original entry on oeis.org

1, 2, 4, 1, 5, 6, 4, 5, 1, 4, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1

Offset: 0

Henry Bottomley, May 12 2002

			a(4)=5 since Product_{k<=4} (x^k + 1) = 1 + x + x^2 + 2x^3 + 2x^4 + 2x^5 + 2x^6 + 2x^7 + x^8 + x^9 + x^10 and 2 appears as a coefficient 5 times.

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

If n mod 4 = 0 or 3 then a(n) odd, otherwise a(n) even.
For n > 9: a(n) = A014695(n).
From Chai Wah Wu, Apr 10 2021: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) for n > 12.
G.f.: (2*x^12 - 2*x^11 + 6*x^10 - 4*x^9 + 6*x^8 - 2*x^7 - 2*x^6 + 2*x^5 - 6*x^4 + 2*x^3 - 3*x^2 - x - 1)/((x - 1)*(x^2 + 1)). (End)

A222740 Denominators of 1/16 - 1/(4 + 8*n)^2.

Original entry on oeis.org

1, 18, 50, 49, 81, 242, 338, 225, 289, 722, 882, 529, 625, 1458, 1682, 961, 1089, 2450, 2738, 1521, 1681, 3698, 4050, 2209, 2401, 5202, 5618, 3025, 3249, 6962, 7442, 3969, 4225, 8978, 9522, 5041, 5329, 11250, 11858, 6241

Offset: 0

Paul Curtz, May 29 2013

Denominators of the reduced fractions A064038(n)/a(n) = 0/1, 1/18, 3/50, 3/49, 5/81, 15/242, 21/338, 14/225, 18/289, ... .

Also, A064038 and a(n) are related to the sequence of period 4: repeat 1, 2, 2, 1.

			a(0) = 1*1, a(1) = 2*9 = 18, a(2) = 2*25 = 50, a(3) = 1*49 = 49.
a(0) = 16*0 + 1 = 1, a(1) = 16*1 + 2 = 18, a(2) = 16*3 + 2 = 50, a(3) = 16*3 + 1 = 49.

Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Table[1/16-1/(4+8n)^2,{n,0,40}]//Denominator (* or *) LinearRecurrence[ {3,-6,10,-12,12,-10,6,-3,1},{1,18,50,49,81,242,338,225,289},40] (* Harvey P. Dale, Aug 30 2021 *)

a(n) = A014695(n) * A016754(n).
a(n) = 16*A064038(n+1) + A014695(n).
a(n) = A061042(4+8*n).
a(2n+2) - a(2n+1) = 32*A026741(n+1).
G.f.: ( -1 - 15*x - 2*x^2 + 3*x^3 - 66*x^4 + 3*x^5 - 2*x^6 - 15*x^7 - x^8 ) / ( (x-1)^3*(x^2+1)^3 ). - R. J. Mathar, Jun 04 2013
a(n) = (3-sqrt(2)*cos((2*n+1)*Pi/4))*(2*n+1)^2/2. - Wesley Ivan Hurt, Oct 04 2018

cp's OEIS Frontend

A177841 Decimal expansion of (5+sqrt(221))/14.

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A134430 Period 4: repeat [1, -2, -2, 1].

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A227168 a(n) = gcd(2*n, n*(n+1)/2)^2.

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A244840 Denominators of the triangle T(n,k) = (n*(n+1)/2+k+1)/(k+1) for n >= k >= 0.

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A251091 a(n) = n^2 / gcd(n+2, 4).

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A288994 a(n) = n*(n+3) when n is congruent to 0 or 3 (mod 4), and n*(n+3)/2 otherwise.

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A070937 Number of times maximal coefficient (A025591) appears in Product_{k<=n} (x^k + 1), i.e., number of times highest value appears in n-th row of A053632 or n-th column of A070936.

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A222740 Denominators of 1/16 - 1/(4 + 8*n)^2.

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A227168 a(n) = gcd(2n, n(n+1)/2)^2.

A288994 a(n) = n(n+3) when n is congruent to 0 or 3 (mod 4), and n(n+3)/2 otherwise.