A010710 -id:A010710 - cp's OEIS frontend

A239614 a(n) = A239611(n) / A079458(n).

1, 2, 2, 4, 2, 4, 2, 6, 3, 4, 2, 8, 2, 4, 4, 8, 2, 6, 2, 8, 4, 4, 2, 12, 3, 4, 4, 8, 2, 8, 2, 10, 4, 4, 4, 12, 2, 4, 4, 12, 2, 8, 2, 8, 6, 4, 2, 16, 3, 6, 4, 8, 2, 8, 4, 12, 4, 4, 2, 16, 2, 4, 6, 12, 4, 8, 2, 8, 4, 8, 2, 18, 2, 4, 6, 8, 4, 8, 2, 16, 5, 4, 2, 16, 4, 4, 4, 12, 2, 12, 4, 8, 4, 4, 4, 20, 2, 6, 6, 12, 2, 8, 2, 12, 8

Offset: 1

José María Grau Ribas, Jun 25 2014

Related to Menon's identity. See Conclusions and further work section of the arXiv file linked.

Multiplicative because both A239611 and A079458 are. - Andrew Howroyd, Aug 07 2018

Antti Karttunen, Table of n, a(n) for n = 1..2101 (computed from the b-files of A079458 and A239611)
C. Calderón, J. M. Grau, A. Oller-Marcen, and Laszlo Toth, Counting invertible sums of squares modulo n and a new generalization of Euler totient function, arXiv:1403.7878 [math.NT], 2014.

Cf. A239611, A239612, A239613, A239615, A079458, A053191, A227499.

a239611[n_] := Sum[If[GCD[x^2 + y^2, n] == 1, GCD[x^2 + y^2 - 1, n], 0], {x, 1, n}, {y, 1, n}];
a079458[n_] := Product[{p, e} = pe; Which[p==2, 2^(2e-1), Mod[p, 4]==3, (p^2-1)p^(2e-2), Mod[p, 4]==1, (p-1)^2 p^(2e-2)], {pe, FactorInteger[n]}];
a[1] = 1; a[n_] := a239611[n]/a079458[n];
Array[a, 105] (* Jean-François Alcover, Dec 04 2018 *)

Conjectures from Ridouane Oudra, Jul 22 2024: (Start)
a(n) = A010710(n)*tau(n) - 2*tau(2n) ;
a(2*n) = 2*tau(n) ;
a(2*n+1) = tau(2*n+1). (End)

More terms from Antti Karttunen, Sep 23 2017

A063289 Dimension of the space of weight n cuspidal newforms for Gamma_1( 16 ).

Original entry on oeis.org

-1, 2, 7, 11, 16, 20, 25, 29, 34, 38, 43, 47, 52, 56, 61, 65, 70, 74, 79, 83, 88, 92, 97, 101, 106, 110, 115, 119, 124, 128, 133, 137, 142, 146, 151, 155, 160, 164, 169, 173, 178, 182, 187, 191, 196, 200, 205, 209, 214, 218, 223, 227, 232, 236

Offset: 2

N. J. A. Sloane, Jul 14 2001

It appears that for n > 2 a(n) = floor((9n-22)/2). - Gary Detlefs, Mar 02 2010

William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_1(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (1, 1, -1).

Cf. A063232, A063233, A017185 (bisection), A130880, A332438.

Join[{-1}, Table[9*n/2 + (-1)^n/4 - 45/4, {n, 3, 60}]] (* Amiram Eldar, Jan 12 2024 *)

a(n) = 9*n/2 + (-1)^n/4 - 45/4 for n >= 3, with first differences in A010710. - R. J. Mathar, Dec 06 2010
From M. F. Hasler, Mar 05 2012: (Start)
G.f.: x^2*(-1 + 3*x + 6*x^2 + x^3)/(1 - x - x^2 + x^3).
a(n+2) = a(n)+9 (n>2), a(2n+1) = a(2n)+4 (n>1), a(2n) = a(2n-1)+5 (n>1). (End)
Sum_{n>=3} (-1)^(n+1)/a(n) = cot(2*Pi/9)*Pi/9. - Amiram Eldar, Jan 12 2024
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=3} (1 - (-1)^n/a(n)) = 2*sin(Pi/18) + 1 (= A130880 + 1).
Product_{n>=3} (1 + (-1)^n/a(n)) = (1/2) * sec(Pi/9) (= A332438 - 3). (End)

A321483 a(n) = 7*2^n + (-1)^n.

Original entry on oeis.org

8, 13, 29, 55, 113, 223, 449, 895, 1793, 3583, 7169, 14335, 28673, 57343, 114689, 229375, 458753, 917503, 1835009, 3670015, 7340033, 14680063, 29360129, 58720255, 117440513, 234881023, 469762049, 939524095, 1879048193, 3758096383, 7516192769, 15032385535

Offset: 0

Paul Curtz, Nov 11 2018

Difference table:
   8, 13, 29, 55, 113, 223, 449, ...
   5, 16, 26, 58, 110, 226, 446, 898, ...
  11, 10, 32, 52, 116, 220, 452, 892, 1796, ...
  -1, 22, 20, 64, 104, 232, 440, 904, 1784, 3592, ...
      -2, 44, 40, 128, 208, 464, 880, 1808, 3568, 7184, ...
etc.
Every diagonal is a sequence of the form k*2^m.
a(n) is divisible by
.  5 if n is a term of A004767,
. 11 if n is a term of A016885,
. 13 if n is a term of A017533.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A000079, A001045, A005029, A010710, A014551, A062092, A062510, A070366, A175805, A199207, A206372.
Cf. A033999, A005009.
Subsequence of A001651, A160543, A160544.

a[n_] := 7*2^n + (-1)^n ; Array[a, 32, 0] (* Amiram Eldar, Nov 12 2018 *)
CoefficientList[Series[E^-x + 7 E^(2 x), {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* Stefano Spezia, Nov 12 2018 *)
LinearRecurrence[{1,2},{8,13},40] (* Harvey P. Dale, Mar 18 2022 *)

Vec((8 + 5*x) / ((1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Nov 11 2018

O.g.f.: (8 + 5*x) / ((1 + x)*(1 - 2*x)). - Colin Barker, Nov 11 2018
E.g.f.: exp(-x) + 7*exp(2*x). - Stefano Spezia, Nov 12 2018
a(n) = a(n-1) + 2*a(n-2).
a(n) = 2*a(n-1) + 3*(-1)^n for n>0, a(0)=8.
a(2*k) = 7*4^k + 1, a(2*k+1) = 14*4^k - 1.
a(n) = A014551(n) + A014551(n-1) + A014551(n-2).
a(n) = 2^(n+3) - 3*A001045(n).
a(n) mod 9 = A070366(n+3).
a(n) + a(n+1) = 21*2^n.

Two terms corrected, and more terms added by Colin Barker, Nov 11 2018

A176215 Decimal expansion of (10+2*sqrt(30))/5.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 12 2010

Continued fraction expansion of (10+2*sqrt(30))/5 is A010710.

			(10+2*sqrt(30))/5 = 4.19089023002066445382...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010485 (decimal expansion of sqrt(30)), A010710 (repeat 4, 5).

RealDigits[(10+2Sqrt[30])/5,10,120][[1]] (* Harvey P. Dale, Nov 28 2011 *)

A176662 a(0)=2, a(1)=7, and a(n) = (3n+1)2^(n-1) if n > 1.

Original entry on oeis.org

2, 7, 14, 40, 104, 256, 608, 1408, 3200, 7168, 15872, 34816, 75776, 163840, 352256, 753664, 1605632, 3407872, 7208960, 15204352, 31981568, 67108864, 140509184, 293601280, 612368384, 1275068416, 2650800128, 5502926848, 11408506880, 23622320128, 48855252992

Offset: 0

Paul Curtz, Apr 23 2010

The sequence appears on the main diagonal of the array defined by A123167 in the first row and successive differences in followup rows:
2, 3, 10, 7, 18, 11, 26, 15, 34, 19, ... A123167
1, 7, -3, 11, -7, 15, -11, 19, -15, 23, ... first diff
6, -10, 14, -18, 22 -26, 30, -34, 38, ... second diff
-16, 24, -32, 40, -48, 56, -64, 72, -80, ... third diff

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4).

LinearRecurrence[{4,-4},{2,7,14,40},40] (* or *) Join[{2,7},Table[ (3n+1) 2^(n-1),{n,2,40}]] (* Harvey P. Dale, Oct 05 2019 *)

a(n) mod 9 = A010710(n-1), n > 2.
a(2n) + a(2n+1) = 9, 54, 360, 2016, ...
a(n) - 2*a(n-1) = 12*A131577(n-2), n > 1.
a(n) = 4*a(n-1) - 4*a(n-2), n > 3.
G.f.: (-6*x^2+12*x^3+2-x)/(1-2*x)^2.

Edited by R. J. Mathar, Jun 30 2010

A184418 Convolution square of A040001.

Original entry on oeis.org

1, 2, 5, 6, 10, 10, 15, 14, 20, 18, 25, 22, 30, 26, 35, 30, 40, 34, 45, 38, 50, 42, 55, 46, 60, 50, 65, 54, 70, 58, 75, 62, 80, 66, 85, 70, 90, 74, 95, 78, 100, 82, 105, 86, 110, 90, 115, 94, 120, 98, 125, 102, 130, 106, 135, 110, 140, 114, 145, 118, 150, 122, 155, 126

Offset: 0

Michael Somos, Feb 14 2011

nonn

			G.f. = 1 + 2*x + 5*x^2 + 6*x^3 + 10*x^4 + 10*x^5 + 15*x^6 + 14*x^7 + 20*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

I:=[2,5,6,10]; [1] cat [n le 4 select I[n] else 2*Self(n-2) - Self(n-4): n in [1..30]]; // G. C. Greubel, Aug 14 2018

LinearRecurrence[{0,2,0,-1},{1,2,5,6,10},80] (* Harvey P. Dale, Jul 03 2017 *)

{a(n) = (n==0) + n * ([5/2, 2] [n%2 + 1])};

{a(n) = if( n==0, 1, sign(n) * polcoeff( (1 + x + x^2)^2 / (1 - x^2)^2 + x * O(x^abs(n)), abs(n)))};

G.f.: (1 + x + x^2)^2 / (1 - x^2)^2 = 1 + x * (x + 2) * (2*x + 1) / (1 - x^2)^2. a(-n) = -a(n) except a(0) = 2.
Euler transform of length 3 sequence [2, 2, -2].
a(n) = 2 * b(n) where b() is multiplicative with b(2^e) = 5 * 2^(e-2) if e>0, b(p^e) = p^e if p>2.
a(2*n + 1) = 4*n + 2, a(2*n) = 5*n except a(0) = 2.
a(n) = (9+(-1)^n)*n/4 = (n/2)*A010710(n+1) for n>0. - Bruno Berselli, Mar 24 2011

A280173 a(0) = 1, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [5, -4].

Original entry on oeis.org

1, 7, 10, 25, 46, 97, 190, 385, 766, 1537, 3070, 6145, 12286, 24577, 49150, 98305, 196606, 393217, 786430, 1572865, 3145726, 6291457, 12582910, 25165825, 50331646, 100663297, 201326590, 402653185, 805306366, 1610612737, 3221225470, 6442450945, 12884901886

Offset: 0

Paul Curtz, Dec 28 2016

a(n) mod 9 = period 2: repeat [1, 7].
The last digit from 7 is of period 4: repeat [7, 0, 5, 6].
The bisection A096045 = 1, 10, 46, ... is based on Bernoulli numbers.
a(n) is a companion to A051049(n).
With an initial 0, A051049(n) is an autosequence of the first kind.
With an initial 2, this sequence is an autosequence of the second kind.
See the reference.
Difference table:
1,   7, 10, 25, 46,  97, ... = this sequence.
6,   3, 15, 21, 51,  93, ... = 3*A014551(n)
-3, 12,  6, 30, 42, 102, ... = -3 followed by 6*A014551(n).
The main diagonal of the difference table gives A003945: 1, 3, 6, 12, 24, ...

			a(0) = 1, a(1) = 2*1 + 5 = 7, a(2) = 2*7 - 4 = 10, a(3) = 2*10 + 5 = 25.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A003945, A005010, A010688, A010710, A014551, A051049, A096045, A199116.

seq(3*2^n-(-1)^n*(1+irem(n+1,2)),n=0..32); # Peter Luschny, Dec 29 2016

LinearRecurrence[{2,1,-2},{1,7,10},50] (* Paolo Xausa, Nov 13 2023 *)

Vec((1 + 5*x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Dec 28 2016

a(2n) = 3*4^n - 2, a(2n+1) = 6*4^n + 1.
a(n+2) = a(n) + 9*2^n, a(0) = 1, a(1) = 7.
a(n) = 2*A051049(n+1) - A051049(n).
From Colin Barker, Dec 28 2016: (Start)
a(n) = 3*2^n - 2 for n even.
a(n) = 3*2^n + 1 for n odd.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2.
G.f.: (1 + 5*x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).
(End)

A176319 Decimal expansion of (5+sqrt(30))/2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 15 2010

Continued fraction expansion of (5+sqrt(30))/2 is A010710 preceded by 5.

			(5+sqrt(30))/2 = 5.23861278752583056728...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A010485 (decimal expansion of sqrt(30)), A010710 (repeat 4, 5).

SetDefaultRealField(RealField(120)); (5 + Sqrt(30))/2; // G. C. Greubel, Nov 26 2019

evalf( (5+sqrt(30))/2, 120); # G. C. Greubel, Nov 26 2019

RealDigits[(5+Sqrt[30])/2,10,120][[1]]  (* Harvey P. Dale, Apr 24 2011 *)

default(realprecision, 120); (5+sqrt(30))/2 \\ G. C. Greubel, Nov 26 2019

numerical_approx((5+sqrt(30))/2, digits=120) # G. C. Greubel, Nov 26 2019

A207260 Triangle read by rows: T(n,k) = k^2 + (1-(-1)^(n-k))/2.

Original entry on oeis.org

0, 1, 1, 0, 2, 4, 1, 1, 5, 9, 0, 2, 4, 10, 16, 1, 1, 5, 9, 17, 25, 0, 2, 4, 10, 16, 26, 36, 1, 1, 5, 9, 17, 25, 37, 49, 0, 2, 4, 10, 16, 26, 36, 50, 64, 1, 1, 5, 9, 17, 25, 37, 49, 65, 81, 0, 2, 4, 10, 16, 26, 36, 50, 64, 82, 100, 1, 1, 5, 9, 17, 25, 37, 49, 65, 81, 101, 121

Offset: 0

Philippe Deléham, Feb 16 2012

Row sums are A171218(n).

			Triangle begins:
  0;
  1, 1;
  0, 2, 4;
  1, 1, 5,  9;
  0, 2, 4, 10, 16;
  1, 1, 5,  9, 17, 25;
  0, 2, 4, 10, 16, 26, 36;
  1, 1, 5,  9, 17, 25, 37, 49;
  0, 2, 4, 10, 16, 26, 36, 50, 64;
  1, 1, 5,  9, 17, 25, 37, 49, 65, 81;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Cf. A000034, A000035, A000290, A002522, A007590, A010710, A010735, A080335, A171218, A137928.

/* As triangle */ [[ k^2 + (1-(-1)^(n-k))/2: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Nov 09 2024

Table[k^2 + (1-(-1)^(n-k))/2, {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Nov 13 2024 *)

T(n+k, n) = A002522(n) if k is odd.
T(n+k, n) = n^2 = A000290(n) if k is even.
T(2*n, n) = A137928(n), n>0.
T(2*n+1, n+1) = A080335(n).
T(n,0) = A000035(n).
T(n+1,1) = A000034(n).
T(n+2,2) = A010710(n).
T(n+3,3) = A010735(n).
Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A007590(n), A000035(n), A171218(n)
for x = -1, 0, 1 respectively.
G.f.: x*(1 + y - x*y + x*(1 + 2*x)*y^2)/((1 - x^2)*(1 - x*y)^3). - Stefano Spezia, Nov 12 2024

A021026 Decimal expansion of 1/22.

Original entry on oeis.org

0, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5

Offset: 0

N. J. A. Sloane

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

Cf. A010710. [From Jaume Oliver Lafont, Mar 20 2009]

Join[{0},RealDigits[1/22,10,120][[1]]] (* or *) CoefficientList[ Series[ x (4+5x)/(1-x^2),{x,0,120}],x] (* Harvey P. Dale, Apr 29 2011 *)

G.f.: x*(4+5*x)/(1-x^2). [From Jaume Oliver Lafont, Mar 20 2009]

cp's OEIS Frontend

A239614 a(n) = A239611(n) / A079458(n).

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A063289 Dimension of the space of weight n cuspidal newforms for Gamma_1( 16 ).

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A321483 a(n) = 7*2^n + (-1)^n.

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A176215 Decimal expansion of (10+2*sqrt(30))/5.

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A176662 a(0)=2, a(1)=7, and a(n) = (3*n+1)*2^(n-1) if n > 1.

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A184418 Convolution square of A040001.

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A280173 a(0) = 1, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [5, -4].

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A176319 Decimal expansion of (5+sqrt(30))/2.

A176662 a(0)=2, a(1)=7, and a(n) = (3n+1)2^(n-1) if n > 1.