A090771 -id:A090771 - cp's OEIS frontend

A195142 Concentric 10-gonal numbers.

0, 1, 10, 21, 40, 61, 90, 121, 160, 201, 250, 301, 360, 421, 490, 561, 640, 721, 810, 901, 1000, 1101, 1210, 1321, 1440, 1561, 1690, 1821, 1960, 2101, 2250, 2401, 2560, 2721, 2890, 3061, 3240, 3421, 3610, 3801, 4000, 4201, 4410, 4621, 4840, 5061, 5290

Offset: 0

Omar E. Pol, Sep 17 2011

Also concentric decagonal numbers. Also sequence found by reading the line from 0, in the direction 0, 10, ..., and the same line from 1, in the direction 1, 21, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. Main axis, perpendicular to A028895 in the same spiral.

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

A033583 and A069133 interleaved.
Cf. A028895, A032527, A032528, A077221, A085787, A195042, A195143, A195145, A195146, A195147, A195148, A195149.
Cf. A090771 (first differences).
Column 10 of A195040. - Omar E. Pol, Sep 28 2011

a195142 n = a195142_list !! n
a195142_list = scanl (+) 0 a090771_list
-- Reinhard Zumkeller, Jan 07 2012

[(10*n^2+3*(-1)^n-3)/4: n in [0..50]]; // Vincenzo Librandi, Sep 27 2011

RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-2]+10(n-1)},a[n],{n,50}] (* or *) LinearRecurrence[{2,0,-2,1},{0,1,10,21},50] (* Harvey P. Dale, Sep 29 2011 *)

G.f.: -x*(1+8*x+x^2) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Sep 18 2011
a(n) = -a(n-1) + 5*n^2 - 5*n + 1, a(0)=0. - Vincenzo Librandi, Sep 27 2011
From Bruno Berselli, Sep 27 2011: (Start)
a(n) = a(-n) = (10*n^2 + 3*(-1)^n - 3)/4.
a(n) = a(n-2) + 10*(n-1). (End)
a(n) = 2*a(n-1) + 0*a(n-2) - 2*a(n-3) + a(n-4); a(0)=0, a(1)=1, a(2)=10, a(3)=21. - Harvey P. Dale, Sep 29 2011
Sum_{n>=1} 1/a(n) = Pi^2/60 + tan(sqrt(3/5)*Pi/2)*Pi/(2*sqrt(15)). - Amiram Eldar, Jan 16 2023

A175886 Numbers that are congruent to {1, 12} mod 13.

Original entry on oeis.org

1, 12, 14, 25, 27, 38, 40, 51, 53, 64, 66, 77, 79, 90, 92, 103, 105, 116, 118, 129, 131, 142, 144, 155, 157, 168, 170, 181, 183, 194, 196, 207, 209, 220, 222, 233, 235, 246, 248, 259, 261, 272, 274, 285, 287, 298, 300, 311, 313, 324, 326, 337, 339, 350

Offset: 1

Bruno Berselli, Oct 08 2010 - Nov 17 2010

Cf. property described by Gary Detlefs in A113801: 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 this case, a(n)^2-1 == 0 (mod 13).

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

Cf. A000217, A091998, A113801, A005408, A047209, A007310, A047336, A047522, A056020, A090771, A175885, A175887.

Cf. A195045 (partial sums).

a175886 n = a175886_list !! (n-1)
a175886_list = 1 : 12 : map (+ 13) a175886_list
-- Reinhard Zumkeller, Jan 07 2012

[n: n in [1..350] | n mod 13 in [1, 12]]; // Bruno Berselli, Feb 29 2012

[(26*n+9*(-1)^n-13)/4: n in [1..55]]; // Vincenzo Librandi, Aug 19 2013

Select[Range[1, 350], MemberQ[{1, 12}, Mod[#, 13]]&] (* Bruno Berselli, Feb 29 2012 *)
CoefficientList[Series[(1 + 11 x + x^2) / ((1 + x) (1 - x)^2), {x, 0, 55}], x] (* Vincenzo Librandi, Aug 19 2013 *)
LinearRecurrence[{1,1,-1},{1,12,14},60] (* Harvey P. Dale, Oct 23 2015 *)

a(n)=(26*n+9*(-1)^n-13)/4 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: x*(1+11*x+x^2)/((1+x)*(1-x)^2).
a(n) = (26*n+9*(-1)^n-13)/4.
a(n) = -a(-n+1) = a(n-1)+a(n-2)-a(n-3).
a(n) = a(n-2)+13.
a(n) = 13*A000217(n-1)+1 - 2*Sum_{i=1..n-1} a(i) for n>1.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/13)*cot(Pi/13). - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((26*x - 13)*exp(x) + 9*exp(-x))/4. - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/13).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/13)*cosec(Pi/13). (End)

A175887 Numbers that are congruent to {1, 14} mod 15.

Original entry on oeis.org

1, 14, 16, 29, 31, 44, 46, 59, 61, 74, 76, 89, 91, 104, 106, 119, 121, 134, 136, 149, 151, 164, 166, 179, 181, 194, 196, 209, 211, 224, 226, 239, 241, 254, 256, 269, 271, 284, 286, 299, 301, 314, 316, 329, 331, 344, 346, 359, 361, 374, 376, 389, 391, 404

Offset: 1

Bruno Berselli, Oct 08 2010 - Nov 17 2010

Cf. property described by Gary Detlefs in A113801: 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 this case, a(n)^2-1==0 (mod 15).

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

Cf. A000217, A019693, A019976, A113801, A175886, A091998, A175885, A090771, A056020, A047522, A047336, A007310, A047209, A005408, A001651.

Cf. A132240 (primes).

a175887 n = a175887_list !! (n-1)
a175887_list = 1 : 14 : map (+ 15) a175887_list
-- Reinhard Zumkeller, Jan 07 2012

[n: n in [1..450] | n mod 15 in [1,14]];

[(30*n+11*(-1)^n-15)/4: n in [1..55]]; // Vincenzo Librandi, Aug 19 2013

Select[Range[1, 450], MemberQ[{1,14}, Mod[#, 15]]&]
CoefficientList[Series[(1 + 13 x + x^2) / ((1 + x) (1 - x)^2), {x, 0, 55}], x] (* Vincenzo Librandi, Aug 19 2013 *)

a(n)=(30*n+11*(-1)^n-15)/4 \\ Charles R Greathouse IV, Sep 28 2015

G.f.: x*(1+13*x+x^2)/((1+x)*(1-x)^2).
a(n) = (30*n+11*(-1)^n-15)/4.
a(n) = -a(-n+1) = a(n-1)+a(n-2)-a(n-3).
a(n) = 15*A000217(n-1) -2*sum(a(i), i=1..n-1) +1 for n>1.
a(n) = A047209(A047225(n+1)).
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/15)*cot(Pi/15) = A019693 * A019976 / 10. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((30*x - 15)*exp(x) + 11*exp(-x))/4. - David Lovler, Sep 05 2022
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = (Pi/15)*cosec(Pi/15).
Product_{n>=2} (1 + (-1)^n/a(n)) = 2*cos(Pi/15). (End)

A261327 a(n) = (n^2 + 4) / 4^((n + 1) mod 2).

Original entry on oeis.org

1, 5, 2, 13, 5, 29, 10, 53, 17, 85, 26, 125, 37, 173, 50, 229, 65, 293, 82, 365, 101, 445, 122, 533, 145, 629, 170, 733, 197, 845, 226, 965, 257, 1093, 290, 1229, 325, 1373, 362, 1525, 401, 1685, 442, 1853, 485, 2029, 530, 2213, 577, 2405, 626, 2605, 677

Offset: 0

Paul Curtz, Aug 15 2015

Using (n+sqrt(4+n^2))/2, after the integer 1 for n=0, the reduced metallic means are b(1) = (1+sqrt(5))/2, b(2) = 1+sqrt(2), b(3) = (3+sqrt(13))/2, b(4) = 2+sqrt(5), b(5) = (5+sqrt(29))/2, b(6) = 3+sqrt(10), b(7) = (7+sqrt(53))/2, b(8) = 4+sqrt(17), b(9) = (9+sqrt(85))/2, b(10) = 5+sqrt(26), b(11) = (11+sqrt(125))/2 = (11+5*sqrt(5))/2, ... . The last value yields the radicals in a(n) or A013946.
b(2) = 2.41, b(3) = 3.30, b(4) = 4.24, b(5) = 5.19 are "good" approximations of fractal dimensions corresponding to dimensions 3, 4, 5, 6: 2.48, 3.38, 4.33 and 5.45 based on models. See "Arbres DLA dans les espaces de dimension supérieure: la théorie des peaux entropiques" in Queiros-Condé et al. link. DLA: beginning of the title of the Witten et al. link.
Consider the symmetric array of the half extended Rydberg-Ritz spectrum of the hydrogen atom:
   0,    1/0,     1/0,     1/0,     1/0,      1/0,      1/0,     1/0, ...
-1/0,      0,     3/4,     8/9,   15/16,    24/25,    35/36,   48/49, ...
-1/0,   -3/4,       0,    5/36,    3/16,   21/100,      2/9,  45/196, ...
-1/0,   -8/9,   -5/36,       0,   7/144,   16/225,     1/12,  40/441, ...
-1/0, -15/16,   -3/16,  -7/144,       0,    9/400,    5/144,  33/784, ...
-1/0, -24/25, -21/100, -16/225,  -9/400,        0,   11/900, 24/1225, ...
-1/0, -35/36,    -2/9,   -1/12,  -5/144,  -11/900,        0, 13/1764, ...
-1/0, -48/49, -45/196, -40/441, -33/784, -24/1225, -13/1764,       0, ... .
The numerators are almost A165795(n).
Successive rows: A000007(n)/A057427(n), A005563(n-1)/A000290(n), A061037(n)/A061038(n), A061039(n)/A061040(n), A061041(n)/A061042(n), A061043(n)/A061044(n), A061045(n)/A061046(n), A061047(n)/A061048(n), A061049(n)/A061050(n).
A144433(n) or A195161(n+1) are the numerators of the second upper diagonal (denominators: A171522(n)).
c(n+1) = a(n) + a(n+1) = 6, 7, 15, 18, 34, 39, 63, 70, 102, 111, ... .
c(n+3) - c(n+1) = 9, 11, 19, 21, 29, 31, ... = A090771(n+2).
The final digit of a(n) is neither 4 nor 8. - Paul Curtz, Jan 30 2019

Colin Barker, Table of n, a(n) for n = 0..1000
Diogo Queiros-Condé, Jean Chaline, and Jacques Dubois, Le monde des fractales La Nature trans-échelles, 478p., ellipses, Paris, 2015, page 220.
T. A. Witten, Jr. and L. M. Sander, Diffusion-Limited Aggregation, a Kinetic Critical Phenomenom, Phys. Rev. Lett., Vol. 47 (Nov 09 1981), pp. 1400-1403.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000007, A000290, A001622, A002522, A005563, A010685, A010698, A013946, A014176, A057427, A061035-A061050, A078370, A087475, A090771, A098316, A098317, A098318, A144433, A168077, A171621, A176398, A176439, A176458, A176522, A176537, A195161.

Cf. A069894.

[Numerator(1+n^2/4): n in [0..60]]; // Vincenzo Librandi, Aug 15 2015

A261327:=n->numer((4 + n^2)/4); seq(A261327(n), n=0..60); # Wesley Ivan Hurt, Aug 15 2015

LinearRecurrence[{0, 3, 0, -3, 0, 1}, {1, 5, 2, 13, 5, 29}, 60] (* Vincenzo Librandi, Aug 15 2015 *)
a[n_] := (n^2 + 4) / 4^Mod[n + 1, 2]; Table[a[n], {n, 0, 52}] (* Peter Luschny, Mar 18 2022 *)

vector(60, n, n--; numerator(1+n^2/4)) \\ Michel Marcus, Aug 15 2015

Vec((1+5*x-x^2-2*x^3+2*x^4+5*x^5)/(1-x^2)^3 + O(x^60)) \\ Colin Barker, Aug 15 2015

a(n)=if(n%2,n^2+4,(n/2)^2+1) \\ Charles R Greathouse IV, Oct 16 2015

[(n*n+4)//4**((n+1)%2) for n in range(60)] # Gennady Eremin, Mar 18 2022

[numerator(1+n^2/4) for n in (0..60)] # G. C. Greubel, Feb 09 2019

a(n) = numerator(1 + n^2/4). (Previous name.) See A010685 (denominators).
a(2*k) = 1 + k^2.
a(2*k+1) = 5 + 4*k*(k+1).
a(2*k+1) = 4*a(2*k) + 4*k + 1.
a(4*k+2) = A069894(k). - Paul Curtz, Jan 30 2019
a(-n) = a(n).
a(n+2) = a(n) + A144433(n) (or A195161(n+1)).
a(n) = A168077(n) + period 2: repeat 1, 4.
a(n) = A171621(n) + period 2: repeat 2, 8.
From Colin Barker, Aug 15 2015: (Start)
a(n) = (5 - 3*(-1)^n)*(4 + n^2)/8.
a(n) = n^2/4 + 1 for n even;
a(n) = n^2 + 4 for n odd.
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>5.
G.f.: (1 + 5*x - x^2 - 2*x^3 + 2*x^4 + 5*x^5)/ (1 - x^2)^3. (End)
E.g.f.: (5/8)*(x^2 + x + 4)*exp(x) - (3/8)*(x^2 - x + 4)*exp(-x). - Robert Israel, Aug 18 2015
Sum_{n>=0} 1/a(n) = (4*coth(Pi)+tanh(Pi))*Pi/8 + 1/2. - Amiram Eldar, Mar 22 2022

New name by Peter Luschny, Mar 18 2022

A090298 Permutation of natural numbers generated by 5-row array shown below.

Original entry on oeis.org

1, 9, 2, 11, 8, 3, 19, 12, 7, 4, 21, 18, 13, 6, 5, 29, 22, 17, 14, 10, 31, 28, 23, 16, 15, 39, 32, 27, 24, 20, 41, 38, 33, 26, 25, 49, 42, 37, 34, 30, 51, 48, 43, 36, 35, 59, 52, 47, 44, 40, 61, 58, 53, 46, 45, 69, 62, 57, 54, 50, 71, 68, 63, 56, 55, 79, 72, 67, 64, 60, 81, 78

Offset: 1

Giovanni Teofilatto, Jan 25 2004

9 11 19 21 29 31 39... (A090771)
8 12 18 22 28 32 38... (A090772)
7 13 17 23 27 33 37... (A063226)
6 14 16 24 26 34 36... (A090773)
10 15 20 25 30 35 40... (A008587, excluding initial term)
-----------------------------------------------------------
For such arrays A_k, here A_5, see a W. Lang comment on A113807, the A_7 case. However, in order to obtain A_5 one should take the last row as the first one after adding a 0 in front (thus getting a permutation of the nonnegative integers). - Wolfdieter Lang, Feb 02 2012

Cf. A088520, A090243, A092260, A113807.

More terms from Ray Chandler, Feb 01 2004

A188593 Decimal expansion of (diagonal)/(shortest side) of a golden rectangle.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 04 2011

A rectangle of length L and width W is a golden rectangle if L/W = r = (1+sqrt(5))/2. The diagonal has length D = sqrt(L^2+W^2), so D/W = sqrt(r^2+1) = sqrt(r+2).
Largest root of x^4 - 5x^2 + 5. - Charles R Greathouse IV, May 07 2011
This is the case n=10 of (Gamma(1/n)/Gamma(2/n))*(Gamma((n-1)/n)/Gamma((n-2)/n)) = 2*cos(Pi/n). - Bruno Berselli, Dec 13 2012
Edge length of a pentagram (regular star pentagon) with unit circumradius. - Stanislav Sykora, May 07 2014
The ratio diagonal/side of the shortest diagonal in a regular 10-gon. - Mohammed Yaseen, Nov 04 2020

			1.902113032590307144232878666758764286811397268251...

Chai Wah Wu, Table of n, a(n) for n = 1..10001
Michael Penn, On the fifth root of the identity matrix, YouTube video, 2022.
Eric Weisstein's World of Mathematics, Golden Rectangle.
Eric Weisstein's World of Mathematics, Pentagram.
Index entries for algebraic numbers, degree 4.

Cf. A001622 (decimal expansion of the golden ratio), A019881.
Cf. A188594 (D/W for the silver rectangle, r=1+sqrt(2)).
Cf. A090771, A195693, A195723, A296184.

SetDefaultRealField(RealField(100)); Sqrt((5+Sqrt(5))/2); // G. C. Greubel, Nov 02 2018

r = (1 + 5^(1/2))/2; RealDigits[(2 + r)^(1/2), 10, 130][[1]]
RealDigits[Sqrt[GoldenRatio+2],10,130][[1]] (* Harvey P. Dale, Oct 27 2023 *)

PARI
```
sqrt((5+sqrt(5))/2)
```

Equals 2*A019881. - Mohammed Yaseen, Nov 04 2020
Equals csc(A195693) = sec(A195723). - Amiram Eldar, May 28 2021
Equals i^(1/5) + i^(-1/5). - Gary W. Adamson, Jul 08 2022
Equals sqrt(2 + phi) = sqrt(A296184), with phi = A001622. - Wolfdieter Lang, Aug 28 2022
Equals Product_{k>=0} ((10*k + 2)*(10*k + 8))/((10*k + 1)*(10*k + 9)). - Antonio Graciá Llorente, Feb 24 2024
Equals Product_{k>=1} (1 - (-1)^k/A090771(k)). - Amiram Eldar, Nov 23 2024

A019916 Decimal expansion of tan(Pi/10) (angle of 18 degrees).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

In a regular pentagon inscribed in a unit circle this is the cube of the length of the side divided by 5: (1/5)*(sqrt(3 - phi))^3 with phi from A001622. - Wolfdieter Lang, Jan 08 2018
Quartic number of denominator 5 and minimal polynomial 5x^4 - 10x^2 + 1. - Charles R Greathouse IV, May 13 2019
The other positive root of the minimal polynomial is A019952. - R. J. Mathar, Sep 06 2025

			0.3249196962329063261558714122151344649549034715214751003078047191...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 4.

Cf. A001622, A019827 (sin(Pi/10)), A019881 (cos(Pi/10)).

RealDigits[Tan[18 Degree],10,120][[1]] (* Harvey P. Dale, Mar 07 2012 *)

tan(Pi/10) \\ Michel Marcus, Jan 08 2018

polrootsreal(5*x^4-10*x^2+1)[3] \\ Charles R Greathouse IV, Feb 04 2025

Equals A019827/A019881 = 1/A019970 = 1/sqrt(5+2*sqrt(5)). - R. J. Mathar, Jul 26 2010
Equals tan((phi - 1)/sqrt(2 + phi)) = (1/5)*(sqrt(3 - phi))^3 = (3 - phi)*sqrt(3 - phi)/5 = sqrt(7 - 4*phi)/(2*phi - 1), with phi from A001622. - Wolfdieter Lang, Jan 08 2018
Equals Product_{k>=0} ((5*k + 1)/(5*k + 4))^(-1)^(k) = Product_{k>=0} A090771(k)/A090773(k). - Antonio Graciá Llorente, Mar 24 2024
Equals A019845/(1+A019863). - R. J. Mathar, Sep 06 2025

A113428 Expansion of f(-x^2, -x^3) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 0, -1, -1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Oct 31 2005

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

See the Hardy-Wright reference for the identity given as g.f. formula below. - Wolfdieter Lang, Oct 28 2016

			G.f. = 1 - x^2 - x^3 + x^9 + x^11 - x^21 - x^24 + x^38 + x^42 - x^60 - x^65 + ...
G.f. = q - q^81 - q^121 + q^361 + q^441 - q^841 - q^961 + q^1521 + q^1681 + ...

G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 93.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, Theorem 356, p. 284.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities

Cf. A003114, A057569, A090771, A010815, A113429.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5] QPochhammer[ x^5], {x, 0, n}]; (* Michael Somos, Jan 06 2016 *)

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 - x^k * [1, 0, 1, 1, 0][k%5 + 1], 1 + x * O(x^n)), n))};

Expansion of G(x) * f(-x) in powers of x where G() is the g.f. of A003114.
Euler transform of period 5 sequence [ 0, -1, -1, 0, -1, ...].
|a(n)| is the characteristic function of the numbers in A057569.
The exponents in the q-series q * A(q^40) are the square of the numbers in A090771.
G.f.: Sum_{k in Z} (-1)^k * x^((5*k^2 + k)/2) = Prod_{k>0} (1 - x^(5*k)) * (1 - x^(5*k - 2)) * (1 - x^(5*k - 3)).
Convolution of A003114 and A010815.
From Wolfdieter Lang, Oct 30 2016: (Start)
a(n) = (-1)^k if n = b(2*k+1) for k >= 0, a(n) = (-1)^k if n = b(2*k), for k >= 1, and a(n) = 0 otherwise, where b(n) = A057569(n). See the third formula.
G.f.: Sum_{n>=0} (-1)^n*x^(n*(5*n+1)/2)*(1-x^(2*(2*n+1))). See the Hardy reference, p. 93, eq. (6.11.1) with k=2, a=x and C_n = 1.
(End)
G.f.: Sum_{n >= 0} x^(n^2)*Product_{k >= n+1} 1 - x^k. Cf. A113429. - Peter Bala, Feb 12 2021

A094888 Decimal expansion of 2Piphi, where phi = (1+sqrt(5))/2.

Original entry on oeis.org

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

Offset: 2

N. J. A. Sloane, Jun 15 2004

			10.16640738463051963161901802648439768366367858644230824...

Ivan Panchenko, Table of n, a(n) for n = 2..1000
John Baez, Pi and the Golden Ratio, Azimuth, Mar 07 2017.
Index entries for transcendental numbers.

Cf. A000796, A001622, A090771, A094886, A135155.

Integral_{x>=0} 1/(1+x^m) dx: A019669 (m=2), A248897 (m=3), A093954 (m=4), A352324 (m=5), A019670 (m=6), A352125 (m=8), this sequence (m=10).

evalf(Pi*(1+sqrt(5)), 121);  # Alois P. Heinz, May 16 2022

RealDigits[2 * Pi * GoldenRatio, 10, 100][[1]] (* Amiram Eldar, May 18 2021 *)

From Peter Bala, Nov 03 2019: (Start)
Equals 10*Integral_{x >= 0} cosh(4*x)/cosh(5*x) dx = Integral_{x = 0..1} (1 + x^8)/(1 + x^10) dx .
Equals 100*Sum_{n >= 0} (-1)^n*(2*n + 1)/( (10*n + 1)*(10*n + 9) ). (End)
Equals 10 * Product_{k>=2} 2/sqrt(2 + sqrt(2 + ... sqrt(2 + phi)...)), with k nested radicals (Baez, 2017). - Amiram Eldar, May 18 2021
Equals Integral_{x>=0} 1/(1 + x^10) dx = (Pi/10) * csc(Pi/10). - Bernard Schott, May 15 2022
Equals Gamma(1/10)*Gamma(9/10). - Andrea Pinos, Jul 03 2023
Equals 10 * Product_{k >= 1} (10*k)^2/((10*k)^2 - 1). - Antonio Graciá Llorente, Mar 15 2024
Equals 10 * Product_{k>=2} (1 + (-1)^k/A090771(k)). - Amiram Eldar, Nov 23 2024
Equals 2*A094886 = 10*A135155/e. - Hugo Pfoertner, Nov 23 2024

A219190 Numbers of the form k(5k+1), where k = 0,-1,1,-2,2,-3,3,...

Original entry on oeis.org

0, 4, 6, 18, 22, 42, 48, 76, 84, 120, 130, 174, 186, 238, 252, 312, 328, 396, 414, 490, 510, 594, 616, 708, 732, 832, 858, 966, 994, 1110, 1140, 1264, 1296, 1428, 1462, 1602, 1638, 1786, 1824, 1980, 2020, 2184, 2226, 2398, 2442, 2622, 2668, 2856, 2904, 3100

Offset: 1

Bruno Berselli, Nov 14 2012

Equivalently, numbers m such that 20*m+1 is a square.
Also, integer values of h*(h+1)/5.
More generally, for the numbers of the form n*(k*n+1) with n in A001057, we have:
. generating function (offset 1): x^2*(k-1+2*x+(k-1)*x^2)/((1+x)^2*(1-x)^3);
. n-th term: b(n) = (2*k*n*(n-1)+(k-2)*(-1)^n*(2*n-1)+k-2)/8;
. first differences: (n-1)*((-1)^n*(k-2)+k)/2;
. b(2n+1)-b(2n) = 2*n (independent from k);
. (4*k)*b(n)+1 = (2*k*n+(k-2)*(-1)^n-k)^2/4.

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

Subsequence of A011858.
Cf. A090771: square roots of 20*a(n)+1 (see the first comment).
Cf. numbers of the form n*(k*n+1) with n in A001057: k=0, A001057; k=1, A110660; k=2, A000217; k=3, A152749; k=4, A074378; k=5, this sequence; k=6, A036498; k=7, A219191; k=8, A154260.
Cf. similar sequences listed in A219257.

k:=5; f:=func; [0] cat [f(n*m): m in [-1,1], n in [1..25]];

I:=[0,4,6,18,22]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

Rest[Flatten[{# (5 # - 1), # (5 # + 1)} & /@ Range[0, 25]]]
CoefficientList[Series[2 x (2 + x + 2 x^2) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,4,6,18,22},50] (* Harvey P. Dale, Jan 21 2015 *)

G.f.: 2*x^2*(2 + x + 2*x^2)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n+1) = (10*n*(n-1) + 3*(-1)^n*(2*n - 1) + 3)/8.
a(n) = 2*A057569(n) = A008851(n+1)*A047208(n)/5.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Harvey P. Dale, Jan 21 2015
Sum_{n>=2} 1/a(n) = 5 - sqrt(1+2/sqrt(5))*Pi. - Amiram Eldar, Mar 15 2022
a(n) = A132356(n-1)/2, n >= 1. - Bernard Schott, Mar 15 2022

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