A047209 -id:A047209 - cp's OEIS frontend

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

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)

A094550 Numbers n such that there are integers a < b with a+(a+1)+...+(n-1) = (n+1)+(n+2)+...+b.

Original entry on oeis.org

4, 6, 9, 11, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 34, 35, 36, 38, 39, 40, 41, 43, 44, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 59, 61, 64, 66, 68, 69, 70, 71, 72, 74, 76, 77, 79, 81, 82, 83, 84, 86, 87, 89, 91, 93, 94, 95, 96, 97, 98, 99, 100, 101, 104

Offset: 1

T. D. Noe, May 10 2004

Liljestrom shows that n is in this sequence if and only if 4n^2+1 is composite.
Complement of A001912.
From Hermann Stamm-Wilbrandt, Sep 16 2014: (Start)
For n > 1, A047209 is a subset of this sequence [ 4*n^2+1 is divisible by 5 if n is (1 or 4) mod 5].
A092464 is a subset of this sequence [4*n^2+1 is divisible by 13 if n is (4 or 9) mod 13].
The above are for divisibility by 5, 13; notation (1,4,5), (4,9,13). Divisibility by p for a and p-a; notation (a,p-a,p). These are the next tuples: (2,15,17), (6,23,29), (3,34,37), (16,25,41), ... . The corresponding sequences are a subset of this sequence [ 2,15,19,32,36,49,... for (2,15,17) ]. These sequences have no entries in the OEIS yet. For any prime of the form 4*k+1 there is exactly one of these tuples/sequences [solution to 4*a^2+1=0 (mod p)].
For n>1, A000290 (squares) is a subset of this sequence (4,9,16,25,...) [ 4*(m^2)^2+1 is divisible by  m^2+(m+1)^2, tuple (m^2, (m+1)^2, m^2+(m+1)^2) ].
(End)

			6 is in this sequence because 1+2+3+4+5 = 7+8.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
R. J. Liljestrom and Richard Zucker, Numerical Fulcrums (PowerPoint Format)

Cf. A094551, A094552, A094553.

Cf. A001912.

[n: n in [1..100] |not IsPrime(4*n^2 + 1)]; // Vincenzo Librandi, Sep 27 2012

lst={}; Do[i1=n-1; i2=n+1; s1=i1; s2=i2; While[i1>1 && s1!=s2, If[s1T. D. Noe, Nov 12 2010 *)

A352324 Decimal expansion of 4Pi / (5sqrt(10-2*sqrt(5))).

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Mar 12 2022

Cauchy's residue theorem implies that Integral_{x=0..oo} 1/(1 + x^m) dx = (Pi/m) * csc(Pi/m); this is the case m = 5.

The area of a circle circumscribing a unit-area regular decagon.

			1.0689593321155951134251843725068826399014509252665...

Jean-François Pabion, Éléments d'Analyse Complexe, licence de Mathématiques, page 111, Ellipses, 1995.

Index entries for transcendental numbers.

Cf. A019694, A047209, A121570, A371604, A377405.

Integral_{x=0..oo} 1/(1+x^m) dx: A019669 (m=2), A248897 (m=3), A093954 (m=4), this sequence (m=5), A019670 (m=6), A352125 (m=8), A094888 (m=10).

evalf(4*Pi / (5*(sqrt(10-2sqrt(5)))), 100);

First[RealDigits[N[4Pi/(5Sqrt[10-2Sqrt[5]]), 93]]] (* Stefano Spezia, Mar 12 2022 *)

Equals Integral_{x=0..oo} 1/(1 + x^5) dx.
Equals (Pi/5) *csc(Pi/5).
Equals (1/2) * A019694 * A121570.
Equals 1/Product_{k>=1} (1 - 1/(5*k)^2). - Amiram Eldar, Mar 12 2022
Equals Product_{k>=2} (1 + (-1)^k/A047209(k)). - Amiram Eldar, Nov 22 2024
Equals 1/A371604 = A377405/5. - Hugo Pfoertner, Nov 22 2024

A358244 Number of n-regular, N_0-weighted pseudographs on 2 vertices with total edge weight 4, up to isomorphism.

Original entry on oeis.org

1, 6, 13, 27, 38, 55, 67, 85, 97, 115, 127, 145, 157, 175, 187, 205, 217, 235, 247, 265, 277, 295, 307, 325, 337, 355, 367, 385, 397, 415, 427, 445, 457, 475, 487, 505, 517, 535, 547, 565, 577, 595, 607, 625, 637, 655, 667, 685, 697, 715, 727, 745, 757, 775

Offset: 1

Lars Göttgens, Nov 04 2022

nonn

Pseudographs are finite graphs with undirected edges without identity, where parallel edges between the same vertices and loops are allowed.

			For n = 2 the a(2) = 6 such pseudographs are:
  1. two vertices connected by a 4-edge and a 0-edge,
  2. two vertices connected by a 3-edge and a 1-edge,
  3. two vertices connected by two 2-edges,
  4. two vertices where one has a 4-loop and the other one has a 0-loop,
  5. two vertices where one has a 3-loop and the other one has a 1-loop,
  6. two vertices with a 2-loop each.

Lars Göttgens, Table of n, a(n) for n = 1..10000
J. Flake and V. Mackscheidt, Interpolating PBW Deformations for the Orthosymplectic Groups, arXiv:2206.08226 [math.RT], 2022.
Eric Weisstein's World of Mathematics, Pseudograph.

Other total edge weights: A358243 (3), A358245 (5), A358246 (6), A358247 (7), A358248 (8), A358249 (9).

Cf. A047209.

using Combinatorics
function A(n::Int)
    sum_total = 4
    result = 0
    for num_loops in 0:div(n, 2)
        num_cross = n - 2 * num_loops
        for sum_cross in 0:sum_total
            for sum_loop1 in 0:sum_total-sum_cross
                sum_loop2 = sum_total - sum_cross - sum_loop1
                if sum_loop2 == sum_loop1
                    result +=
                        div(
                            npartitions_with_zero(sum_loop2, num_loops) *
                            (npartitions_with_zero(sum_loop2, num_loops) + 1),
                            2,
                        ) * npartitions_with_zero(sum_cross, num_cross)
                elseif sum_loop2 > sum_loop1
                    result +=
                        npartitions_with_zero(sum_loop2, num_loops) *
                        npartitions_with_zero(sum_loop1, num_loops) *
                        npartitions_with_zero(sum_cross, num_cross)
                end
            end
        end
    end
    return result
end
function npartitions_with_zero(n::Int, m::Int)
    if m == 0
        if n == 0
            return 1
        else
            return 0
        end
    else
        return Combinatorics.npartitions(n + m, m)
    end
end
print([A(n) for n in 1:54])

Apparently a(n) = 6*A047209(n-2) + 1 for n >= 6, i.e., terms satisfy the linear recurrence a(n) = a(n-1) + a(n-2) - a(n-3) for n >= 9. - Hugo Pfoertner, Dec 02 2022

A090772 Numbers that are congruent to {2, 8} mod 10.

Original entry on oeis.org

2, 8, 12, 18, 22, 28, 32, 38, 42, 48, 52, 58, 62, 68, 72, 78, 82, 88, 92, 98, 102, 108, 112, 118, 122, 128, 132, 138, 142, 148, 152, 158, 162, 168, 172, 178, 182, 188, 192, 198, 202, 208, 212, 218, 222, 228, 232, 238, 242, 248, 252, 258, 262, 268, 272, 278, 282

Offset: 1

Giovanni Teofilatto, Feb 07 2004

Their square ends in the digit 4. - Kausthub Gudipati, Sep 08 2011

10*a(n) = 20, 80, 120, 180, 220, ... are the only numbers written in French ending in "vingt(s)". - Paul Curtz, Aug 02 2018

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

Cf. A019952, A047209, A090298, A179290, A226294, A317633.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(2*x*(1+3*x+x^2)/((1+x)*(1-x)^2))); // G. C. Greubel, Aug 08 2018

Union@ Flatten@ Outer[Plus, {2, 8}, 10 Range[0, 28]] (* or *)
CoefficientList[Series[2 (1 + 3x + x^2)/((1 + x) (1 - x)^2), {x, 0, 57}], x] (* Michael De Vlieger, Aug 02 2018 *)
LinearRecurrence[{1, 1, -1}, {2, 8, 12}, 61] (* Robert G. Wilson v, Aug 08 2018 *)

is(n) = #setintersect([2, 8], [n%10]) > 0 \\ Felix Fröhlich, Aug 02 2018

Vec(2*x*(1+3*x+x^2)/((1+x)*(1-x)^2) + O(x^60)) \\ Felix Fröhlich, Aug 02 2018

a(n) = 2 * A047209(n).
a(n) = 10*n - a(n-1) - 10 (with a(1)=2). - Vincenzo Librandi, Nov 16 2010
G.f.: 2*x*(1+3*x+x^2)/((1+x)*(1-x)^2). - Bruno Berselli, Sep 08 2011
a(1) = 2. For n > 1, a(n) = a(n-1) + A226294(n). - Felix Fröhlich, Aug 02 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(1+2/sqrt(5))*Pi/10. - Amiram Eldar, Dec 28 2021
E.g.f.: 2 + ((10*x - 5)*exp(x) + exp(-x))/2. - David Lovler, Sep 03 2022
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = tan(3*Pi/10) (A019952).
Product_{n>=1} (1 + (-1)^n/a(n)) = cosec(2*Pi/5)/2 (= A179290 / 2). (End)

Edited and extended by Ray Chandler, Feb 10 2004

A273375 Squares ending in digit 4.

Original entry on oeis.org

4, 64, 144, 324, 484, 784, 1024, 1444, 1764, 2304, 2704, 3364, 3844, 4624, 5184, 6084, 6724, 7744, 8464, 9604, 10404, 11664, 12544, 13924, 14884, 16384, 17424, 19044, 20164, 21904, 23104, 24964, 26244, 28224, 29584, 31684, 33124, 35344, 36864, 39204, 40804, 43264

Offset: 1

Vincenzo Librandi, May 24 2016

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

Cf. A000290, A047209.
Cf. A017317 (numbers ending in 4), A017319 (cubes ending in 4).
Cf. similar sequences listed in A273373.

/* By definition: */ [n^2: n in [0..200] | Modexp(n, 2, 10) eq 4];

Magma
```
[(10*n+(-1)^n-5)^2/4: n in [1..50]];
```

Table[(10 n + (-1)^n - 5)^2/4, {n, 1, 50}] (* or *) LinearRecurrence[{1, 2, -2, -1, 1}, {4, 64, 144, 324, 484}, 50]
Select[Range[200]^2,Mod[#,10]==4&] (* or *) LinearRecurrence[{1,1,-1},{2,8,12},40]^2(* Harvey P. Dale, Aug 06 2017 *)

G.f.: 4*x*(1 + 15*x + 18*x^2 + 15*x^3 + x^4) /((1+x)^2*(1-x)^3).
a(n) = 4*A047209(n)^2 = (10*n + (-1)^n - 5)^2/4.
Sum_{n>=1} 1/a(n) = 2*Pi^2/(25*(5-sqrt(5))). - Amiram Eldar, Feb 16 2023
E.g.f.: (4 - 5*x + 25*x^2)*cosh(x) + (9 + 5*x + 25*x^2)*sinh(x) - 4. - Stefano Spezia, Feb 21 2025

Edited by Bruno Berselli, May 24 2016

A008253 Coordination sequence for diamond.

Original entry on oeis.org

1, 4, 12, 24, 42, 64, 92, 124, 162, 204, 252, 304, 362, 424, 492, 564, 642, 724, 812, 904, 1002, 1104, 1212, 1324, 1442, 1564, 1692, 1824, 1962, 2104, 2252, 2404, 2562, 2724, 2892, 3064, 3242, 3424, 3612, 3804, 4002, 4204, 4412, 4624, 4842, 5064, 5292, 5524

Offset: 0

N. J. A. Sloane, Ralf W. Grosse-Kunstleve

Inorganic Crystal Structure Database: Collection Code 9327.

N. J. A. Sloane, Table of n, a(n) for n = 0..1000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
R. W. Grosse-Kunstleve, G. O. Brunner and N. J. A. Sloane, Algebraic Description of Coordination Sequences and Exact Topological Densities for Zeolites, Acta Cryst., A52 (1996), pp. 879-889.
Sean A. Irvine, Generating Functions for Coordination Sequences of Zeolites, after Grosse-Kunstleve, Brunner, and Sloane
M. O'Keeffe, M. A. Peskov, S. J. Ramsden, and O. M. Yaghi, The reticular chemistry structure resource (RCSR) database of, and symbols for, crystal nets, Accounts of Chemical Research, (2008), 1782-1789. See p. 1786.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A007904, A047209.

{1}~Join~Table[2 (2 + Sum[Floor[(5 k + 3)/2], {k, n - 1}]), {n, 50}] (* Alexander Adamchuk, May 23 2006, edited by Michael De Vlieger, May 31 2022 *)

Vec((1 + 2*x + 4*x^2 + 2*x^3 + x^4) / ((1 - x)^3*(1 + x)) + O(x^60)) \\ Colin Barker, Mar 21 2017

G.f.: (1 + 2*x + 4*x^2 + 2*x^3 + x^4) / ((1 - x)^3*(1 + x)).
a(2*m) = 10*m^2+2, a(2*m+1) = 10*m^2+10*m+4 (N. J. A. Sloane).
Apart from first term, first differences of A007904(n). - Alexander Adamchuk, May 23 2006
a(n) = 2* ( 2 + Sum_{k=1..n-1} floor((5*k+3)/2) ). - Alexander Adamchuk, May 23 2006
From Colin Barker, Mar 21 2017: (Start)
a(n) = (5*n^2 + 4)/2 for n>0 and even.
a(n) = (5*n^2 + 3)/2 for n odd.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>4.
(End)

A032769 Numbers that are congruent to {0, 1, 2, 4} mod 5.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 71, 72, 74, 75, 76, 77, 79, 80, 81, 82, 84, 85

Offset: 1

Patrick De Geest, May 15 1998

Also, numbers m such that m*(m+1)*(m+2)*(m+3)*(m+4)/(m+(m+1)+(m+2)+(m+3)+(m+4)) is an integer.

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

Cf. A001622, A032768, A032770, A047209, A047215.

[Floor((5*n - 4)/4) : n in [1..80]]; // Wesley Ivan Hurt, May 30 2016

seq(floor((5*n-4)/4), n=1..69); # Gary Detlefs, Mar 06 2010

Table[Floor[(5n - 4)/4], {n, 80}] (* Wesley Ivan Hurt, May 30 2016 *)

a(n)=5*n\4-1 \\ Charles R Greathouse IV, Jan 02 2025

a(n) = (1/8)*(10*n-11+(-1)^n+2*(-1)^floor(n/2)). - Ralf Stephan, Jun 09 2005
a(n) = floor((5*n-4)/4). - Gary Detlefs, Mar 06 2010
G.f.: x^2*(1+x+2*x^2+x^3) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 30 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (10*n-11+i^(2*n)+(1+i)*I^(-n)+(1-i)*i^n)/8 where i=sqrt(-1).
a(2k) = A047209(k), a(2k-1) = A047215(k). (End)
E.g.f.: (4 + sin(x) + cos(x) + (5*x - 6)*sinh(x) + 5*(x - 1)*cosh(x))/4. - Ilya Gutkovskiy, May 31 2016
Sum_{n>=2} (-1)^n/a(n) = log(5)/4 + 3*sqrt(5)*log(phi)/10 - sqrt(1-2/sqrt(5))*Pi/10, where phi is the golden ratio (A001622). - Amiram Eldar, Dec 10 2021

Better description from Michael Somos, Jun 08 2000

A219259 Numbers k such that 25*k+1 is a square.

Original entry on oeis.org

0, 23, 27, 96, 104, 219, 231, 392, 408, 615, 635, 888, 912, 1211, 1239, 1584, 1616, 2007, 2043, 2480, 2520, 3003, 3047, 3576, 3624, 4199, 4251, 4872, 4928, 5595, 5655, 6368, 6432, 7191, 7259, 8064, 8136, 8987, 9063, 9960, 10040, 10983, 11067, 12056, 12144

Offset: 1

Bruno Berselli, Nov 19 2012

Equivalently, numbers of the form m*(25*m+2), where m = 0,-1,1,-2,2,-3,3,...
Also, integer values of h*(h+2)/25.
Exponents in the expansion of Product_{n >= 1} (1 - q^(50*n))*(1 - q^(50*n-23))*(1 - q^(50*n-27)) = 1 - q^23 - q^27 + q^96 + q^104 - q^219 - q^231 + + - - .... - Peter Bala, Dec 18 2024

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).

Cf. similar sequences listed in A219257.

Cf. A047209, A197652.

[n: n in [0..13000] | IsSquare(25*n+1)];

I:=[0,23,27,96,104]; [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

A219259:=proc(q)
local n;
for n from 1 to q do if type(sqrt(25*n+1), integer) then print(n);
fi; od; end:
A219259(1000); # Paolo P. Lava, Feb 19 2013

Select[Range[0, 13000], IntegerQ[Sqrt[25 # + 1]] &]
CoefficientList[Series[x (23 + 4 x + 23 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)

G.f.: x^2*(23 + 4*x + 23*x^2)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n+1) = (50*n*(n-1) + 21*(-1)^n*(2*n - 1) + 5)/8 + 2.
25*a(n)+1 = A047209(A197652(n+1))^2.
Sum_{n>=2} 1/a(n) = 25/4 - cot(2*Pi/25)*Pi/2. - Amiram Eldar, Mar 17 2022

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

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Magma

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

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Magma

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A094550 Numbers n such that there are integers a < b with a+(a+1)+...+(n-1) = (n+1)+(n+2)+...+b.

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A352324 Decimal expansion of 4*Pi / (5*sqrt(10-2*sqrt(5))).

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Maple

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A358244 Number of n-regular, N_0-weighted pseudographs on 2 vertices with total edge weight 4, up to isomorphism.

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Julia

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A090772 Numbers that are congruent to {2, 8} mod 10.

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A273375 Squares ending in digit 4.

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A352324 Decimal expansion of 4Pi / (5sqrt(10-2*sqrt(5))).