A058331 -id:A058331 - cp's OEIS frontend

A173133 a(n) = Sinh[(2n-1) ArcSinh[n]].

0, 1, 38, 4443, 1166876, 546365045, 400680904674, 423859315570607, 611038907405197432, 1151555487914640463209, 2748476184146759127540190, 8102732939160371170806346243, 28915133156938367486730067779348

Offset: 0

Artur Jasinski, Feb 10 2010

nonn

Cf. A058331, A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130, A173131.

Table[Round[Sinh[(2 n - 1) ArcSinh[n]]], {n, 0, 20}] (* Artur Jasinski *)
Round[Table[1/2 (n - Sqrt[1 + n^2])^(2 n - 1) + 1/2 (n + Sqrt[1 + n^2])^(2 n - 1), {n, 0, 10}]] (* Artur Jasinski, Feb 14 2010 *)

a(n) = 1/2 (n - sqrt(1 + n^2))^(2 n - 1) + 1/2 (n + sqrt(1 + n^2))^(2 n - 1). - Artur Jasinski, Feb 14 2010

Minor edits by Vaclav Kotesovec, Apr 05 2016

A190405 Decimal expansion of Sum_{k>=1} (1/2)^T(k), where T=A000217 (triangular numbers); based on column 1 of the natural number array, A000027.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, May 10 2011

See A190404.
Binary expansion is .1010010001... (A023531). - Rick L. Shepherd, Jan 05 2014
From Amiram Eldar, Dec 07 2020: (Start)
This constant is not a quadratic irrational (Duverney, 1995).
The Engel expansion of this constant are the powers of 2 (A000079) above 1. (End)

			0.64163256065515386629...

Danny Rorabaugh, Table of n, a(n) for n = 0..10000
Daniel Duverney, Sommes de deux carrés et irrationalité de valeurs de fonctions thêta, Comptes rendus de l'Académie des sciences, Série 1, Mathématique, Vol. 320, No. 9 (1995), pp. 1041-1044.

A190404: (1/2)(1 + Sum_{k>=1} (1/2)^T(k)), where T = A000217 (triangular numbers).
A190405: Sum_{k>=1} (1/2)^T(k), where T = A000217 (triangular numbers).
A190406: Sum_{k>=1} (1/2)^S(k-1), where S = A001844 (centered square numbers).
A190407: Sum_{k>=1} (1/2)^V(k), where V = A058331 (1 + 2*k^2).
Cf. A000079.

RealDigits[EllipticTheta[2, 0, 1/Sqrt[2]]/2^(7/8) - 1, 10, 120] // First (* Jean-François Alcover, Feb 12 2013 *)
RealDigits[Total[(1/2)^Accumulate[Range[50]]],10,120][[1]] (* Harvey P. Dale, Oct 18 2013 *)
(* See also A190404 *)

th2(x)=2*x^.25 + 2*suminf(n=1,x^(n+1/2)^2)
th2(sqrt(.5))/2^(7/8)-1 \\ Charles R Greathouse IV, Jun 06 2016

def A190405(b):  # Generate the constant with b bits of precision
    return N(sum([(1/2)^(j*(j+1)/2) for j in range(1,b)]),b)
A190405(409) # Danny Rorabaugh, Mar 25 2015

A173134 a(n) = Sinh[(2n-1)ArcCosh[n]]^2.

Original entry on oeis.org

-1, 0, 675, 11309768, 878801253135, 208241673295152024, 118270071682117442287235, 137788343929239264227213170608, 295355309179742652677310128859789375

Offset: 0

Artur Jasinski, Feb 10 2010

sign

Cf. A058331, A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130, A173131, A173133.

Table[Round[Sinh[(2 n - 1) ArcCosh[n]]^2], {n, 0, 20}]

a(n) ~ 2^(4*n-4) * n^(4*n-2). - Vaclav Kotesovec, Apr 05 2016

A268581 a(n) = 2n^2 + 8n + 5.

Original entry on oeis.org

5, 15, 29, 47, 69, 95, 125, 159, 197, 239, 285, 335, 389, 447, 509, 575, 645, 719, 797, 879, 965, 1055, 1149, 1247, 1349, 1455, 1565, 1679, 1797, 1919, 2045, 2175, 2309, 2447, 2589, 2735, 2885, 3039, 3197, 3359, 3525, 3695, 3869, 4047, 4229, 4415, 4605

Offset: 0

Juri-Stepan Gerasimov, Apr 10 2016

Also, numbers m such that 2*m + 6 is a square.

All the terms end with a digit in {5, 7, 9}, or equivalently, are congruent to {5, 7, 9} mod 10. - Stefano Spezia, Aug 05 2021

Stefano Spezia, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. numbers n such that 2*n + k is a perfect square: A093328 (k=-6), A097080 (k=-5), no sequence (k=-4), A051890 (k=-3), A058331 (k=-2), A001844 (k=-1), A001105 (k=0), A046092 (k=1), A056222 (k=2), A142463 (k=3), A054000 (k=4), A090288 (k=5), this sequence (k=6), A059993 (k=7), A147973 (k=8), A139570 (k=9), no sequence (k=10), A222182 (k=11), A152811 (k=12), A181570 (k=13).

Magma
```
[2*n^2+8*n+5: n in [0..60]];
```
Magma
```
[n: n in [0..6000] | IsSquare(2*n+6)];
```

Table[2 n^2 + 8 n + 5, {n, 0, 50}] (* Vincenzo Librandi, Apr 13 2016 *)
LinearRecurrence[{3,-3,1},{5,15,29},50] (* Harvey P. Dale, Jan 18 2017 *)

lista(nn) = for(n=0, nn, print1(2*n^2+8*n+5, ", ")); \\ Altug Alkan, Apr 10 2016

[2*n^2 + 8*n + 5 for n in [0..46]] # Stefano Spezia, Aug 04 2021

From Vincenzo Librandi, Apr 13 2016: (Start)
G.f.: (5-x^2)/(1-x)^3.
a(n) = 2*(n+2)^2 - 3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
E.g.f.: exp(x)*(5 + 10*x + 2*x^2). - Stefano Spezia, Aug 03 2021

Changed offset from 1 to 0, adapted formulas and programs by Bruno Berselli, Apr 13 2016

A271625 a(n) = = 2*(n+1)^2 - 5.

Original entry on oeis.org

3, 13, 27, 45, 67, 93, 123, 157, 195, 237, 283, 333, 387, 445, 507, 573, 643, 717, 795, 877, 963, 1053, 1147, 1245, 1347, 1453, 1563, 1677, 1795, 1917, 2043, 2173, 2307, 2445, 2587, 2733, 2883, 3037, 3195, 3357, 3523, 3693, 3867, 4045, 4227, 4413, 4603, 4797, 4995, 5197, 5403, 5613, 5827

Offset: 1

Juri-Stepan Gerasimov, Apr 11 2016

Numbers n such that 2*n + 10 is a perfect square.

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

Cf. A000290, A201713.

Numbers h such that 2*h + k is a perfect square: A294774 (k=-9), A255843 (k=-8), A271649 (k=-7), A093328 (k=-6), A097080 (k=-5), A271624 (k=-4), A051890 (k=-3), A058331 (k=-2), A001844 (k=-1), A001105 (k=0), A046092 (k=1), A056222 (k=2), A142463 (k=3), A054000 (k=4), A090288 (k=5), A268581 (k=6), A059993 (k=7), (-1)*A147973 (k=8), A139570 (k=9), this sequence (k=10), A222182 (k=11), A152811 (k=12), A181510 (k=13), A161532 (k=14), no sequence (k=15).

Magma
```
[ 2*n^2 + 4*n - 3: n in [1..60]];
```

[ n: n in [1..6000] | IsSquare(2*n+10)];

Table[2 n^2 + 4 n - 3, {n, 53}] (* Michael De Vlieger, Apr 11 2016 *)
LinearRecurrence[{3,-3,1},{3,13,27},60] (* Harvey P. Dale, Jun 08 2023 *)
2*Range[2,60]^2 -5 (* G. C. Greubel, Jan 21 2025 *)

x='x+O('x^99); Vec(x*(3+4*x-3*x^2)/(1-x)^3) \\ Altug Alkan, Apr 11 2016

def A271625(n): return 2*pow(n+1,2) - 5
print([A271625(n) for n in range(1,61)]) # G. C. Greubel, Jan 21 2025

G.f.: x*(3 + 4*x - 3*x^2)/(1 - x)^3. - Ilya Gutkovskiy, Apr 11 2016
Sum_{n>=1} 1/a(n) = 13/30 - Pi*cot(sqrt(5/2)*Pi)/(2*sqrt(10)) = 0.5627678459924... . - Vaclav Kotesovec, Apr 11 2016
From Elmo R. Oliveira, Nov 17 2024: (Start)
E.g.f.: exp(x)*(2*x^2 + 6*x - 3) + 3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
a(n) = 2*A000290(n+1) - 5. - G. C. Greubel, Jan 21 2025

Name simplified by G. C. Greubel, Jan 21 2025

A166926 A000004 preceded by 1, 2, 4.

Original entry on oeis.org

1, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Klaus Brockhaus, Oct 23 2009

Inverse binomial transform of A058331.

Cf. A000004 (zero sequence), A058331 (2*n^2+1), A130706 (1, 2, 0, 0, 0, 0, ...), A130779 (1, 1, 2, 0, 0, 0, 0, ...).

PARI
```
{concat([1,2,4],vector(102))}
```

a(0) = 1, a(1) = 2, a(2) = 4, a(n) = 0 for n > 2.
G.f.: (1+2*x+4*x^2).
a(n) = 2^n mod 8. - Ridouane Oudra, Apr 08 2025

A212656 a(n) = 5*n^2 + 1.

Original entry on oeis.org

1, 6, 21, 46, 81, 126, 181, 246, 321, 406, 501, 606, 721, 846, 981, 1126, 1281, 1446, 1621, 1806, 2001, 2206, 2421, 2646, 2881, 3126, 3381, 3646, 3921, 4206, 4501, 4806, 5121, 5446, 5781, 6126, 6481, 6846, 7221, 7606, 8001, 8406, 8821, 9246, 9681, 10126, 10581, 11046, 11521, 12006, 12501

Offset: 0

Alonso del Arte, May 23 2012

Z[sqrt(-5)] is not a unique factorization domain, and some of the numbers in this sequence have two different factorizations in that domain, e.g., 21 = 3 * 7 = (1 + 2*sqrt(-5))*(1 - 2*sqrt(-5)). And of course some primes in Z are composite in Z[sqrt(-5)], like 181 = (1 + 6*sqrt(-5))*(1 - 6*sqrt(-5)).

These are pentagonal-star numbers. - Mario Cortés, Oct 26 2020

Benjamin Fine & Gerhard Rosenberger, Number Theory: An Introduction via the Distribution of Primes, Boston: Birkhäuser, 2007, page 268.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033429, A134538, A002522, A053755, A056107, A058331.

Cf. A137530 (primes of the form 1+5*n^2).

[5*n^2 + 1: n in [0..50]]; // Vincenzo Librandi, Jul 10 2012

Table[5n^2 + 1, {n, 0, 49}]
LinearRecurrence[{3,-3,1},{1,6,21},60] (* Harvey P. Dale, Apr 04 2017 *)

a(n)=5*n^2+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 5*n^2 + 1 = (1 + n*sqrt(-5))*(1 - n*sqrt(-5)).
G.f.: (1+3*x+6*x^2)/(1-x)^3. - Bruno Berselli, May 23 2012
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Vincenzo Librandi, Jul 10 2012
From Amiram Eldar, Jul 15 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(5))*coth(Pi/sqrt(5)))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(5))*csch(Pi/sqrt(5)))/2. (End)
a(n) = A005891(n-1) + 5*A000217(n). - Mario Cortés, Oct 26 2020
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(5))*sinh(sqrt(2/5)*Pi).
Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(5))*csch(Pi/sqrt(5)).(End)
E.g.f.: exp(x)*(1 + 5*x + 5*x^2). - Stefano Spezia, Feb 05 2021

A271624 a(n) = 2n^2 - 4n + 4.

Original entry on oeis.org

2, 4, 10, 20, 34, 52, 74, 100, 130, 164, 202, 244, 290, 340, 394, 452, 514, 580, 650, 724, 802, 884, 970, 1060, 1154, 1252, 1354, 1460, 1570, 1684, 1802, 1924, 2050, 2180, 2314, 2452, 2594, 2740, 2890, 3044, 3202, 3364, 3530, 3700, 3874, 4052, 4234, 4420, 4610, 4804, 5002, 5204, 5410, 5620

Offset: 1

Juri-Stepan Gerasimov, Apr 11 2016

Numbers n such that 2*n - 4 is a perfect square.

For n > 2, the number of square a(n)-gonal numbers is finite. - Muniru A Asiru, Oct 16 2016

			a(1) = 2*1^2 - 4*1 + 4 = 2.

Muniru A Asiru, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002522, numbers n such that 2*n + k is a perfect square: no sequence (k = -9), A255843 (k = -8), A271649 (k = -7), A093328 (k = -6), A097080 (k = -5), this sequence (k = -4), A051890 (k = -3), A058331 (k = -2), A001844 (k = -1), A001105 (k = 0), A046092 (k = 1), A056222 (k = 2), A142463 (k = 3), A054000 (k = 4), A090288 (k = 5), A268581 (k = 6), A059993 (k = 7), (-1)*A147973 (k = 8), A139570 (k = 9), A271625 (k = 10), A222182 (k = 11), A152811 (k = 12), A181510 (k = 13), A161532 (k = 14), no sequence (k = 15).

Cf. A005893, A160457.

Magma
```
[ 2*n^2 - 4*n + 4: n in [1..60]];
```

[ n: n in [1..6000] | IsSquare(2*n-4)];

Table[2 n^2 - 4 n + 4, {n, 54}] (* Michael De Vlieger, Apr 11 2016 *)
LinearRecurrence[{3,-3,1},{2,4,10},60] (* Harvey P. Dale, Jul 18 2023 *)

x='x+O('x^99); Vec(2*x*(1-x+2*x^2)/(1-x)^3) \\ Altug Alkan, Apr 11 2016

a(n)=2*n^2-4*n+4 \\ Charles R Greathouse IV, Apr 11 2016

a(n) = 2*A002522(n-1).
G.f.: 2*x*(1 - x + 2*x^2)/(1 - x)^3. - Ilya Gutkovskiy, Apr 11 2016
Sum_{n>=1} 1/a(n) = (1 + Pi*coth(Pi))/4 = 1.038337023734290587067... . - Vaclav Kotesovec, Apr 11 2016
a(n) = A005893(n-1), n > 1. - R. J. Mathar, Apr 12 2016
a(n) = 2 + 2*(n-1)^2. - Tyler Skywalker, Jul 21 2016
From Elmo R. Oliveira, Nov 17 2024: (Start)
E.g.f.: 2*(exp(x)*(x^2 - x + 2) - 2).
a(n) = 2*A160457(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A037235 a(n) = n(2n^2 - 3*n + 4)/3.

Original entry on oeis.org

0, 1, 4, 13, 32, 65, 116, 189, 288, 417, 580, 781, 1024, 1313, 1652, 2045, 2496, 3009, 3588, 4237, 4960, 5761, 6644, 7613, 8672, 9825, 11076, 12429, 13888, 15457, 17140, 18941, 20864, 22913, 25092, 27405, 29856, 32449, 35188, 38077, 41120, 44321, 47684, 51213

Offset: 0

N. J. A. Sloane

Row sums of triangle A134249. Also, binomial transform of (1, 3, 6, 4, 0, 0, 0, ...). - Gary W. Adamson, Oct 15 2007
Binomial transform of a(n) starts: 0, 1, 6, 28, 112, 400, 1312, 4032, ... . - Wesley Ivan Hurt, Oct 21 2014
Number of equivalence classes of n-tuples from the set {1,0,-1} where at the number of nonzero elements is 1,2, or 3 and two n-tuples are equivalent if they are negatives of each other. - Michael Somos, Oct 19 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A058331, A134249.

[n*(2*n^2-3*n+4)/3: n in [0..40]]; // Vincenzo Librandi, Jun 15 2011

A037235:=n->n*(2*n^2-3*n+4)/3: seq(A037235(n), n=0..50); # Wesley Ivan Hurt, Oct 21 2014

Table[n (2 n^2 - 3 n + 4)/3, {n, 0, 50}] (* Wesley Ivan Hurt, Oct 21 2014 *)

A037235(n) = n*(2*n^2-3*n+4)/3 \\ Michael B. Porter, Dec 07 2009

a <- c(0, 1, 4, 13)
for(n in (length(a)+1):30) a[n] <- 4*a[n-1] -6*a[n-2] +4*a[n-3] -a[n-4]
a
# Yosu Yurramendi, Sep 03 2013

G.f.: x*(1+3*x^2)/(1-x)^4.
a(n) = Sum_{k=0..n-1} (2*k^2 + 1). - Mike Warburton, Sep 08 2007
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) with n>3, a(0)=0, a(1)=1, a(2)=4, a(3)=13. - Yosu Yurramendi, Sep 03 2013
a(n+1) = a(n) + A058331(n). - Michael Somos, Oct 19 2022

A037237 Expansion of (3 + x^2) / (1 - x)^4.

Original entry on oeis.org

3, 12, 31, 64, 115, 188, 287, 416, 579, 780, 1023, 1312, 1651, 2044, 2495, 3008, 3587, 4236, 4959, 5760, 6643, 7612, 8671, 9824, 11075, 12428, 13887, 15456, 17139, 18940, 20863, 22912, 25091, 27404

Offset: 0

N. J. A. Sloane

This sequence is the partial sums of A058331. - J. M. Bergot, May 31 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1)

I:=[3, 12, 31, 64]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)- Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 21 2012

CoefficientList[Series[(3+x^2)/(1-x)^4,{x,0,50}],x]  (* Harvey P. Dale, Mar 06 2011 *)
LinearRecurrence[{4,-6,4,-1},{3,12,31,64},40] (* Vincenzo Librandi Jun 21 2012 *)

x='x+O('x^50); Vec((3+x^2)/(1-x)^4) \\ G. C. Greubel, Jul 22 2017

a(n) = Sum_{k=0..n} (2*(k+1)^2 + 1). - Mike Warburton, Jul 07 2007, Sep 07 2007
a(n) = (n+1)*(2*n^2 + 7*n + 9)/3. - R. J. Mathar, Mar 29 2010
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 21 2012
E.g.f.: (1/3)*(9 + 27*x + 15*x^2 + 2*x^3)*exp(x). - G. C. Greubel, Jul 22 2017

cp's OEIS Frontend

A173133 a(n) = Sinh[(2n-1) ArcSinh[n]].

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A190405 Decimal expansion of Sum_{k>=1} (1/2)^T(k), where T=A000217 (triangular numbers); based on column 1 of the natural number array, A000027.

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A173134 a(n) = Sinh[(2n-1)ArcCosh[n]]^2.

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A268581 a(n) = 2*n^2 + 8*n + 5.

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Magma

Magma

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A271625 a(n) = = 2*(n+1)^2 - 5.

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Magma

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A166926 A000004 preceded by 1, 2, 4.

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PARI

Formula

A212656 a(n) = 5*n^2 + 1.

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Magma

Mathematica

A268581 a(n) = 2n^2 + 8n + 5.

A271624 a(n) = 2n^2 - 4n + 4.

A037235 a(n) = n(2n^2 - 3*n + 4)/3.