A156164 -id:A156164 - cp's OEIS frontend

A156160 a(n) = 34*a(n-1)-a(n-2)-2312 for n > 2; a(1)=169, a(2)=2809.

169, 2809, 93025, 3157729, 107267449, 3643933225, 123786459889, 4205095700689, 142849467361225, 4852676794578649, 164848161548310529, 5599984815847977025, 190234635577282906009, 6462377624811770824969

Offset: 1

Klaus Brockhaus, Feb 09 2009

nonn

lim_{n -> infinity} a(n)/a(n-1) = (17+12*sqrt(2)).

			a(3) = 34*a(2)-a(1)-2312 = 34*2809-169-2312 = 93025.

Index entries for linear recurrences with constant coefficients, signature (35, -35, 1).

First trisection of A156159.

Cf. A156164 (decimal expansion of (17+12*sqrt(2))).

LinearRecurrence[{35,-35,1},{169,2809,93025},20] (* Harvey P. Dale, Nov 15 2014 *)

{m=14; v=concat([169, 2809], vector(m-2)); for(n=3, m, v[n]=34*v[n-1]-v[n-2]-2312); v}

a(n) = (578+ (2211-1550*sqrt(2))*(17+12*sqrt(2))^n+(2211+1550*sqrt(2))*(17-12*sqrt(2))^n)/8.

G.f.: x*(169-3106*x+625*x^2)/((1-x)*(1-34*x+x^2)).

G.f. corrected by Klaus Brockhaus, Sep 23 2009

A156161 a(n) = 34*a(n-1)-a(n-2)-2312 for n > 2; a(1)=289, a(2)=7225.

Original entry on oeis.org

289, 7225, 243049, 8254129, 280395025, 9525174409, 323575532569, 10992042930625, 373405884106369, 12684808016683609, 430910066683134025, 14638257459209870929, 497269843546452475249, 16892536423120174285225

Offset: 1

Klaus Brockhaus, Feb 09 2009

nonn

lim_{n -> infinity} a(n)/a(n-1) = (17+12*sqrt(2)).

			a(3) = 34*a(2)-a(1)-2312 = 34*7225-289-2312 = 243049.

Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Second trisection of A156159.
Equals 289*A008844. - Klaus Brockhaus, Sep 23 2009
Cf. A156164 (decimal expansion of (17+12*sqrt(2))).

RecurrenceTable[{a[1]==289,a[2]==7225,a[n]==34a[n-1]-a[n-2]-2312},a,{n,20}] (* or *) LinearRecurrence[{35,-35,1},{289,7225,243049},20] (* Harvey P. Dale, Dec 11 2013 *)

{m=14; v=concat([289, 7225], vector(m-2)); for(n=3, m, v[n]=34*v[n-1]-v[n-2]-2312); v}

a(n) = (578+(867-578*sqrt(2))*(17+12*sqrt(2))^n+(867+578*sqrt(2))*(17-12*sqrt(2))^n)/8.
G.f.: x*(289-2890*x+289*x^2)/((1-x)*(1-34*x+x^2)). [corrected by Klaus Brockhaus, Sep 23 2009]
a(1)=289, a(2)=7225, a(3)=243049, a(n) = 35*a(n-1)-35*a(n-2)+a(n-3). - Harvey P. Dale, Dec 11 2013

A156162 a(n) = 34*a(n-1)-a(n-2)-2312 for n > 2; a(1)=625, a(2)=18769.

Original entry on oeis.org

625, 18769, 635209, 21576025, 732947329, 24898630849, 845820499225, 28732998340489, 976076123075089, 33157855186210225, 1126391000208070249, 38264136151888175929, 1299854238163989909025, 44156779961423768728609

Offset: 1

Klaus Brockhaus, Feb 09 2009

nonn

lim_{n -> infinity} a(n)/a(n-1) = (17+12*sqrt(2)).

			a(3) = 34*a(2)-a(1)-2312 = 34*18769-625-2312 = 635209.

Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Third trisection of A156159.

Cf. A156164 (decimal expansion of (17+12*sqrt(2))).

RecurrenceTable[{a[1]==625,a[2]==18769,a[n]==34a[n-1]-a[n-2]-2312},a,{n,20}] (* or *) LinearRecurrence[{35,-35,1},{625,18769,635209},20] (* Harvey P. Dale, Sep 29 2016 *)

{m=14; v=concat([625 ,18769], vector(m-2)); for(n=3, m, v[n]=34*v[n-1]-v[n-2]-2312); v}

a(n) = (578+(387-182*sqrt(2))*(17+12*sqrt(2))^n+(387+182*sqrt(2))*(17-12*sqrt(2))^n)/8.

G.f.: x*(625-3106*x+169*x^2)/((1-x)*(1-34*x+x^2)).

G.f. corrected by Klaus Brockhaus, Sep 23 2009

A222393 Nonnegative integers m such that 18m(m+1)+1 is a square.

Original entry on oeis.org

0, 4, 12, 152, 424, 5180, 14420, 175984, 489872, 5978292, 16641244, 203085960, 565312440, 6898944364, 19203981732, 234361022432, 652370066464, 7961375818340, 22161378278060, 270452416801144, 752834491387592, 9187420795420572, 25574211328900084

Offset: 1

Bruno Berselli, Feb 19 2013

a(n+2)/a(n) tends to A156164.

a(n) is congruent to {0,2,4} (mod 5, 6 and 10).

Bruno Berselli, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (1,34,-34,-1,1).

Cf. nonnegative integers n such that k*n*(n+1)+1 is a square: A001652 (k=2), A001921 (k=3), A001477 (k=4), A053606 (k=5), A105038 (k=6), A105040 (k=7), A053141 (k=8), A222390 (k=10), A105838 (k=11), A061278 (k=12), A104240 (k=13); A105063 (k=17), this sequence (k=18), A101180 (k=19), A077259 (k=20) [incomplete list].

m:=22; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(4*(1+x)^2/((1-x)*(1-6*x+x^2)*(1+6*x+x^2))));

I:=[0,4,12,152,424]; [n le 5 select I[n] else Self(n-1)+34*Self(n-2)-34*Self(n-3)-Self(n-4)+Self(n-5): n in [1..25]]; // Vincenzo Librandi, Aug 18 2013

LinearRecurrence[{1, 34, -34, -1, 1}, {0, 4, 12, 152, 424}, 23]
CoefficientList[Series[4 x (1 + x)^2 / ((1 - x) (1 - 6 x + x^2) (1 + 6 x + x^2)), {x, 0, 25}], x] (* Vincenzo Librandi, Aug 18 2013 *)

makelist(expand(-1/2+((3+sqrt(2)*(-1)^n)*(3-2*sqrt(2))^(2*floor(n/2))+(3-sqrt(2)*(-1)^n)*(3+2*sqrt(2))^(2*floor(n/2)))/12), n, 1, 23);

x='x+O('x^30); concat([0], Vec(4*x*(1+x)^2/((1-x)*(1-6*x+x^2)*(1+6*x+x^2)))) \\ G. C. Greubel, Jul 15 2018

G.f.: 4*x*(1+x)^2/((1-x)*(1-6*x+x^2)*(1+6*x+x^2)).
a(n) = a(-n+1) = a(n-1)+34*a(n-2)-34*a(n-3)-a(n-4)+a(n-5).
a(n) = -1/2+((3+t*(-1)^n)*(3-2*t)^(2*floor(n/2))+(3-t*(-1)^n)*(3+2*t)^(2*floor(n/2)))/12, where t=sqrt(2).

A383734 Numbers k such that 2+k and 2*k are squares.

Original entry on oeis.org

2, 98, 3362, 114242, 3880898, 131836322, 4478554082, 152139002498, 5168247530882, 175568277047522, 5964153172084898, 202605639573839042, 6882627592338442562, 233806732499933208098, 7942546277405390632802, 269812766699283348307202, 9165691521498228451812098

Offset: 1

Emilio Martín, May 07 2025

The limit of a(n+1)/a(n) is 33.97056... = 17+12*sqrt(2) = (3+2*sqrt(2))^2 (see A156164).

			98 is a term becouse 98+2=100 is a square and 98*2=196 is a square.

Eric Weisstein's World of Mathematics, NSW numbers.
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Cf. A002315, A088165, A008843, A008844, A075870, A031396, A156164.
Cf. A382209 (10+k and 10*k are squares).
Cf. A245226 (m such that k+m and k*m are squares).

LinearRecurrence[{35, -35, 1}, {2, 98, 3362}, 20] (* Amiram Eldar, May 07 2025 *)

from itertools import islice
def A383734_gen(): # generator of terms
    x, y = 1, 7
    while True:
        yield 2*x**2
        x, y = y, 6*y - x
A383734_list = list(islice(A383734_gen(), 100))

a(n) = (1/2) * ((3+2*sqrt(2))^(2*n-1) + (3-2*sqrt(2))^(1-2*n)) - 1.
a(n) = -2*sqrt(2)*sinh(n*log(17+12*sqrt(2))) + 3*cosh(n*log(17+12*sqrt(2))) - 1.
a(n) = 2*A002315(n-1)^2.
a(n) = A075870(n)^2 - 2.
a(n) = 34*a(n-1) - a(n-2) + 32.
G.f.: 2 * (1 + 14*x + x^2) / ((1 - x)*(1 - 34*x + x^2)). - Stefano Spezia, May 08 2025

A280181 Indices of centered 9-gonal numbers (A060544) that are also squares (A000290).

Original entry on oeis.org

1, 17, 561, 19041, 646817, 21972721, 746425681, 25356500417, 861374588481, 29261379507921, 994025528680817, 33767606595639841, 1147104598723073761, 38967788749988868017, 1323757712900898438801, 44968794449880558051201, 1527615253583038075302017

Offset: 1

Colin Barker, Dec 28 2016

Also positive integers y in the solutions to 2*x^2 - 9*y^2 + 9*y - 2 = 0, the corresponding values of x being A046176.

Consider all ordered triples of consecutive integers (k, k+1, k+2) such that k is a square and k+1 is twice a square; then the values of k are the squares of the NSW numbers (A002315), the values of k+1 are twice the squares of the odd Pell numbers (A001653), and the values of k+2 are thrice the terms of this sequence. (See the Example section.) - Jon E. Schoenfield, Sep 06 2019

			17 is in the sequence because the 17th centered 9-gonal number is 1225, which is also the 35th square.
From _Jon E. Schoenfield_, Sep 06 2019: (Start)
The following table illustrates the relationship between the NSW numbers (A002315), the odd Pell numbers (A001653), and the terms of this sequence:
.
  |  A002315(n-1)^2  |   2*A001653(n)^2  |
n |   = 3*a(n) - 2   |    = 3*a(n) - 1   |       3*a(n)
--+------------------+-------------------+-------------------
1 |    1^2 =       1 |   1^2*2 =       2 |      1*3 =       3
2 |    7^2 =      49 |   5^2*2 =      50 |     17*3 =      51
3 |   41^2 =    1681 |  29^2*2 =    1682 |    561*3 =    1683
4 |  239^2 =   57121 | 169^2*2 =   57122 |  19041*3 =   57123
5 | 1393^2 = 1940449 | 985^2*2 = 1940450 | 646817*3 = 1940451
(End)

Colin Barker, Table of n, a(n) for n = 1..650
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Cf. A000290, A001653, A002315, A046176, A046177, A056771, A060544, A077420.

Cf. A156164.

LinearRecurrence[{35, -35, 1}, {1, 17, 561}, 50] (* G. C. Greubel, Dec 28 2016 *)

Vec(x*(1 - 18*x + x^2) / ((1 - x)*(1 - 34*x + x^2)) + O(x^20))

a(n) = (6 + (3-2*sqrt(2))*(17+12*sqrt(2))^(-n) + (3+2*sqrt(2))*(17+12*sqrt(2))^n) / 12.
a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3) for n>3.
G.f.: x*(1 - 18*x + x^2) / ((1 - x)*(1 - 34*x + x^2)).
a(n) = (A002315(n-1)^2 + 2)/3 = (2*A001653(n)^2 + 1)/3. - Jon E. Schoenfield, Sep 06 2019
a(n) = A077420(floor((n-1)/2)) * A056771(floor(n/2)). - Jon E. Schoenfield, Sep 08 2019
E.g.f.: -1+(1/12)*(6*exp(x)+(3-2*sqrt(2))*exp((17-12*sqrt(2))*x)+(3+2*sqrt(2))*exp((17+12*sqrt(2))*x)). - Stefano Spezia, Sep 08 2019
Limit_{n->oo} a(n+1)/a(n) = 17 + 12*sqrt(2) = A156164. - Andrea Pinos, Oct 07 2022

A381686 Decimal expansion of the isoperimetric quotient of a truncated cube.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 04 2025

For the definition of isoperimetric quotient of a solid, references and links, see A381684.

			0.61302821107928032110240558144714079708976169223933...

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A377298 (surface area), A377299 (volume).

Cf. A000796, A002194, A010524, A156164, A381684.

First[RealDigits[49*Pi*(17 + 12*Sqrt[2])/(2*(6 + 6*Sqrt[2] + Sqrt[3])^3), 10, 100]]

Equals 36*Pi*A377299^2/(A377298^3).

Equals 49*Pi*(17 + 12*sqrt(2))/(2*(6 + 6*sqrt(2) + sqrt(3))^3) = 49*A000796*A156164/(2*(6 + A010524 + A002194)^3).

cp's OEIS Frontend

A156160 a(n) = 34*a(n-1)-a(n-2)-2312 for n > 2; a(1)=169, a(2)=2809.

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A156161 a(n) = 34*a(n-1)-a(n-2)-2312 for n > 2; a(1)=289, a(2)=7225.

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A156162 a(n) = 34*a(n-1)-a(n-2)-2312 for n > 2; a(1)=625, a(2)=18769.

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A222393 Nonnegative integers m such that 18*m*(m+1)+1 is a square.

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A383734 Numbers k such that 2+k and 2*k are squares.

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A280181 Indices of centered 9-gonal numbers (A060544) that are also squares (A000290).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A381686 Decimal expansion of the isoperimetric quotient of a truncated cube.

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Author

A222393 Nonnegative integers m such that 18m(m+1)+1 is a square.