A160682 -id:A160682 - cp's OEIS frontend

A238379 Expansion of (1 - x)/(1 - 36*x + x^2).

1, 35, 1259, 45289, 1629145, 58603931, 2108112371, 75833441425, 2727895778929, 98128414600019, 3529895029821755, 126978092658983161, 4567681440693572041, 164309553772309610315, 5910576254362452399299, 212616435603275976764449

Offset: 0

Bruno Berselli, Feb 25 2014

First bisection of A041611.

Bruno Berselli, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (36,-1).

Cf. similar sequences with g.f. (1-x)/(1-k*x+x^2): A122367 (k=3), A079935 (k=4), A004253 (k=5), A001653 (k=6), A049685 (k=7), A070997 (k=8), A070998 (k=9), A138288 (k=10), A078922 (k=11), A077417 (k=12), A085260 (k=13), A001570 (k=14), A160682 (k=15), A157456 (k=16), A161595 (k=17). From 18 to 38, even k only, except k=27 and k=31: A007805 (k=18), A075839 (k=20), A157014 (k=22), A159664 (k=24), A153111 (k=26), A097835 (k=27), A159668 (k=28), A157877 (k=30), A111216 (k=31), A159674 (k=32), A077420 (k=34), this sequence (k=36), A097315 (k=38).

[n le 2 select 35^(n-1) else 36*Self(n-1)-Self(n-2): n in [1..20]];

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1 - x)/(1 - 36*x + x^2))); // Marius A. Burtea, Jan 14 2020

CoefficientList[Series[(1 - x)/(1 - 36 x + x^2), {x, 0, 20}], x] (* or *) LinearRecurrence[{36, -1}, {1, 35}, 20]

a(n)=([0,1; -1,36]^n*[1;35])[1,1] \\ Charles R Greathouse IV, May 10 2016

m = 20; L. = PowerSeriesRing(ZZ, m); f = (1-x)/(1-36*x+x^2)
print(f.coefficients())

G.f.: (1 - x)/(1 - 36*x + x^2).
a(n) = a(-n-1) = 36*a(n-1) - a(n-2).
a(n) = ((19-sqrt(323))/38)*(1+(18+sqrt(323))^(2*n+1))/(18+sqrt(323))^n.
a(n+1) - a(n) = 34*A144128(n+1).
323*a(n+1)^2 - ((a(n+2)-a(n))/2)^2 = 34.
Sum_{n>0} 1/(a(n) - 1/a(n)) = 1/34.
See also Tanya Khovanova in Links field:
a(n) = 35*a(n-1) + 34*Sum_{i=0..n-2} a(i).
a(n+2)*a(n) - a(n+1)^2 = 36-2 = 34 = 34*1,
a(n+3)*a(n) - a(n+1)*a(n+2) = 36*(36-2) = 1224 = 34*36.
Generalizing:
a(n+4)*a(n) - a(n+1)*a(n+3) = 44030 = 34*1295,
a(n+5)*a(n) - a(n+1)*a(n+4) = 1583856 = 34*46584,
a(n+6)*a(n) - a(n+1)*a(n+5) = 56974786 = 34*1675729, etc.,
where 1, 36, 1295, 46584, 1675729, ... is the sequence A144128, which is the second bisection of A041611.
a(n)^2 - 36*a(n)*a(n+1) + a(n+1)^2 + 34 = 0 (see comments by Colin Barker in similar sequences).

A160695 Integers m such that 3m+1 and 7m+1 are both perfect squares.

Original entry on oeis.org

0, 5, 120, 2760, 63365, 1454640, 33393360, 766592645, 17598237480, 403992869400, 9274237758725, 212903475581280, 4887505700610720, 112199727638465285, 2575706229984090840, 59129043561995624040, 1357392295695915262085, 31160893757444055403920

Offset: 1

Paul Weisenhorn, May 24 2009

The ansatz 3*a(n)+1=A^2, 7*a(n)+1=B^2 is equivalent to the Pell equation x^2-21*y^2=1 (see A077232 for d=21), with x=(21*a(n)+5)/2 and y=A*B/2.
The associated A are in A004253, the B in A030221.
Bisection of A089927. - R. J. Mathar, Jul 10 2009

Michael De Vlieger, Table of n, a(n) for n = 1..736
Francesca Arici and Jens Kaad, Gysin sequences and SU(2)-symmetries of C*-algebras, arXiv:2012.11186 [math.OA], 2020.
Index entries for linear recurrences with constant coefficients, signature (24,-24,1).

Cf. A004253, A004254, A030221, A089927, A160682, A245031, A334673.

j:=0: for n from 0 to 1000000 do a:=sqrt(3*n+1): b:=sqrt(7*n+1):
if (trunc(a)=a) and (trunc(b)=b) then j:=j+1: print(j,n,a,b): end if:
end do:

LinearRecurrence[{24,-24,1},{0,5,120},30] (* Harvey P. Dale, Dec 17 2013 *)

a(n) = 24*a(n-1) - 24*a(n-2) + a(n-3).
a(n) = (A004253(n)^2 - 1)/3 = (A030221(n)^2 - 1)/7.
a(n) = ((5+w)/2*((23+5*w)/2)^(n-1) + (5-w)/2*((23-5*w)/2)^(n-1) - 5)/21; where w=sqrt(21). [Corrected by Kevin Ryde, Sep 11 2020]
G.f.: 5*x^2/((1-x)*(x^2-23*x+1)). - R. J. Mathar, Jul 10 2009
From Francesca Arici, Sep 12 2020: (Start)
a(n) = 23*a(n-1) - a(n-2) + 5.
a(n) = A004254(n)* A004254(n+1). (End)
a(n) = 5*A334673(n-1). - Hugo Pfoertner, Apr 07 2021

Edited and extended by R. J. Mathar, Jul 10 2009

Name edited by Michel Marcus, Sep 12 2020

A161591 The list of the B values in the common solutions to the 2 equations 13k + 1 = A^2, 17k + 1 = B^2.

Original entry on oeis.org

1, 16, 239, 3569, 53296, 795871, 11884769, 177475664, 2650250191, 39576277201, 590993907824, 8825332340159, 131788991194561, 1968009535578256, 29388354042479279, 438857301101610929, 6553471162481684656, 97863210136123658911, 1461394680879373199009

Offset: 1

Paul Weisenhorn, Jun 14 2009

nonn

The 2 equations are equivalent to the Pell equation x^2 - 221*y^2 = 1, with x = (221*k+15)/2 and y = A*B/2, case C=13 in A160682.

Andersen, K., Carbone, L. and Penta, D., Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
Index entries for linear recurrences with constant coefficients, signature (15,-1).

Cf. A160682 (sequence of A), A161584 (sequence of k).

t:=0: for b from 1 to 1000000 do a:=sqrt((13*b^2+4)/17):
if (trunc(a)=a) then t:=t+1: n:=(b^2-1)/17: print(t,a,b,n): end if: end do:

LinearRecurrence[{15,-1},{1,16},30] (* Harvey P. Dale, Dec 04 2015 *)

[(lucas_number2(n,15,1)-lucas_number2(n-1,15,1))/13 for n in range(1, 20)] # Zerinvary Lajos, Nov 10 2009

B(t+2) = 15*B(t+1) - B(t).
B(t) = ((221+17*w)*((15+w)/2)^(t-1) + (221-17*w)*((15-w)/2)^(t-1))/442 where w=sqrt(221).
B(t) = floor of ((221+17*w)*((15+w)/2)^(t-1))/442 = A078364(t-2) + A078364(t-1).
G.f.: x*(1+x)/(1-15*x+x^2).

Edited, extended by R. J. Mathar, Sep 02 2009

A161595 The list of the A values in the common solutions to the 2 equations 15k+1=A^2, 19k+1=B^2.

Original entry on oeis.org

1, 16, 271, 4591, 77776, 1317601, 22321441, 378146896, 6406175791, 108526841551, 1838550130576, 31146825378241, 527657481299521, 8939030356713616, 151435858582831951, 2565470565551429551, 43461563755791470416, 736281113282903567521, 12473317362053569177441

Offset: 1

Paul Weisenhorn, Jun 14 2009

The 2 equations are equivalent to the Pell equation x^2- 285*y^2=1,
with x=(285*k+17)/2 and y=A*B/2, case C=15 in A160682.
Also: the first differences of A078366.
Positive values of x (or y) satisfying x^2 - 17xy + y^2 + 15 = 0. - Colin Barker, Feb 14 2014

J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.
Index entries for linear recurrences with constant coefficients, signature (17,-1).

Cf. A078366, A160682, A161599 (sequence of B), A161583 (sequence of k).

Cf. similar sequences listed in A238379.

t:=0: for a from 1 to 1000000 do b:=sqrt((19*a^2-4)/15):
if (trunc(b)=b) then t:=t+1: n:=(a^2-1)/15: print(t,a,b,n): end if: end do:

Rest[CoefficientList[Series[x (1-x)/(1-17x+x^2),{x,0,40}],x]] (* or *) LinearRecurrence[{17,-1},{1,16},20] (* Harvey P. Dale, Oct 12 2012 *)

Vec(x*(1-x)/(1-17*x+x^2) + O(x^100)) \\ Colin Barker, Feb 14 2014

a(t+2) = 17*a(t+1)-a(t).
a(t) = ((285+15*w)*((17+w)/2)^(t-1)+(285-15*w)*((17-w)/2)^(t-1))/570, where w=sqrt(285).
a(t) = ceiling of ((285+15*w)*((17+w)/2)^(t-1))/570.
G.f.: x*(1-x)/(1-17*x+x^2).
a(n) = 17*a(n-1)-a(n-2). - Colin Barker, Feb 14 2014

Edited, extended by R. J. Mathar, Sep 02 2009

A161583 The list of the k values in the common solutions to the 2 equations 15k+1=A^2, 19k+1=B^2.

Original entry on oeis.org

0, 17, 4896, 1405152, 403273745, 115738159680, 33216448554432, 9533004996962321, 2735939217679631712, 785205022469057339040, 225351105509401776672785, 64674982076175840847750272, 18561494504756956921527655296, 5327084247883170460637589319697

Offset: 1

Paul Weisenhorn, Jun 14 2009

nonn

The 2 equations are equivalent to the Pell equation x^2-285*y^2=1,

with x=(285*k+17)/2 and y=A*B/2, case C=15 in A160682.

Index entries for linear recurrences with constant coefficients, signature (288, -288, 1).

Cf. A160682, A161595 (sequence of A), A161599 (sequence of B)

t:=0: for n from 0 to 1000000 do a:=sqrt(15*n+1): b:=sqrt(19*n+1):
if (trunc(a)=a) and (trunc(b)=b) then t:=t+1: print(t,n,a,b): end if: end do:

k(t+3)=288*(k(t+2)-k(t+1))+k(t).
k(t)=((17+w)*((287+17*w)/2)^(t-1)+(17-w)*((287-17*w)/2)^(t-1))/570 where w=sqrt(285).
k(t) = floor of ((17+w)*((287+17*w)/2)^(t-1))/570;
G.f.: -17*x^2/((x-1)*(x^2-287*x+1)).

Edited, extended by R. J. Mathar, Sep 02 2009

A161599 The list of the B values in the common solutions to the 2 equations 15k + 1 = A^2, 19k + 1 = B^2.

Original entry on oeis.org

1, 18, 305, 5167, 87534, 1482911, 25121953, 425590290, 7209912977, 122142930319, 2069219902446, 35054595411263, 593858902089025, 10060546740102162, 170435435679647729, 2887341859813909231, 48914376181156809198, 828657053219851847135, 14038255528556324592097

Offset: 1

Paul Weisenhorn, Jun 14 2009

nonn

The case C=15 of finding k such that C*k+1 and (C+4)*k+2 are both perfect squares (A160682).

The 2 equations are equivalent to the Pell equation x^2 - 285*y^2 = 1, with x = (285*k+17)/2 and y = A*B/2.

Andersen, K., Carbone, L. and Penta, D., Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
Index entries for linear recurrences with constant coefficients, signature (17, -1).

Cf. A160682, A161595 (sequence of A), A161583 (sequence of k).

t:=0: for b from 1 to 1000000 do a:=sqrt((15*b^2+4)/19):
if (trunc(a)=a) then t:=t+1: n:=(b^2-1)/19: print(t,a,b,n): end if: end do:

LinearRecurrence[{17,-1},{1,18},30] (* Harvey P. Dale, Jan 30 2024 *)

[(lucas_number2(n,17,1)-lucas_number2(n-1,17,1))/15 for n in range(1, 20)] # Zerinvary Lajos, Nov 10 2009

B(t+2) = 17*B(t+1) - B(t).
B(t) = ((285+19*w)*((17+w)/2)^(t-1)+(285-19*w)*((17-w)/2)^(t-1))/570 where w=sqrt(285).
G.f.: (1+x)*x/(1-17*x+x^2).

Edited, extended by R. J. Mathar, Sep 02 2009

A161582 The list of the k values in the common solutions to the 2 equations 5k+1=A^2, 9k+1=B^2.

Original entry on oeis.org

0, 7, 336, 15792, 741895, 34853280, 1637362272, 76921173511, 3613657792752, 169764995085840, 7975341111241735, 374671267233275712, 17601574218852716736, 826899317018844410887, 38846666325666834594960, 1824966417989322381552240, 85734574979172485098360327

Offset: 1

Paul Weisenhorn, Jun 14 2009

The 2 equations are equivalent to the Pell equation x^2-45*y^2=1, with x=(45*k+7)/2 and y= A*B/2, case C=5 in A160682.

Index entries for linear recurrences with constant coefficients, signature (48,-48,1).

Cf. A160682, A049685 (sequence of A), A033890 (sequence of B).

t:=0: for n from 0 to 1000000 do a:=sqrt(5*n+1); b:=sqrt(9*n+1);
if (trunc(a)=a) and (trunc(b)=b) then t:=t+1; print(t,n,a,b): end if: end do:

LinearRecurrence[{48,-48,1},{0,7,336},30] (* or *) Rest[CoefficientList[ Series[ -7x^2/((x-1)(x^2-47x+1)),{x,0,30}],x]] (* Harvey P. Dale, Mar 21 2013 *)

k(t+3) = 48*(k(t+2)-k(t+1))+k(t).
With w = sqrt(5),
k(t) = ((7+3*w)*((47+21*w)/2)^(t-1)+(7-3*w)*((47-21*w)/2)^(t-1))/90.
k(t) = floor((7+3*w)*((47+21*w)/2)^(t-1)/90) = 7*|A156093(t-1)|.
G.f.: -7*x^2/((x-1)*(x^2-47*x+1)).
a(1)=0, a(2)=7, a(3)=336, a(n) = 48*a(n-1)-48*a(n-2)+a(n-3). - Harvey P. Dale, Mar 21 2013

Edited, extended by R. J. Mathar, Sep 02 2009

A161584 The list of the k values in the common solutions to the 2 equations 13k+1=A^2, 17k+1=B^2.

Original entry on oeis.org

0, 15, 3360, 749280, 167086095, 37259449920, 8308690246080, 1852800665425935, 413166239699737440, 92134218652376023200, 20545517593240153436175, 4581558289073901840243840, 1021666952945886870220940160

Offset: 1

Paul Weisenhorn, Jun 14 2009

nonn

The 2 equations are equivalent to the Pell equation x^2-221*y^2=1,

with x=(221*k+15)/2 and y= A*B/2, case C=13 of A160682.

Index entries for linear recurrences with constant coefficients, signature (224, -224, 1).

Cf. A160682 (sequence of A), A161591 (sequence of B)

t:=0: for n from 0 to 1000000 do a:=sqrt(13*n+1): b:=sqrt(17*n+1):
if (trunc(a)=a) and (trunc(b)=b) then t:=t+1: print(t,n,a,b): end if: end do:

LinearRecurrence[{224,-224,1},{0,15,3360},20] (* Harvey P. Dale, Nov 22 2013 *)

k(t+3)=224*(k(t+2)-k(t+1))+k(t).
k(t)=((15+w)*((223+15*w)/2)^(t-1)+(15-w)*((223-15*w)/2)^(t-1))/442 where w=sqrt(221).
k(t) = floor of ((15+w)*((223+15*w)/2)^(t-1))/442;
G.f.: -15*x^2/((x-1)*(x^2-223*x+1)).
a(1)=0, a(2)=15, a(3)=3360, a(n)=224*a(n-1)-224*a(n-2)+a(n-3). - Harvey P. Dale, Nov 22 2013

Edited, extended by R. J. Mathar, Sep 02 2009

A161585 The list of the k values in the common solutions to the 2 equations 7k+1=A^2, 11k+1=B^2.

Original entry on oeis.org

0, 9, 720, 56880, 4492809, 354875040, 28030635360, 2214065318409, 174883129518960, 13813553166679440, 1091095817038156809, 86182755992847708480, 6807346627617930813120, 537694200825823686528009, 42471034518612453304899600, 3354674032769557987400540400

Offset: 1

Paul Weisenhorn, Jun 14 2009

The 2 equations are equivalent to the Pell equation x^2-77*y^2=1,

with x=(77*k+9)/2 and y= A*B/2, case C=7 in A160682.

Index entries for linear recurrences with constant coefficients, signature (80,-80,1)

Cf. A160682, A070998 (sequence of A), A057081 (sequence of B)

t:=0: for n from 0 to 1000000 do a:=sqrt(7*n+1): b:=sqrt(11*n+1):
if (trunc(a)=a) and (trunc(b)=b) then t:=t+1: print(t,n,a,b): end if: end do:

LinearRecurrence[{80,-80,1},{0,9,720},20] (* Harvey P. Dale, Jun 07 2023 *)

k(t+3)=80*(k(t+2)-k(t+1))+k(t).
k(t)=((9+w)*((79+9*w)/2)^(t-1)+(9-w)*((79-9*w)/2)^(t-1))/154 where w=sqrt(77).
k(t) = floor of ((9+w)*((79+9*w)/2)^(t-1))/154.
G.f.: -9*x^2/((x-1)*(x^2-79*x+1)).

Edited, extended by R. J. Mathar, Sep 02 2009

A161586 The list of the k values in the common solutions to the 2 equations 9k+1=A^2, 13k+1=B^2.

Original entry on oeis.org

0, 11, 1320, 157080, 18691211, 2224097040, 264648856560, 31490989833611, 3747163141343160, 445880922830002440, 53056082653628947211, 6313227954859014715680, 751221070545569122218720, 89388994166967866529312011, 10636539084798630547865910600

Offset: 1

Paul Weisenhorn, Jun 14 2009

The 2 equations are equivalent to the Pell equation x^2 - 117*y^2 = 1, with x = (117*k+11)/2 and y = A*B/2, case C = 9 in A160682.

Index entries for linear recurrences with constant coefficients, signature (120,-120,1).

Cf. A004190, A078922 (sequence of A), A097783 (sequence of B), A085550, A160682.

t:=0: for n from 0 to 1000000 do a:=sqrt(9*n+1): b:=sqrt(13*n+1):
if (trunc(a)=a) and (trunc(b)=b) then t:=t+1: print(t,n,a,b): end if: end do:

LinearRecurrence[{120, -120, 1}, {0, 11, 1320}, 20] (* Harvey P. Dale, Apr 01 2024, corrected by Amiram Eldar, Dec 02 2024 *)

a(n) = 120*(a(n-1) - a(n-2)) + a(n-3).
a(n) = ((11+w)*((119+11*w)/2)^(n-1) + (11-w)*((119-11*w)/2)^(n-1) - 22)/234 where w = sqrt(117). [corrected by Amiram Eldar, Dec 02 2024]
a(n) = floor((11+w)*((119+11*w)/2)^(n-1) - 21)/234. [corrected by Amiram Eldar, Dec 02 2024]
G.f.: -11*x^2/((x-1)*(x^2-119*x+1)).
From Amiram Eldar, Dec 02 2024: (Start)
a(n) == 0 (mod 11).
a(n) = A004190(n-2)*A004190(n-1), for n >= 2.
Sum_{n>=2} 1/a(n) = ((sqrt(13)-3)/2)^2 = A085550^2. (End)

Edited and extended by R. J. Mathar, Sep 02 2009

Missing term a(2) = 11 inserted by Amiram Eldar, Dec 02 2024

cp's OEIS Frontend

A238379 Expansion of (1 - x)/(1 - 36*x + x^2).

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A160695 Integers m such that 3*m+1 and 7*m+1 are both perfect squares.

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A161591 The list of the B values in the common solutions to the 2 equations 13*k + 1 = A^2, 17*k + 1 = B^2.

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A161595 The list of the A values in the common solutions to the 2 equations 15*k+1=A^2, 19*k+1=B^2.

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A161583 The list of the k values in the common solutions to the 2 equations 15*k+1=A^2, 19*k+1=B^2.

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A161599 The list of the B values in the common solutions to the 2 equations 15*k + 1 = A^2, 19*k + 1 = B^2.

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A161582 The list of the k values in the common solutions to the 2 equations 5*k+1=A^2, 9*k+1=B^2.

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A160695 Integers m such that 3m+1 and 7m+1 are both perfect squares.

A161591 The list of the B values in the common solutions to the 2 equations 13k + 1 = A^2, 17k + 1 = B^2.

A161595 The list of the A values in the common solutions to the 2 equations 15k+1=A^2, 19k+1=B^2.

A161583 The list of the k values in the common solutions to the 2 equations 15k+1=A^2, 19k+1=B^2.

A161599 The list of the B values in the common solutions to the 2 equations 15k + 1 = A^2, 19k + 1 = B^2.

A161582 The list of the k values in the common solutions to the 2 equations 5k+1=A^2, 9k+1=B^2.

A161584 The list of the k values in the common solutions to the 2 equations 13k+1=A^2, 17k+1=B^2.

A161585 The list of the k values in the common solutions to the 2 equations 7k+1=A^2, 11k+1=B^2.

A161586 The list of the k values in the common solutions to the 2 equations 9k+1=A^2, 13k+1=B^2.