A097731 -id:A097731 - cp's OEIS frontend

A097733 Pell equation solutions (7b(n))^2 - 2(5a(n))^2 = -1 with b(n)=A097732(n), n >= 0. Note that D=50=25^2 is not squarefree.

1, 197, 39005, 7722793, 1529074009, 302748930989, 59942759261813, 11868363584907985, 2349876047052519217, 465263588952813896981, 92119840736610099083021, 18239263202259846804541177

Offset: 0

Wolfdieter Lang, Aug 31 2004

			(x,y) = (7,1), (1393,197), (275807,39005), ... give the positive integer solutions to x^2 - 50*y^2 =-1.

Indranil Ghosh, Table of n, a(n) for n = 0..434
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (198,-1).

Cf. A097731 for S(n, 198).
Row 7 of array A188647.
Cf. A000129, A100047.

a:=[1,197];; for n in [3..20] do a[n]:=198*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

I:=[1,197]; [n le 2 select I[n] else 198*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019

LinearRecurrence[{198, -1},{1, 197},20] (* Ray Chandler, Aug 11 2015 *)

my(x='x+O('x^20)); Vec((1-x)/(1-198*x+x^2)) \\ G. C. Greubel, Aug 01 2019

((1-x)/(1-198*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

a(n) = S(n, 2*99) - S(n-1, 2*99) = T(2*n+1, 5*sqrt(2))/(5*sqrt(2)), with Chebyshev polynomials of the 2nd and first kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.
a(n) = ((-1)^n)*S(2*n, 14*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
G.f.: (1-x)/(1-198*x+x^2).
a(n) = 198*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=197. - Philippe Deléham, Nov 18 2008
a(n) = k^n + k^(-n) - a(n-1) = A003499(3n) - a(n-1), where k = (sqrt(2)+1)^6 = 99 + 70*sqrt(2) and a(0)=1. - Charles L. Hohn, Apr 05 2011
From Peter Bala, Mar 23 2015: (Start)
a(n) = ( Pell(6*n + 6 - 2*k) - Pell(6*n + 2*k) )/( Pell(6 - 2*k) - Pell(2*k) ), for k an arbitrary integer.
a(n) = ( Pell(6*n + 6 - 2*k - 1) + Pell(6*n + 2*k + 1) )/( Pell(6 - 2*k - 1) + Pell(2*k + 1) ), for k an arbitrary integer.
The aerated sequence (b(n))n>=1 = [1, 0, 197, 0, 39005, 0, 7722793, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -200, Q = 1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for the connection with Chebyshev polynomials. (End)
Sum_{n >= 1} 1/( a(n) - 1/a(n) ) = 1/196. - Peter Bala, Mar 26 2015

A097732 Pell equation solutions (7a(n))^2 - 2(5b(n))^2 = -1 with b(n):=A097733(n), n >= 0. Note that D=50=25^2 is not squarefree.

Original entry on oeis.org

1, 199, 39401, 7801199, 1544598001, 305822602999, 60551330795801, 11988857674965599, 2373733268312392801, 469987198268178808999, 93055091523831091789001, 18424438134520287995413199, 3647945695543493192000024401, 722274823279477131728009418199

Offset: 0

Wolfdieter Lang, Aug 31 2004

Also numbers k such that (7*k+1)^2 + (7*k-1)^2 is a square. - Bruno Berselli, Oct 11 2019

			(x,y) = (7,1), (1393,197), (275807,39005), ... give the positive integer solutions to x^2 - 50*y^2 =-1.

Indranil Ghosh, Table of n, a(n) for n = 0..434
Christian Aebi and Grant Cairns, Lattice equable quadrilaterals III: tangential and extangential cases, Integers (2023) Vol. 23, #A48.
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (198,-1).

Cf. A097731 for S(n, 2*99), A100047.

Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

LinearRecurrence[{198, -1}, {1, 199}, 12] (* Ray Chandler, Aug 11 2015 *)

x='x+O('x^99); Vec((1+x)/(1-2*99*x+x^2)) \\ Altug Alkan, Apr 05 2018

G.f.: (1 + x)/(1 - 2*99*x + x^2).
a(n) = S(n, 2*99) + S(n-1, 2*99) = S(2*n, 10*sqrt(2)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).
a(n) = ((-1)^n)*T(2*n+1, 7*i)/(7*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
a(n) = 198*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=199. - Philippe Deléham, Nov 18 2008
From Peter Bala, Mar 23 2015: (Start)
a(n) = ( Pell(6*n + 6 - 2*k) + Pell(6*n + 2*k) )/( Pell(6 - 2*k) + Pell(2*k) ), for k an arbitrary integer.
a(n) = ( Pell(6*n + 6 - 2*k - 1) - Pell(6*n + 2*k + 1) )/( Pell(6 - 2*k - 1) - Pell(2*k + 1) ), for k an arbitrary integer, k != 1.
The aerated sequence (b(n))n>=1 = [1, 0, 199, 0, 39401, 0, 7801199, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -196, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for the connection with Chebyshev polynomials. (End)
a(n) = (1/7)*sinh((2*n + 1)*arcsinh(7)). - Bruno Berselli, Apr 03 2018

A097735 Pell equation solutions (8a(n))^2 - 65b(n)^2 = -1 with b(n):=A097736(n), n >= 0.

Original entry on oeis.org

1, 259, 66821, 17239559, 4447739401, 1147499525899, 296050429942541, 76379863425649679, 19705708713387674641, 5083996468190594407699, 1311651383084459969511701, 338400972839322481539611159, 87306139341162115777250167321, 22524645549046986548049003557659

Offset: 0

Wolfdieter Lang, Aug 31 2004

			(x,y) = (8,1), (2072,257), (534568,66305), ... give the positive integer solutions to x^2 - 65*y^2 =-1.

Indranil Ghosh, Table of n, a(n) for n = 0..413
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (258,-1).

Cf. A097731 for S(n, 2*129).

Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

LinearRecurrence[{258, -1}, {1, 259}, 20] (* Harvey P. Dale, Oct 30 2011 *)

x='x+O('x^99); Vec((1+x)/(1-2*129*x+x^2)) \\ Altug Alkan, Apr 05 2018

G.f.: (1 + x)/(1 - 2*129*x + x^2).
a(n) = S(n, 2*129) + S(n-1, 2*129) = S(2*n, 2*sqrt(65)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x) = U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).
a(n) = ((-1)^n)*T(2*n+1, 8*i)/(8*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
a(n) = 258*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=259. - Philippe Deléham, Nov 18 2008
a(n) = (1/8)*sinh((2*n + 1)*arcsinh(8)). - Bruno Berselli, Apr 03 2018

A111647 a(n) = A001541(n)A001653(n+1)A002315(n).

Original entry on oeis.org

1, 105, 20213, 3998709, 791704585, 156753394977, 31036379835581, 6145046450172525, 1216688160731724433, 240898110778299543129, 47696609245941810082565, 9443687732585695622131557

Offset: 0

Charlie Marion, Aug 24 2005

			a(1) = 105 = 3*5*7.

G. C. Greubel, Table of n, a(n) for n = 0..434
Index entries for linear recurrences with constant coefficients, signature (204,-1190,204,-1).

Cf. A001541, A001653, A002315, A111648, A111649.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x^3-17*x^2-99*x)/((x^2-6*x+1)*(x^2-198*x+1)))); // G. C. Greubel, Jul 15 2018

CoefficientList[Series[(1+3*x^3-17*x^2-99*x)/((x^2-6*x+1)*(x^2-198*x+1)), {x, 0, 30}], x] (* G. C. Greubel, Jul 15 2018 *)

x='x+O('x^30); Vec((1+3*x^3-17*x^2-99*x)/((x^2-6*x+1)*(x^2-198*x+1))) \\ G. C. Greubel, Jul 15 2018

2*a(n) = A001109(3*n+1) + A001109(n+1).
a(n) = sqrt(A011900(2*n)*A046090(2*n)*A001109(2*n+1)).
a(n) = A001541(3*n) + 2*A001109(n)*A001541(n-1)*A001541(n).
For n>0, a(n) = A001652(3*n) - A053141(2*n)*A002315(n-1) - A001652(n-1).
G.f.: (1+3*x^3-17*x^2-99*x)/((x^2-6*x+1)*(x^2-198*x+1)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
2*a(n) = A001109(n+1) + A097731(n) + 6*A097731(n-1). - R. J. Mathar, Jan 31 2024

A383742 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of g.f. x/(1 - A002203(k)x + (-1)^kx^2).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 6, 5, 4, 0, 1, 14, 35, 12, 5, 0, 1, 34, 197, 204, 29, 6, 0, 1, 82, 1155, 2772, 1189, 70, 7, 0, 1, 198, 6725, 39236, 39005, 6930, 169, 8, 0, 1, 478, 39203, 551532, 1332869, 548842, 40391, 408, 9, 0, 1, 1154, 228485, 7761996, 45232349, 45278310, 7722793, 235416, 985, 10

Offset: 0

Seiichi Manyama, May 07 2025

			Square array begins:
  0,  0,    0,     0,       0,        0, ...
  1,  1,    1,     1,       1,        1, ...
  2,  2,    6,    14,      34,       82, ...
  3,  5,   35,   197,    1155,     6725, ...
  4, 12,  204,  2772,   39236,   551532, ...
  5, 29, 1189, 39005, 1332869, 45232349, ...

Columns k=0..6 give A001477, A000129, A001109, A041085(n-1), A091761, A292423, A097731(n-1).
Rows n=0..5 give A000004, A000012, A002203, A383720, A383740, A383741.
Main diagonal gives A380083.
Cf. A028412.

A[n_, k_] := Fibonacci[k*n, 2]/Fibonacci[k, 2]; A[n_, 0] := n; Table[A[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 08 2025 *)

pell(n) = ([2, 1; 1, 0]^n)[2, 1];
a(n, k) = if(k==0, n, pell(k*n)/pell(k));

A(0,k) = 0, A(1,k) = 1; A(n,k) = A002203(k) * A(n-1,k) - (-1)^k * A(n-2,k) for n > 1.

A(n,k) = Pell(k*n)/Pell(k) for k > 0.

A003773 Number of spanning trees in K_4 X P_n.

Original entry on oeis.org

16, 3456, 686000, 135834624, 26894628304, 5325000912000, 1054323287943536, 208750686023540736, 41331581509440922000, 8183444388183674181504, 1620280657278860350213424, 320807386696826179092096000

Offset: 1

Frans J. Faase

nonn

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Paul Raff, Jun 04 2008, Table of n, a(n) for n = 1..15
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
P. Raff, Spanning Trees in Grid Graphs, arXiv:0809.2551 [math.CO], 2008.
P. Raff, Analysis of the Number of Spanning Trees of K_3 x P_n. Contains sequence, recurrence, generating function, and more.
Index entries for sequences related to trees
Index entries for linear recurrences with constant coefficients, signature (204,-1190,204,-1).

LinearRecurrence[{204, -1190, 204, -1},{16, 3456, 686000, 135834624},12] (* Ray Chandler, Aug 11 2015 *)

a(1) = 16,
a(2) = 3456,
a(3) = 686000,
a(4) = 135834624,
a(5) = 26894628304 and
a(n) = 205a(n-1) - 1394a(n-2) + 1394a(n-3) - 205a(n-4) + a(n-5).
a(n) = 204*a(n-1) - 1190*a(n-2) + 204*a(n-3) - a(n-4). - Paul Raff, Jun 04 2008
G.f.: 16x(1+12x+x^2)/((1-6x+x^2)(x^2-198x+1)). a(n) = 35*A097731(n-1)/2 - 3*A001109(n)/2. - R. J. Mathar, Dec 16 2008
a(n)=16*(A001109(n))^3=16*A001109(n)*A001110(n). [R. Guy, seqfan list, Mar 28 2009] - R. J. Mathar, Jun 03 2009

More terms from Paul Raff, Jun 04 2008

Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009

cp's OEIS Frontend

A097733 Pell equation solutions (7*b(n))^2 - 2*(5*a(n))^2 = -1 with b(n)=A097732(n), n >= 0. Note that D=50=2*5^2 is not squarefree.

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A097732 Pell equation solutions (7*a(n))^2 - 2*(5*b(n))^2 = -1 with b(n):=A097733(n), n >= 0. Note that D=50=2*5^2 is not squarefree.

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A097735 Pell equation solutions (8*a(n))^2 - 65*b(n)^2 = -1 with b(n):=A097736(n), n >= 0.

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A111647 a(n) = A001541(n)*A001653(n+1)*A002315(n).

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A383742 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of g.f. x/(1 - A002203(k)*x + (-1)^k*x^2).

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A003773 Number of spanning trees in K_4 X P_n.

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A097733 Pell equation solutions (7b(n))^2 - 2(5a(n))^2 = -1 with b(n)=A097732(n), n >= 0. Note that D=50=25^2 is not squarefree.

A097732 Pell equation solutions (7a(n))^2 - 2(5b(n))^2 = -1 with b(n):=A097733(n), n >= 0. Note that D=50=25^2 is not squarefree.

A097735 Pell equation solutions (8a(n))^2 - 65b(n)^2 = -1 with b(n):=A097736(n), n >= 0.

A111647 a(n) = A001541(n)A001653(n+1)A002315(n).

A383742 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of g.f. x/(1 - A002203(k)x + (-1)^kx^2).