A156164 -id:A156164 - cp's OEIS frontend

A008844 Squares of sequence A001653: y^2 such that x^2 - 2*y^2 = -1 for some x.

1, 25, 841, 28561, 970225, 32959081, 1119638521, 38034750625, 1292061882721, 43892069261881, 1491038293021225, 50651409893459761, 1720656898084610641, 58451683124983302025, 1985636569351347658201, 67453191674820837076801, 2291422880374557112953025

Offset: 0

N. J. A. Sloane

Numbers simultaneously square and centered square. E.g., a(1)=25 because 25 is the fourth centered square number and the fifth square number. - Steven Schlicker, Apr 24 2007
Solutions to A007913(x)=A007913(2x-1). - Benoit Cloitre, Apr 07 2002
From Ant King, Nov 09 2011: (Start)
Indices of positive hexagonal numbers that are also perfect squares.
As n increases, this sequence is approximately geometric with common ratio r = lim_{n -> infinity} a(n)/a(n-1) = (1 + sqrt(2))^4 = 17 + 12 * sqrt(2).
(End)
Also indices of hexagonal numbers (A000384) which are also centered octagonal numbers (A016754). - Colin Barker, Jan 25 2015
Also positive integers x in the solutions to 4*x^2 - 8*y^2 - 2*x + 8*y - 2 = 0, the corresponding values of y being A253826. - Colin Barker, Jan 25 2015
Squares that are sum of two consecutive squares: y^2 = (k + 1)^2 + k^2 is equivalent to x^2 - 2*y^2 = -1 with x = 2*k + 1. - Jean-Christophe Hervé, Nov 11 2015
Squares in the main diagonal of the natural number array, A000027. - Clark Kimberling, Mar 12 2023

			From _Ravi Kumar Davala_, May 26 2013: (Start)
A001333(0)=1, A001333(4)=17, A001333(8)=577, A000129(0)=0, A000129(2)=2, A000129(4)=12, A000129(8)=408 so clearly
a(n+m)=A001333(4*m)*a(n)-(A000129(2*m))^2+A000129(4*m)*sqrt(2*a(n)^2-a(n)), with m=1,2 is true.
A002203(0)=2, A002203(4)=34, A002203(8)=1154 so clearly
a(n+m)=(1/2)*A002203(4*m)*a(n)-(A000129(2*m))^2+A000129(4*m)*sqrt(2*a(n)^2-a(n)) is true for m=1,2
a(n+1)*a(n-1) = (a(n)+4)^2 , with n=1 is 841*1=(25+4)^2, for n=2 , 28561*25=(841+4)^2.
(End)
1 = 1 + 0, 25 = 16 + 9, 841 = 29^2 = 21^2 + 20^2 = 441 + 400.

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 256.

Muniru A Asiru, Table of n, a(n) for n = 0..100
Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.
M. A. Gruber, Artemas Martin, A. H. Bell, J. H. Drummond, A. H. Holmes, and H. C. Wilkes, Problem 47, Amer. Math. Monthly, 4 (1897), 25-28.
Thomas Koshy, Products Involving Reciprocals of Gibonacci Polynomials, The Fibonacci Quarterly, Vol. 60, No. 1 (2022), pp. 15-24.
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Circular Segment, Forum Geometricorum, Vol. 18 (2018), 47-55.
Steven C. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, Mathematics Magazine, Vol. 84, No. 5, December 2011
Eric Weisstein's World of Mathematics, Hexagonal Square Number.
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Cf. A000290, A001844, A007913, A000384, A046177, A016754, A253826.

a := [1, 25, 841];; for i in [4..10^2] do a[i] := 35*a[i-1] - 35*a[i-2] + a[i-3]; od; a;  # Muniru A Asiru, Jan 17 2018

I:=[1,25,841]; [n le 3 select I[n] else 35*Self(n-1)-35*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jan 20 2018

CP := n -> 1+1/2*4*(n^2-n): N:=10: u:=3: v:=1: x:=4: y:=1: k_pcp:=[1]: for i from 1 to N do tempx:=x; tempy:=y; x:=tempx*u+8*tempy*v: y:=tempx*v+tempy*u: s:=(y+1)/2: k_pcp:=[op(k_pcp),CP(s)]: end do: k_pcp; # Steven Schlicker, Apr 24 2007

LinearRecurrence[{35, -35, 1}, {1, 25, 841}, 15] (* Ant King, Nov 09 2011 *)
CoefficientList[Series[(1 - 10 x + x^2) / ((1 - x) (1 - 34 x + x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Jan 20 2018 *)

a(n)=if(n<0,0,sqr(subst(poltchebi(n+1)+poltchebi(n),x,3)/4))

vector(40, n, n--; (([5, 2; 2, 1]^n)[1, 1])^2) \\ Altug Alkan, Nov 11 2015

From Benoit Cloitre, Jan 19 2003: (Start)
a(n) = A078522(n) + 1.
a(n) = ceiling(A*B^n) where A = (3 + 2*sqrt(2))/8 and B = 17 + 12*sqrt(2). (End)
G.f.: (1-10x+x^2)/((1-x)(1-34x+x^2)).
a(n) = ceiling(A046176(n)/sqrt(2)). - Helge Robitzsch (hrobi(AT)math.uni-goettingen.de), Jul 28 2000
a(n+1) = 17*a(n) - 4 + 12*sqrt(2*a(n)^2 - a(n)). - Richard Choulet, Sep 14 2007
Define x(n) + y(n)*sqrt(8) = (4+sqrt(8))*(3+sqrt(8))^n, s(n) = (y(n)+1)/2; then a(n) = (1/2)*(2+4*(s(n)^2 - s(n))). - Steven Schlicker, Apr 24 2007
From Ant King, Nov 09 2011: (Start)
a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3).
a(n) = 34*a(n-1) - a(n-2) - 8.
a(n) = 1/8 * ((1 + sqrt(2))^(4*n-2) + (1 - sqrt(2))^(4*n-2) + 2).
a(n) = ceiling((1/8) * (1 + sqrt(2))^(4*n-2)). (End)
From Ravi Kumar Davala, May 26 2013: (Start)
a(n+2) = 577*a(n) - 144 + 408*sqrt(2*a(n)^2 - a(n)).
a(n+m) = A001333(4*m)*a(n) - (A000129(2*m))^2 + A000129(4*m)*sqrt(2*a(n)^2 - a(n)).
a(n+m) = (1/2)*A002203(4*m)*a(n) - (A000129(2*m))^2 + A000129(4*m)*sqrt(2*a(n)^2 - a(n)).
a(n+1)*a(n-1) = (a(n)+4)^2. (End)
a(n) = A001652(n)^2 + A046090(n)^2. - César Aguilera, Jan 15 2018
Limit_{n -> infinity} a(n)/a(n-1) = A156164. - César Aguilera, Jan 28 2018
sqrt(2*a(n))-1 = A002315(n). - Ezhilarasu Velayutham, Apr 05 2019
4*a(n) = 1 +3*A077420(n). - R. J. Mathar, Mar 05 2024
Product_{n>=0} (1 + 4/a(n)) = 2*sqrt(2) + 3 (Koshy, 2022, section 3, p. 19). - Amiram Eldar, Jan 23 2025

Entry edited by N. J. A. Sloane, Sep 14 2007

A156163 Decimal expansion of (19+6*sqrt(2))/17.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Feb 09 2009

lim_{n -> infinity} b(n)/b(n-1) = (19+6*sqrt(2))/17 for n mod 3 = {0, 2}, b = A155923.

lim_{n -> infinity} b(n)/b(n-1) = ((19+6*sqrt(2))/17)^2 for n mod 3 = {0, 2}, b = A156159.

			(19+6*sqrt(2))/17 = 1.61678125730815119369...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 2

Cf. A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A156164 (decimal expansion of 17+12*sqrt(2)).

RealDigits[(19 + 6*Sqrt[2])/17, 10, 100][[1]] (* G. C. Greubel, Jul 05 2017 *)

(6*sqrt(2)+19)/17 \\ Charles R Greathouse IV, Apr 25 2016

A278310 Numbers m such that T(m) + 3*T(m+1) is a square, where T = A000217.

Original entry on oeis.org

3, 143, 4899, 166463, 5654883, 192099599, 6525731523, 221682772223, 7530688524099, 255821727047183, 8690408031080163, 295218051329678399, 10028723337177985443, 340681375412721826703, 11573138040695364122499, 393146012008229658338303, 13355391270239113019379843

Offset: 1

Bruno Berselli, Nov 17 2016

Equivalently, both m+1 and 2*m+3 are squares for nonnegative m.
Corresponding triangular numbers T(m): 6, 10296, 12002550, 13855048416, 15988853699286, 18451128064030200, 21292585958400815526, ...
Square roots of T(m) + 3*T(m+1) are listed by A082405 (after 0).
Negative values of m for which T(m) + 3*T(m+1) is a square: -1, -2, -26, -842, -28562, -970226, -32959082, ...

			3 is in the sequence because T(3) + 3*T(4) = 6 + 3*10 = 6^2.
For n=5 is a(5) = 5654883, therefore floor(sqrt(5654883)) = 2377 = A182189(5) - 2 = 2379 - 2.

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

Subsequence of A000466.
Cf. A000217, A001109, A029546, A046176, A082405, A156164, A182189.
Cf. A278438: numbers m such that T(m) + 2*T(m+1) is a square.
Cf. A078522: numbers m such that 3*T(m) + T(m+1) is a square.
Cf. similar sequences with closed form ((1 + sqrt(2))^(4*r) + (1 - sqrt(2))^(4*r))/8 + k/4: A084703 (k=-1), A076218 (k=3), this sequence (k=-5).

Iv:=[3,143]; [n le 2 select Iv[n] else 34*Self(n-1)-Self(n-2)+40: n in [1..20]];

P:=proc(q) local n; for n from 3 to q do if type(sqrt(2*n^2+5*n+3),integer) then print(n); fi; od; end: P(10^9); # Paolo P. Lava, Nov 18 2016

Table[((1 + Sqrt[2])^(4 n) + (1 - Sqrt[2])^(4 n))/8 - 5/4, {n, 1, 20}]
RecurrenceTable[{a[1] == 3, a[2] == 143, a[n] == 34 a[n - 1] - a[n - 2] + 40}, a, {n, 1, 20}]
LinearRecurrence[{35, -35, 1}, {3, 143, 4899}, 50] (* G. C. Greubel, Nov 20 2016 *)

Vec(x*(3 + 38*x - x^2)/((1 - x)*(1 - 34*x + x^2)) + O(x^50)) \\ G. C. Greubel, Nov 20 2016

def A278310():
    a, b = 3, 143
    yield a
    while True:
        yield b
        a, b = b, 34*b - a + 40
a = A278310(); print([next(a) for  in range(18)]) # _Peter Luschny, Nov 18 2016

O.g.f.: x*(3 + 38*x - x^2)/((1 - x)*(1 - 34*x + x^2)).
E.g.f.: (exp((1-sqrt(2))^4*x) + exp((1+sqrt(2))^4*x) - 10*exp(x))/8 + 1.
a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3) for n>3.
a(n) = 34*a(n-1) - a(n-2) + 40 for n>2.
a(n) = a(-n) = ((1 + sqrt(2))^(4*n) + (1 - sqrt(2))^(4*n))/8 - 5/4.
a(n) = 4*A001109(n)^2 - 1.
a(n) = -A029546(n) + 38*A029546(n-1) + 3*A029546(n-2) for n>1.
Lim_{n -> infinity} a(n)/a(n-1) = A156164.
Floor(sqrt(a(n))) = A182189(n) - 2.
a(n) - a(n-1) = 4*A046176(n) for n>1.

A156571 Decimal expansion of (27 + 10*sqrt(2))/23.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Feb 10 2009

Lim_{n -> infinity} a(n)/a(n-1) = (27+10*sqrt(2))/23 for n mod 3 = {1, 2}, b = A118337, A156567.

Lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))/((27+10*sqrt(2))/23)^2 for n mod 3 = 0, b = A118337, A156567.

			(27 + 10*sqrt(2))/23 = 1.78878850537960654295...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 2.

Cf. A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A156164 (decimal expansion of 17+12*sqrt(2)).

(27+10*Sqrt(2))/23; // G. C. Greubel, Jan 27 2018

RealDigits[(27 + 10*Sqrt[2])/23, 10, 100][[1]] (* G. C. Greubel, Jan 28 2018 *)

(27+10*sqrt(2))/23 \\ G. C. Greubel, Jan 27 2018

A156159 Squares of the form k^2+(k+17)^2 with integer k.

Original entry on oeis.org

169, 289, 625, 2809, 7225, 18769, 93025, 243049, 635209, 3157729, 8254129, 21576025, 107267449, 280395025, 732947329, 3643933225, 9525174409, 24898630849, 123786459889, 323575532569, 845820499225, 4205095700689

Offset: 1

Klaus Brockhaus, Feb 09 2009

Square roots of k^2+(k+17)^2 are in A155923, values k (except for -5) are in A118120.

			625 = 25^2 is of the form k^2+(k+17)^2 with k = 7: 7^2+24^2 = 625. Hence 625 is in the sequence.

Index entries for linear recurrences with constant coefficients, signature (1,0,34,-34,0,-1,1).

Equals A155923^2. Cf. A156160 (first trisection), A156161 (second trisection), A156162 (third trisection).

Cf. A118120, A156035 (decimal expansion of 3+2*sqrt(2)), A156164 (decimal expansion of 17+12*sqrt(2)), A156163 (decimal expansion of (19+6*sqrt(2))/17), A157649 (decimal expansion of (387+182*sqrt(2))/17^2).

LinearRecurrence[{1,0,34,-34,0,-1,1},{169,289,625,2809,7225,18769,93025},30] (* Harvey P. Dale, Apr 22 2022 *)

{forstep(n=-5, 1600000, [1, 3], if(issquare(a=2*n*(n+17)+289), print1(a, ",")))}

a(n) = 34*a(n-3)-a(n-6)-2312 for n > 6; a(1)=169, a(2)=289, a(3)=625, a(4)=2809, a(5)=7225, a(6)=18769.
G.f.: x*(169+120*x+336*x^2-3562*x^3+336*x^4+120*x^5+169*x^6)/((1-x)*(1-34*x^3+x^6)).
Limit_{n -> oo} a(n)/a(n-3) = (17+12*sqrt(2)).
Limit_{n -> oo} a(n)/a(n-1) = ((19+6*sqrt(2))/17)^2 for n mod 3 = {0, 2}.
Limit_{n -> oo} a(n)/a(n-1) = ((387+182*sqrt(2))/17^2)^2 for n mod 3 = 1.

G.f. corrected, fourth comment and cross-references edited by Klaus Brockhaus, Sep 23 2009

A156572 Squares of the form k^2+(k+23)^2 with integer k.

Original entry on oeis.org

289, 529, 1369, 4225, 13225, 42025, 139129, 444889, 1423249, 4721929, 15108769, 48344209, 160402225, 513249025, 1642275625, 5448949489, 17435353849, 55789022809, 185103876169, 592288777609, 1895184495649, 6288082836025

Offset: 1

Klaus Brockhaus, Feb 11 2009

Square roots of k^2+(k+17)^2 are in A156567, values k are in A118337.

			4225 = 65^2 is of the form k^2+(k+23)^2 with k = 33: 33^2+56^2 = 4225. Hence 4225 is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,34,-34,0,-1,1).

Cf. A156567, A156575 (first trisection), A156573 (second trisection), A156574 (third trisection).

Cf. A118337, A156035 (decimal expansion of 3+2*sqrt(2)), A156164 (decimal expansion of 17+12*sqrt(2)), A156571 (decimal expansion of (27+10*sqrt(2))/23), A157472 (decimal expansion of (627+238*sqrt(2))/23^2).

LinearRecurrence[{1,0,34,-34,0,-1,1}, {289,529,1369,4225,13225,42025,139129}, 30] (* Harvey P. Dale, Mar 21 2020 *)

{forstep(n=-8, 1800000, [1, 3], if(issquare(a=2*n*(n+23)+529), print1(a, ",")))}

def f(n,p,q): return p*chebyshev_U(n,17) - q*chebyshev_U(n-1,17)
def a(n):
    if (n%3==0): return -289*bool(n==0) + (1/4)*(529 + 3*f(n/3, 209, 5457))
    elif (n%3==1): return (1/4)*(529 + 3*f((n-1)/3, 209, 1649))
    else: return (1/4)*(529 + 3*f((n-2)/3, 529, 529))
[a(n) for n in (1..30)] # G. C. Greubel, Jan 04 2022

a(n) = 34*a(n-3) - a(n-6) - 4232 for n > 6; a(1)=289, a(2)=529, a(3)=1369, a(4)=4225, a(5)=13225, a(6)=42025.
a(n) = A156567(n)^2.
G.f.: x*(289 +240*x +840*x^2 -6970*x^3 +840*x^4 +240*x^5 +289*x^6)/((1-x)*(1 -34*x^3 +x^6)).
Limit_{n -> infinity} a(n)/a(n-3) = 17 + 12*sqrt(2).
Limit_{n -> infinity} a(n)/a(n-1) = ((627 + 238*sqrt(2))/23^2)^2 for n mod 3 = 1.
Limit_{n -> infinity} a(n)/a(n-1) = ((27 + 10*sqrt(2))/23)^2 for n mod 3 = {0, 2}.
a(n) = -289*[n=0] + (529/4) + (3/4)*( f(n/3, 209, 5457)*(n mod 3 = 1) + f((n-1)/3, 209, 1649)*(n mod 3 = 1) + f((n-2)/2, 529, 529)*(n mod 3 = 2) ), where f(n, p, q) = p*ChebyshevU(n, 17) - q*ChebyshevU(n-1, 17). - G. C. Greubel, Jan 04 2022

Revised by Klaus Brockhaus, Feb 16 2009

G.f. corrected, third comment and cross-references edited by Klaus Brockhaus, Sep 22 2009

A156573 a(n) = 34*a(n-1) - a(n-2) - 4232 for n > 2; a(1)=529, a(2)=13225.

Original entry on oeis.org

529, 13225, 444889, 15108769, 513249025, 17435353849, 592288777609, 20120383080625, 683500735959409, 23218904639535049, 788759257008228025, 26794595833640213569, 910227499086759029089, 30920940373116166771225

Offset: 1

Klaus Brockhaus, Feb 11 2009

			a(3) = 34*a(2) - a(1) - 4232 = 34*13225 - 529 - 4232 = 444889.

G. C. Greubel, Table of n, a(n) for n = 1..600
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Second trisection of A156572.

Cf. A008844, A156164 (decimal expansion of 17+12*sqrt(2)), A156574, A156575.

LinearRecurrence[{35,-35,1}, {529,13225,444889}, 30] (* G. C. Greubel, Jan 04 2022 *)

{m=14; v=concat([529, 13225], vector(m-2)); for(n=3, m, v[n]=34*v[n-1]-v[n-2]-4232); v}

def a(n): return -529*bool(n==0) + (529/4) + (3/4)*(chebyshev_U(n, 17) - 33*chebyshev_U(n-1, 17))
[a(n) for n in (1..30)] # G. C. Greubel, Jan 04 2022

a(n) = 529*(2 + (3 - 2*sqrt(2))*(17 + 12*sqrt(2))^n + (3 + 2*sqrt(2))*(17 - 12*sqrt(2))^n)/8.
a(n) = 529*A008844(n).
G.f.: 529*x*(1 -10*x +x^2)/((1-x)*(1-34*x+x^2)). [corrected by Klaus Brockhaus, Sep 22 2009]
Limit_{n -> infinity} a(n)/a(n-1) = 17+12*sqrt(2).
a(n) = -529*[n=0] + (529/4) + (1587/4)*(ChebyshevU(n, 17) - 33*ChebyshevU(n-1, 17)). - G. C. Greubel, Jan 04 2022

Revised by Klaus Brockhaus, Feb 16 2009

A156574 a(n) = 34*a(n-1) - a(n-2) - 4232 for n > 2; a(1)=1369, a(2)=42025.

Original entry on oeis.org

1369, 42025, 1423249, 48344209, 1642275625, 55789022809, 1895184495649, 64380483825025, 2187041265550969, 74295022544903689, 2523843725261170225, 85736391636334879729, 2912513471910124736329, 98939721653307906151225

Offset: 1

Klaus Brockhaus, Feb 11 2009

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

			a(3) = 34*a(2) - a(1) - 4232 = 34*42025 - 1369 - 4232 = 1423249.

G. C. Greubel, Table of n, a(n) for n = 1..600
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Third trisection of A156572.

Cf. A156164 (decimal expansion of 17+12*sqrt(2)), A156573, A156575.

LinearRecurrence[{35,-35,1}, {1369, 42025, 1423249}, 30] (* G. C. Greubel, Jan 04 2022 *)

{m=14; v=concat([1369, 42025], vector(m-2)); for(n=3, m, v[n]=34*v[n-1]-v[n-2]-4232); v}

def a(n): return -289*bool(n==0) + (529/4) + (3/4)*(209*chebyshev_U(n, 17) - 5457*chebyshev_U(n-1, 17))
[a(n) for n in (1..30)] # G. C. Greubel, Jan 04 2022

a(n) = (1058 + (627 - 238*sqrt(2))*(17 + 12*sqrt(2))^n + (627 + 238*sqrt(2))*(17 - 12*sqrt(2))^n)/8.
G.f.: x*(1369 -5890*x +289*x^2)/((1-x)*(1-34*x+x^2)).
a(n) = -289*[n=0] + (529/4) + (3/4)*(209*ChebyshevU(n, 17) - 5457*ChebyshevU(n - 1, 17)). - G. C. Greubel, Jan 04 2022

Revised by Klaus Brockhaus, Feb 16 2009

G.f. corrected by Klaus Brockhaus, Sep 22 2009

A156575 a(n) = 34*a(n-1)-a(n-2)-4232 for n > 2; a(1)=289, a(2)=4225.

Original entry on oeis.org

289, 4225, 139129, 4721929, 160402225, 5448949489, 185103876169, 6288082836025, 213609712544449, 7256442143671009, 246505423172265625, 8373927945713356009, 284467044731081834449, 9663505592911069011025

Offset: 1

Klaus Brockhaus, Feb 11 2009

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

			a(4) = 34*a(3) -a(2) -4232 = 34*139129 -4225 -4232 = 4721929.

G. C. Greubel, Table of n, a(n) for n = 1..600
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

First trisection of A156572.

Cf. A156164 (decimal expansion of 17+12*sqrt(2)), A156573, A156574.

RecurrenceTable[{a[1]==289,a[2]==4225,a[n]==34a[n-1]-a[n-2]-4232},a,{n,20}] (* or *) LinearRecurrence[{35,-35,1},{289,4225,139129},20] (* Harvey P. Dale, Dec 15 2011 *)

{m=14; v=concat([289, 4225], vector(m-2)); for(n=3, m, v[n]=34*v[n-1]-v[n-2]-4232); v}

def a(n): return -1369*bool(n==0) + (529/4) + (3/4)*(1649*chebyshev_U(n, 17) - 55857*chebyshev_U(n-1, 17))
[a(n) for n in (1..30)] # G. C. Greubel, Jan 04 2022

a(n) = (1058 + (4947 - 3478*sqrt(2))*(17 + 12*sqrt(2))^n + (4947 + 3478*sqrt(2))*(17 - 12*sqrt(2))^n)/8.
G.f.: x*(289 -5890*x +1369*x^2)/((1-x)*(1-34*x+x^2)).
a(1)=289, a(2)=4225, a(3)=139129, a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3). - Harvey P. Dale, Dec 15 2011
a(n) = -1369*[n=0] + (529/4) + (3/4)*(1649*ChebyshevU(n, 17) - 55857*ChebyshevU(n-1, 17)). - G. C. Greubel, Jan 04 2022

Revised by Klaus Brockhaus, Feb 16 2009

G.f. corrected by Klaus Brockhaus, Sep 22 2009

A011906 If b(n) is A011900(n) and c(n) is A001109(n), then a(n) = b(n)*c(n) = b(n) + (b(n)+1) + (b(n)+2) + ... + c(n).

Original entry on oeis.org

1, 18, 525, 17340, 586177, 19896030, 675781821, 22956120408, 779829016225, 26491211221770, 899921240562957, 30570830315362260, 1038508305678375841, 35278711540581704598, 1198437683944896688125, 40711602541832856049200, 1382996048733983114022337

Offset: 1

Mario Velucchi (mathchess(AT)velucchi.it)

			a(3) = 525 = 15*35 = 15 + 16 + ... + 35.

Mario Velucchi "From the desk of ... Mario Velucchi" in 'Mathematics and Informatics quarterly' volume 7 - 2/1997, p. 81.

G. C. Greubel, Table of n, a(n) for n = 1..650
Index entries for linear recurrences with constant coefficients, signature (41,-246,246,-41,1).

Cf. A001109, A011900, A029547, A091761.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x*(1-23*x+33*x^2-3*x^3)/((1-x)*(1-34*x+x^2)*(1-6*x+x^2)) )); // G. C. Greubel, Oct 18 2024

A011900 := proc(n) coeftayl( (1-4*x+x^2)/((1-x)*(1-6*x+x^2)),x=0,n) ; end proc: A001109 := proc(n) coeftayl( x/(1-6*x+x^2),x=0,n) ; end proc: A011906 := proc(n) A001109(n)*A011900(n-1) ; end proc: seq(A011906(n),n=1..30) ; # R. J. Mathar, Apr 15 2010

LinearRecurrence[{41, -246, 246, -41, 1}, {1, 18, 525, 17340, 586177}, 20] (* Paul Cleary, Dec 05 2015 *)
CoefficientList[Series[(-1 + 23*x - 33*x^2 + 3*x^3)/((x - 1)*(x^2 - 34*x + 1)*(1 - 6*x + x^2)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Sep 16 2017 *)

def A011906_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1-23*x+33*x^2-3*x^3)/((1-x)*(1-34*x+x^2)*(1-6*x+x^2)) ).list()
a=A011906_list(30); a[1:] # G. C. Greubel, Oct 18 2024

From R. J. Mathar, Apr 15 2010: (Start)
G.f.: x*(1-23*x+33*x^2-3*x^3)/((1-x)*(1-34*x+x^2)*(1-6*x+x^2)).
a(n) = 41*a(n-1) -246*a(n-2) +246*a(n-3) -41*a(n-4) +a(n-5). (End)
Lim_{n -> infinity} a(n)/a(n-1) = A156164. - César Aguilera, Jul 17 2020
a(n) = (1/16)*(1 - A029547(n) + 41*A091761(n) + 8*A001109(n)). - G. C. Greubel, Oct 18 2024

More terms from R. J. Mathar, Apr 15 2010

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