A079935 -id:A079935 - cp's OEIS frontend

A102591 a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*3^(n-k).

1, 6, 44, 328, 2448, 18272, 136384, 1017984, 7598336, 56714752, 423324672, 3159738368, 23584608256, 176037912576, 1313964867584, 9807567290368, 73204678852608, 546407161659392, 4078438577864704, 30441879976280064

Offset: 0

Paul Barry, Jan 22 2005

In general, Sum_{k=0..n} binomial(2n+1,2k)*r^(n-k) has g.f. (1-(r-1)x)/(1-2(r+1)+(r-1)^2x^2) and a(n) = ((sqrt(r)-1)^(2n+1) + (sqrt(r)+1)^(2n+1))/(2*sqrt(r)).

Index entries for linear recurrences with constant coefficients, signature (8,-4).

Bisection of A080879 or A002605.

Cf. A066443, A079935, A099156, A102592, A107903.

LinearRecurrence[{8,-4},{1,6},20] (* Harvey P. Dale, Sep 28 2021 *)

G.f.: (1-2x)/(1-8x+4x^2);
a(n) = 8*a(n-1) - 4*a(n-2);
a(n) = sqrt(3)*(sqrt(3)-1)^(2n+1)/6 + sqrt(3)*(sqrt(3)+1)^(2n+1)/6.
a(n) = 2^n*A079935(n). - R. J. Mathar, Sep 20 2012
a(n) = 2^(2*n+1)*Sum_{k >= n} binomial(2*k,2*n)*(1/3)^(k+1). Cf. A099156. - Peter Bala, Nov 29 2021
3*a(n)^2 = A107903(n)^2 + 2^(2*n+1). - Philippe Deléham, Mar 21 2023

A079934 Greedy frac multiples of sqrt(2): a(1)=1, Sum_{n>=0} frac(a(n)*x)=1 at x=sqrt(2).

Original entry on oeis.org

1, 3, 5, 10, 17, 29, 46, 99, 169, 268, 577, 985, 1562, 3363, 5741, 9104, 19601, 33461, 53062, 114243, 195025, 309268, 665857, 1136689, 1802546, 3880899, 6625109, 10506008, 22619537, 38613965, 61233502, 131836323, 225058681, 356895004, 768398401, 1311738121

Offset: 1

Benoit Cloitre and Paul D. Hanna, Jan 20 2003

The n-th greedy frac multiple of x is the smallest integer that does not cause Sum_{k=1..n} frac(a(k)*x) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of x.

			a(4) = 10 since frac(1x) + frac(3x) + frac(5x) + frac(10x) < 1, while frac(1x) + frac(3x) + frac(5x) + frac(k*x) > 1 for all k > 5 and k < 10.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A000129 (Pell numbers), A078343, A079935, A079936.

CoefficientList[Series[(1 + 3*z + 5*z^2 + 4*z^3 - z^4 - z^5 - 13*z^6 + 2*z^9)/(1 - 6*z^3 + z^6), {z, 0, 40}], z] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)

x='x+O('x^50); Vec(x*(2*x^9 -13*x^6 -x^5 -x^4 +4*x^3 +5*x^2 +3*x +1)/(x^6-6*x^3 +1)) \\ G. C. Greubel, Sep 22 2017

For n > 0, a(3*n) = A000129(2*n+1).
a(3*n+2) = a(3*n) + A000129(2*n+2).
a(3*n+4) = a(3*n+2) + a(3*n+3).
a(3*n) = ceiling((3+2*sqrt(2))^n*(2+sqrt(2))/4).
a(3*n+2)/a(3*n+1) -> 1/sqrt(2).
a(3*n+1)/a(3*n) -> 3-sqrt(2).
a(3*n)/a(3*n-1) -> (8+5*sqrt(2))/7.
G.f.: x*(2*x^9 - 13*x^6 - x^5 - x^4 + 4*x^3 + 5*x^2 + 3*x + 1) / (x^6 - 6*x^3 + 1). - Colin Barker, Jun 16 2013

A086405 Row T(n,3) of number array A086404.

Original entry on oeis.org

1, 4, 18, 84, 396, 1872, 8856, 41904, 198288, 938304, 4440096, 21010752, 99423936, 470479104, 2226331008, 10535111424, 49852682496, 235905426432, 1116316463616, 5282466223104, 24996898556928, 118286594002944

Offset: 0

Paul Barry, Jul 19 2003

Binomial transform of A079935.

Number of nonisomorphic graded posets with 0 of rank n+1, with exactly 2 elements of each rank level above 0. Here, we do not assume all maximal elements have maximal rank and thus use graded poset to mean: For every element x, all maximal chains among those with x as greatest element have the same finite length. - David Nacin, Feb 13 2012

R. Stanley, Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

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

Cf. A079935, A086404.

LinearRecurrence[{6, -6}, {1, 4}, 60] (* David Nacin, Feb 27 2012 *)

def a(n, adict={0:1, 1:4}):
    if n in adict:
        return adict[n]
    adict[n]=6*a(n-1)-6*a(n-2)
    return adict[n] # David Nacin, Feb 27 2012

G.f.: (1-2*x)/((1-(3-sqrt(3))*x)*(1-(3+sqrt(3))*x)) = (1-2*x)/(1-6*x+6*x^2);
a(n) = (3-sqrt(3))^n*(1/2 - 1/(2*sqrt(3))) + (3 + sqrt(3))^n*(1/2 + 1/(2*sqrt(3))).
E.g.f.: exp(3*x)*(cosh(sqrt(3)*x) + sinh(sqrt(3)*x)/sqrt(3)). - Paul Barry, Nov 20 2003
a(n) = Sum_{k=1..floor(n/2)} C(n, 2k)*3^(n-k-1). - Paul Barry, Nov 22 2003
a(n) = (((1+sqrt(3))*(3+sqrt(3))^n) - ((1-sqrt(3))*(3-sqrt(3))^n))/sqrt(12). - Al Hakanson (hawkuu(AT)gmail.com), Jun 10 2009
a(n) = Sum_{k=0..n} A117317(n,k)*2^k. - Philippe Deléham, Jan 28 2012
a(n) = 6*(a(n-1) - a(n-2)), a(0)=1, a(1)=4. - David Nacin, Feb 27 2012
G.f.: (1-2*x)/(1-6*x+6*x^2). - Colin Barker, Aug 04 2012

A096147 Prime denominators of the rational convergents to sqrt(3).

Original entry on oeis.org

3, 11, 41, 571, 2131, 110771, 1542841, 15558008491, 808717138331, 1663476485027525263506023431291963826940251, 33648911495192637123958375850447995878147331088460770783226682531

Offset: 1

Cino Hilliard, Jul 24 2004

nonn

Next term is too large to include.
The next term has 79 digits. - Harvey P. Dale, Jul 06 2019
This is the prime subsequence of A002530. - Ray Chandler, Aug 01 2004
Primes p such that 3*p^2 - 2 is a square. - Vincenzo Librandi, May 21 2013

Amiram Eldar, Table of n, a(n) for n = 1..23

Cf. A002530, A079935, A096146.

Select[Denominator[Convergents[Sqrt[3],300]],PrimeQ] (* Harvey P. Dale, Jul 06 2019 *)

Offset corrected by Amiram Eldar, Jul 11 2024

A090317 Row sums of triangle in A090285.

Original entry on oeis.org

1, 2, 7, 28, 118, 510, 2235, 9876, 43870, 195556, 873814, 3911168, 17527904, 78622982, 352911939, 1584927828, 7120769526, 32002212252, 143859840114, 646819996008, 2908670252676, 13081556909292, 58839348572574, 264674150692488, 1190649451348908, 5356483791828840, 24098774900561500

Offset: 0

Philippe Deléham, Jan 25 2004

Apply the inverse of the Riordan array (1/(1-x^2),x/(1+x)^2) to 2^n. - Paul Barry, Mar 13 2009

Hankel transform is A079935. - Paul Barry, Mar 13 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A001700, A090285.

Table[SeriesCoefficient[(1-x^2*((1-Sqrt[1-4*x])/(2*x))^4)/(1-2*x*((1-Sqrt[1-4*x])/(2*x))^2),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

a(n):=if n=0 then 1 else 4*binomial(2*n-1,n)/(n+1)+3*sum(((k+1)*2^(k)*binomial(2*n-1,n-k-1))/(n+k+1),k,1,n-1); /* Vladimir Kruchinin, Feb 21 2019 */

x='x+O('x^66); Vec((1-x^2*((1-sqrt(1-4*x))/(2*x))^4)/(1-2*x*((1-sqrt(1-4*x))/(2*x))^2)) \\ Joerg Arndt, May 11 2013

a(n+1) = A000108(n+1) + Sum_{k=0..n} a(n-k)*A001700(k); a(0) = 1.
G.f.: (1-x^2*c(x)^4)/(1-2x*c(x)^2), where c(x) is the g.f. of the Catalan numbers A000108. - Paul Barry, Mar 13 2009
Recurrence: 2*(n+1)*(n+3)*a(n) = (17*n^2+56*n-21)*a(n-1) - 18*(n+4)*(2*n-3)*a(n-2). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ 9^n/2^(n+2). - Vaclav Kotesovec, Oct 14 2012
a(n) = 4*C(2*n-1,n)/(n+1)+3*Sum_{k=1..n-1}(k+1)*2^k*C(2*n-1,n-k-1)/(n+k+1), n>0, a(0)=1. - Vladimir Kruchinin, Feb 21 2019

Term 15 corrected by Paul Barry, Mar 13 2009

A162557 a(n) = ((3+sqrt(3))(4+sqrt(3))^n+(3-sqrt(3))(4-sqrt(3))^n)/6.

Original entry on oeis.org

1, 5, 27, 151, 857, 4893, 28003, 160415, 919281, 5268853, 30200171, 173106279, 992248009, 5687602445, 32601595443, 186873931759, 1071170713313, 6140004593637, 35194817476027, 201738480090935, 1156375213539129, 6628401467130877, 37994333961038339, 217785452615605311

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009

nonn

Binomial transform of A086405.
Inverse binomial transform of A162558.
4th binomial transform of A108411.
2nd binomial transform of A079935. [R. J. Mathar, Jul 17 2009]
From J. Conrad, Aug 29 2016: (Start)
Partial sum of A136777.
Backward difference of Sum_{k=0..n} A027907(n+1,2k+2)*3^k.
(End)
String length in substitution system {0 -> 1001001, 1 -> 11011} at step n from initial string "1" (1 -> 11011 -> 110111101110010011101111011 -> ...). - Ilya Gutkovskiy, Aug 30 2016

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-13).

Cf. A108411 (powers of 3 repeated), A086405, A162558.

Cf. A162558. [R. J. Mathar, Jul 17 2009]

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((3+r)*(4+r)^n+(3-r)*(4-r)^n)/6: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 13 2009

I:=[1,5]; [n le 2 select I[n]  else 8*Self(n-1)-13*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 30 2016

seq(simplify(((3+sqrt(3))*(4+sqrt(3))^n+(3-sqrt(3))*(4-sqrt(3))^n)*1/6), n = 0..20); # Emeric Deutsch, Jul 14 2009

Table[FullSimplify[((3 + #) (4 + #)^n + (3 - #) (4 - #)^n)/6 &@ Sqrt@ 3], {n, 0, 23}] (* Michael De Vlieger, Aug 30 2016 *)
LinearRecurrence[{8,-13},{1,5},30] (* Harvey P. Dale, Oct 23 2020 *)

a(n) = 8*a(n-1)-13*a(n-2) for n > 1; a(0) = 1, a(1) = 5.

G.f.: (1-3*x)/(1-8*x+13*x^2).

Edited, corrected and extended beyond a(5) by Klaus Brockhaus, Emeric Deutsch and R. J. Mathar, Jul 07 2009

More terms from Vincenzo Librandi, Aug 30 2016

A374602 Array of successive integer solutions to sqrt((d-c)b^2 + c(b+1)^2) for nonsquare integers d >= 2 (d=A000037(n) for n >= 1), where b and c are positive integers and c < d, read by antidiagonals.

Original entry on oeis.org

5, 29, 3, 169, 11, 5, 985, 41, 13, 3, 5741, 153, 34, 7, 4, 33461, 571, 89, 18, 5, 10, 195025, 2131, 233, 29, 11, 11, 4, 1136689, 7953, 610, 69, 28, 23, 5, 7, 6625109, 29681, 1597, 178, 62, 58, 13, 8, 6, 38613965, 110771, 4181, 287, 79, 338, 14, 13, 22, 4

Offset: 1

Charles L. Hohn, Jul 13 2024

T(n,k) is the diagonal lengths of increasingly nearly regular d-dimensional Pythagorean hyperrectangles.
Each row n divides into equal length, geometrically periodic subsequences, each with its own subsequence period length (A377290) and geometric growth factor (A377291); it is conjectured that this is the case for all n, and that all solutions conform as such and that there are no solutions that do not, but these are not proven.
It is also not known if there is an algorithm for generating values for all rows other than testing all possible values for a row until a subsequence pattern emerges.
Square d produce solutions following a different pattern, shown as A375336.

			n=row index; d=nonsquare integer of index n (A000037(n)):
 n    d   T(n,k)
---+----+-------------------------------------------------------------
 1 |  2 |  5, 29, 169, 985, 5741, 33461, 195025, 1136689, 6625109, ...
 2 |  3 |  3, 11,  41, 153,  571,  2131,   7953,   29681,  110771, ...
 3 |  5 |  5, 13,  34,  89,  233,   610,   1597,    4181,   10946, ...
 4 |  6 |  3,  7,  18,  29,   69,   178,    287,     683,    1762, ...
 5 |  7 |  4,  5,  11,  28,   62,    79,    175,     446,     988, ...
 6 |  8 | 10, 11,  23,  58,  338,   373,    781,    1970,   11482, ...
 7 | 10 |  4,  5,  13,  14,   25,    62,    111,     148,     185, ...
 8 | 11 |  7,  8,  13,  32,   57,   139,    158,     259,     638, ...
 9 | 12 |  6, 22,  39,  69,   82,   125,    306,     543,    1142, ...
10 | 13 |  4,  5,   7,  17,   30,    43,     53,      76,     185, ...
11 | 14 |  9, 11,  14,  19,   46,    81,    267,     329,     418, ...
12 | 15 |  6, 10,  21,  23,   30,    39,     94,     165,     362, ...
13 | 17 | 25, 27,  34,  41,   98,   171,    260,    1649,    1779, ...
14 | 18 |  6, 13,  15,  18,   21,    50,     87,     132,     198, ...
15 | 19 |  5,  7,   8,   9,   11,    31,     34,      37,      56, ...
16 | 20 | 10, 26,  68, 125,  159,   178,    197,     466,     807, ...
17 | 21 |  6,  9,  12,  13,   14,    33,     57,      86,     134, ...
18 | 22 |  5,  7,   8,  17,   18,    19,     31,      64,      77, ...
19 | 23 | 16, 19,  27,  28,   29,    68,    117,     176,     764, ...
20 | 24 |  6,  9,  11,  14,   36,    39,     57,      58,      59, ...
...
sqrt((2-1)*1^2 + 1*(1+1)^2) = sqrt(5) -> not an integer so not included.
sqrt((2-1)*3^2 + 1*(3+1)^2) = 5 -> T(1,1).
sqrt((2-1)*20^2 + 1*(20+1)^2) = 29 -> T(1,2).
sqrt((3-2)*1^2 + 2*(1+1)^2) = 3 -> T(2,1).
sqrt((6-2)*7^2 + 2*(7+1)^2) = 18 -> T(4,3).

Row 1 is A001653 starting at n=2.
Row 2 is A079935 starting at n=2.
Bisection of row 2 starting with the first term is A189356 starting at n=1.
Bisection of row 2 starting with the second term is A122769 starting at n=2.
Row 3 is A001519 starting at n=3.
Bisection of row 3 starting with the first term is A033889 starting at n=1.
Bisection of row 3 starting with the second term is A033891 starting at n=1.
Row 4 is A131093 starting at n=3.
Cf. A000037, A373666.
Cf. A377290, A377291, A375336.

row(n, c)=my(v=List(), d=n+floor(sqrt(n)+1/2) /* d=A000037(n) */, t=ceil(sqrt(d))); while(#v

T(n, 1) = A373666(A000037(n)).

A010905 Pisot sequence E(4,15): a(n) = floor(a(n-1)^2/a(n-2)+1/2) for n>1, a(0)=4, a(1)=15.

Original entry on oeis.org

4, 15, 56, 209, 780, 2911, 10864, 40545, 151316, 564719, 2107560, 7865521, 29354524, 109552575, 408855776, 1525870529, 5694626340, 21252634831, 79315912984, 296011017105, 1104728155436, 4122901604639, 15386878263120, 57424611447841, 214311567528244

Offset: 0

Simon Plouffe

nonn

Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, Preprint, 2016.

Colin Barker, Table of n, a(n) for n = 0..1000
D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305
D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (4,-1).

Cf. A010925, A001353, A079935, A195503.

/* By definition: */ [n le 2 select 11*n-7 else Floor(Self(n-1)^2/Self(n-2)+1/2): n in [1..22]]; // Bruno Berselli, Apr 16 2012

a[0] = 4; a[1] = 15; a[n_] := a[n] = Floor[a[n - 1]^2/a[n - 2] + 1/2]; Table[a[n], {n, 0, 24}] (* Michael De Vlieger, Jul 27 2016 *)

pisotE(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]+1/2));
  a
}
pisotE(50, 4, 15) \\ Colin Barker, Jul 27 2016

@cached_function
def A010905(n):
    if n==0: return 4
    elif n==1: return 15
    else: return 4*A010905(n-1) - A010905(n-2)
[A010905(n) for n in range(30)] # G. C. Greubel, Dec 13 2018

a(n) = 4*a(n-1) - a(n-2) for n>=2. (Proved using the PtoRv program of Ekhad-Sloane-Zeilberger.) - N. J. A. Sloane, Sep 09 2016
This was conjectured by Colin Barker, Apr 16 2012, and implies the G.f.: (4-x)/(1-4*x+x^2) and the formula a(n) = ((1+sqrt(3))^(2*n+4)-(1-sqrt(3))^(2*n+4))/(2^(n+3)*sqrt(3)).
Partial sums of A079935. - Erin Pearse, Dec 13 2018

Edited by N. J. A. Sloane, Jul 26 2016 and Sep 09 2016

A079936 Greedy frac multiples of sqrt(5): a(1)=1, sum(n>0,frac(a(n)*x))=1 at x=sqrt(5).

Original entry on oeis.org

1, 2, 5, 13, 17, 34, 305, 610, 1597, 4181, 5473, 10946, 98209, 196418, 514229, 1346269, 1762289, 3524578, 31622993, 63245986, 165580141, 433494437, 567451585, 1134903170, 10182505537, 20365011074, 53316291173, 139583862445

Offset: 1

Benoit Cloitre and Paul D. Hanna, Jan 21 2003

The n-th greedy frac multiple of x is the smallest integer that does not cause sum(k=1..n,frac(a(k)*x)) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of x.

			a(4) = 13 since frac(1x) + frac(2x) + frac(5x) + frac(13x) < 1, while frac(1x) + frac(2x) + frac(5x) + frac(k*x) > 1 for all k>5 and k<13.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,322,0,0,0,0,0,-1).

Cf. A001076 (denominators of convergents to sqrt(5)), A079934, A079935, A079937.

For n>=0, a(6n+1)=A001076(4n+1); a(6n+2)=2a(6n+1); a(6n+3)=A001076(4n+1)+A001076(4n+2); a(6n+4)=A001076(4n+3)-A001076(4n+2); a(6n+5)=A001076(4n+3); a(6n+6)=2a(6n+5). Asymptotics: a(6n) -> 2*sqrt(5)*(tau)^(12n-3); a(6n+2)/a(6n+1) -> (tau)^2; a(6n+3)/a(6n+2) -> (tau)^2; a(6n+4)/a(6n+3) -> (tau)^2/2; a(6n+6)/a(6n+5) -> (tau)^6/2; where tau = (1+sqrt(5))/2.

G.f.: -x*(x -1)*(2*x^10 +3*x^9 +8*x^8 +21*x^7 +55*x^6 +72*x^5 +38*x^4 +21*x^3 +8*x^2 +3*x +1) / (x^12 -322*x^6 +1). - Colin Barker, Jun 16 2013

A086404 Square array of numbers T(n,k) = ((1+sqrt(3))(k+sqrt(3))^n-(1-sqrt(3))(k-sqrt(3))^n)/(2*sqrt(3)), read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 3, 1, 4, 11, 16, 9, 1, 5, 18, 41, 44, 9, 1, 6, 27, 84, 153, 120, 27, 1, 7, 38, 151, 396, 571, 328, 27, 1, 8, 51, 248, 857, 1872, 2131, 896, 81, 1, 9, 66, 381, 1644, 4893, 8856, 7953, 2448, 81, 1, 10, 83, 556, 2889, 10984, 28003, 41904, 29681

Offset: 0

Paul Barry, Jul 19 2003

			Rows begin
  1, 1,  3,   3,   9, ...
  1, 2,  6,  16,  44, ...
  1, 3, 11,  41, 153, ...
  1, 4, 18,  84, 396, ...
  1, 5, 27, 151, 857, ...

Rows include A002605, A079935, A086405. Main diagonal is A086406. Rows are successive binomial transforms of (1, 1, 3, 3, 9, 9, ...).

Cf. A086350.

cp's OEIS Frontend

A102591 a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*3^(n-k).

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A079934 Greedy frac multiples of sqrt(2): a(1)=1, Sum_{n>=0} frac(a(n)*x)=1 at x=sqrt(2).

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A086405 Row T(n,3) of number array A086404.

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A096147 Prime denominators of the rational convergents to sqrt(3).

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A090317 Row sums of triangle in A090285.

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A162557 a(n) = ((3+sqrt(3))*(4+sqrt(3))^n+(3-sqrt(3))*(4-sqrt(3))^n)/6.

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A374602 Array of successive integer solutions to sqrt((d-c)*b^2 + c*(b+1)^2) for nonsquare integers d >= 2 (d=A000037(n) for n >= 1), where b and c are positive integers and c < d, read by antidiagonals.

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Programs

A162557 a(n) = ((3+sqrt(3))(4+sqrt(3))^n+(3-sqrt(3))(4-sqrt(3))^n)/6.

A374602 Array of successive integer solutions to sqrt((d-c)b^2 + c(b+1)^2) for nonsquare integers d >= 2 (d=A000037(n) for n >= 1), where b and c are positive integers and c < d, read by antidiagonals.

A086404 Square array of numbers T(n,k) = ((1+sqrt(3))(k+sqrt(3))^n-(1-sqrt(3))(k-sqrt(3))^n)/(2*sqrt(3)), read by antidiagonals.