A001595 -id:A001595 - cp's OEIS frontend

A128587 Row sums of A128586.

1, 1, 1, -1, 3, -5, 9, -15, 25, -41, 67, -109, 177, -287, 465, -753, 1219, -1973, 3193, -5167, 8361, -13529, 21891, -35421, 57313, -92735, 150049, -242785, 392835, -635621, 1028457, -1664079, 2692537, -4356617, 7049155, -11405773, 18454929

Offset: 1

Gary W. Adamson, Mar 11 2007

sign

Binomial transform of A128587 = A128588: (1, 2, 4, 6, 10, 16, 26, ...).

			a(5) = 3 = ( -3, 8, 0, -7, 5).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Yüksel Soykan, Summing Formulas For Generalized Tribonacci Numbers, arXiv:1910.03490 [math.GM], 2019.
Index entries for linear recurrences with constant coefficients, signature (-2,0,1).

Cf. A128586, A128588.
This is a signed version of A001595. - Franklin T. Adams-Watters, Sep 30 2009
Cf. A000045.

List([1..40], n-> (-1)^(n-1)*(2*Fibonacci(n-2)-1)); # G. C. Greubel, Jul 10 2019

[(-1)^(n-1)*(2*Fibonacci(n-2)-1): n in [1..40]]; // G. C. Greubel, Jul 10 2019

Table[(-1)^(n-1)*(2*Fibonacci[n-2] -1), {n, 40}] (* G. C. Greubel, Jul 10 2019 *)

vector(40, n, f=fibonacci; (-1)^(n-1)*(2*f(n-2)-1)) \\ G. C. Greubel, Jul 10 2019

[(-1)^(n-1)*(2*fibonacci(n-2)-1) for n in (1..40)] # G. C. Greubel, Jul 10 2019

Row sums of triangle A128586, inverse binomial transform of A128588.
From R. J. Mathar, Jun 03 2009: (Start)
a(n) = -2*a(n-1) + a(n-3) = (-1)^n*(1 - A118658(n-1)).
G.f.: x*(1+3*x+3*x^2)/((1+x)*(1+x-x^2)). (End)
a(n+3) = (-1)^n * A001595(n) for all n>=0. - M. F. Hasler and Franklin T. Adams-Watters, Sep 30 2009
a(n) = (-1)^(n-1)*(2*Fibonacci(n-2) - 1). - G. C. Greubel, Jul 10 2019

More terms from R. J. Mathar, Jun 03 2009
Duplicate of a formula removed by R. J. Mathar, Oct 23 2009
Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A382843 Length of the long leg in the unique primitive Pythagorean triple (x,y,z) such that (x-y+z)/2 = A000045(n) and its long leg and hypotenuse are consecutive natural numbers.

Original entry on oeis.org

0, 0, 0, 4, 12, 40, 112, 312, 840, 2244, 5940, 15664, 41184, 108112, 283504, 742980, 1946364, 5097624, 13348944, 34953160, 91516920, 239607940, 627323620, 1642389984, 4299890112, 11257351200, 29472278112, 77159668612, 202007027820, 528861900424, 1384579459120

Offset: 0

Miguel-Ángel Pérez García-Ortega, Apr 06 2025

			Triples begin:
  n=0:     -1,     0,     1;
  n=1:      1,     0,     1;
  n=2:      1,     0,     1;
  n=3:      3,     4,     5.
This sequence gives column 2.

Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas

Cf. A000045, A382844 (area), A382845 (sum of the legs), A095122 (semiperimeter), A001595 (short leg).

a(n) = 2*A000045(n)*(A000045(n) - 1),.

A049112 2-ranks of difference sets constructed from Glynn type I hyperovals.

Original entry on oeis.org

1, 1, 3, 7, 13, 23, 45, 87, 167, 321, 619, 1193, 2299, 4431, 8541, 16463, 31733, 61167, 117903, 227265, 438067, 844401, 1627635, 3137367, 6047469, 11656871, 22469341, 43311047, 83484727, 160921985, 310187099, 597904857, 1152498667

Offset: 1

Christian Krattenthaler (kratt(AT)ap.univie.ac.at)

G. C. Greubel, Table of n, a(n) for n = 1..1000
R. Evans, H. Hollmann, C. Krattenthaler and Q. Xiang, Gauss sums, Jacobi sums, and p-ranks of cyclic difference sets, J. Combin. Theory Ser. A, 87.1 (1999), 74-119.
Ronald Evans, Henk Hollmann, Christian Krattenthaler, and Qing Xiang, Supplement to "Gauss Sums, Jacobi Sums and p-ranks ..."
Q. Xiang, On Balanced Binary Sequences with Two-Level Autocorrelation Functions, IEEE Trans. Inform. Theory 44 (1998), 3153-3156.

Cf. A001595, A049114.

a:=[1,3,7,13];; for n in [5..40] do a[n]:=a[n-1]+a[n-2]+a[n-3] +a[n-4] -1; od; Concatenation([1], a); # G. C. Greubel, Jul 10 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+x^2+x^3-x^4-2*x^5)/(1-2*x+x^5) )); // G. C. Greubel, Jul 10 2019

L := 1,1,3,7,13: for i from 6 to 140 do l := nops([ L ]): L := L,op(l,[ L ])+op(l-1,[ L ])+op(l-2,[ L ])+op(l-3,[ L ])-1: od: [ L ];

Join[{1,1,3,7}, Table[a[1]=3; a[2]=1; a[3]=3; a[4]=7; a[i]=a[i-1]+a[i-2] +a[i-3]+a[i-4] -1, {i,5,40}]]
CoefficientList[Series[(1-x+x^2+x^3-x^4-2*x^5)/(1-2*x+x^5), {x,0,40}], x] (* G. C. Greubel, Jul 10 2019 *)

my(x='x+O('x^40)); Vec((1-x+x^2+x^3-x^4-2*x^5)/(1-2*x+x^5)) \\ G. C. Greubel, Jul 10 2019

((1-x+x^2+x^3-x^4-2*x^5)/(1-2*x+x^5)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jul 10 2019

G.f.: (1-x+x^2+x^3-x^4-2*x^5)/(1-2*x+x^5).

a(n+1) = a(n) + a(n-1) + a(n-2) + a(n-3) - 1, n >= 5.

A049114 2-ranks of difference sets constructed from Glynn type II hyperovals.

Original entry on oeis.org

1, 1, 5, 7, 21, 37, 89, 173, 383, 777, 1665, 3441, 7277, 15159, 31885, 66645, 139865, 292757, 613823, 1285585, 2694433, 5644609, 11828501, 24782311, 51928773, 108802597, 227978105, 477674813, 1000877759, 2097121497, 4394101857

Offset: 1

Christian Krattenthaler (kratt(AT)ap.univie.ac.at)

G. C. Greubel, Table of n, a(n) for n = 1..1000
R. Evans, H. Hollmann, C. Krattenthaler and Q. Xiang, Gauss sums, Jacobi sums, and p-ranks of cyclic difference sets, J. Combin. Theory Ser. A, 87.1 (1999), 74-119.
Ronald Evans, Henk Hollmann, Christian Krattenthaler, and Qing Xiang, Supplement to "Gauss Sums, Jacobi Sums and p-ranks ..."
Q. Xiang, On Balanced Binary Sequences with Two-Level Autocorrelation Functions, IEEE Trans. Inform. Theory 44 (1998), 3153-3156.

Cf. A001595, A049112.

a:=[1,5,7,21];; for n in [5..40] do a[n]:=a[n-1]+3*a[n-2]-a[n-3] -a[n-4] +1; od; Concatenation([1], a); # G. C. Greubel, Jul 10 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+x^2-x^3+x^4)/(1-2*x-2*x^2+4*x^3-x^5) )); // G. C. Greubel, Jul 10 2019

L := 1,1,5,7: for i from 5 to 100 do l := nops([ L ]): L := L,op(l,[ L ])+3*op(l-1,[ L ])-op(l-2,[ L ])-op(l-3,[ L ])+1: od: [ L ];

Join[{1,1,5,7}, Table[a[1]=1; a[2]=1; a[3]=5; a[4]=7; a[i]=a[i-1]+ 3*a[i-2]-a[i-3]-a[i-4] +1, {i, 5, 40}]]
CoefficientList[Series[(1-x+x^2-x^3+x^4)/(1-2*x-2*x^2+4*x^3-x^5), {x, 0, 40}], x] (* G. C. Greubel, Jul 10 2019 *)

my(x='x+O('x^40)); Vec((1-x+x^2-x^3+x^4)/(1-2*x-2*x^2+4*x^3-x^5)) \\ G. C. Greubel, Jul 10 2019

((1-x+x^2-x^3+x^4)/(1-2*x-2*x^2+4*x^3-x^5)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jul 10 2019

G.f.: (1-x+x^2-x^3+x^4)/(1-2*x-2*x^2+4*x^3-x^5).

a(n+1) = a(n) + 3*a(n-1) - a(n-2) - a(n-3) + 1.

A209421 Triangle of coefficients of polynomials u(n,x) jointly generated with A209422; see the Formula section.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 9, 7, 5, 1, 1, 15, 15, 9, 6, 1, 1, 25, 28, 22, 11, 7, 1, 1, 41, 53, 44, 30, 13, 8, 1, 1, 67, 97, 91, 63, 39, 15, 9, 1, 1, 109, 176, 179, 140, 85, 49, 17, 10, 1, 1, 177, 315, 349, 291, 201, 110, 60, 19, 11, 1, 1, 287, 559, 667, 601, 437, 275, 138, 72, 21, 12, 1, 1

Offset: 1

Clark Kimberling, Mar 09 2012

For a discussion and guide to related arrays, see A208510.

			First five rows:
  1
  1 1
  3 1 1
  5 4 1 1
  9 7 5 1 1
First three polynomials v(n,x): 1, 1 + x, 3 + x + x^2.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A001595 (column 1), A209422, A208510, A212804.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209421 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209422 *)
CoefficientList[CoefficientList[Series[(1 - t + t^2)/((1 - t)*(1 - (x + 1)*t + (x - 1)*t^2)), {t, 0, 10}], t], x] // Flatten (* G. C. Greubel, Jan 03 2018 *)

u(n,x) = x*u(n-1,x) + v(n-1,x),
v(n,x) = u(n-1,x) + v(n-1,x) +1,
where u(1,x) = 1, v(1,x) = 1.
Riordan array (f(z), z*g(z)) where f(z) = (1 - z + z^2)/(1 - 2*z + z^3) is the o.g.f. for A001595 and g(z) = (1 - z)/(1 - z - z^2) is the o.g.f. for A212804, a variant of the Fibonacci numbers. - Peter Bala, Dec 30 2015
G.f.: (1 + (1 - x)*t - t^2)/((1 - t)*(1 - (x + 1)*t + (x - 1)*t^2)) = 1 + (1+x)*t + (3+x+x^2)*t^2 + ... . - G. C. Greubel, Jan 03 2018

A209769 Triangle of coefficients of polynomials u(n,x) jointly generated with A209770; see the Formula section.

Original entry on oeis.org

1, 1, 2, 3, 5, 3, 5, 12, 11, 5, 9, 26, 34, 24, 8, 15, 53, 88, 86, 48, 13, 25, 104, 210, 258, 200, 93, 21, 41, 198, 470, 695, 680, 440, 175, 34, 67, 369, 1007, 1737, 2043, 1671, 929, 323, 55, 109, 676, 2085, 4107, 5625, 5529, 3895, 1901, 587, 89, 177

Offset: 1

Clark Kimberling, Mar 15 2012

Column 1: A001595
Row n ends with F(n+1), where F=A000045 (Fibonacci numbers).
Row sums: 1,3,11,33,101,303,911,2733,... A081250
Alternating row sums: 1,-1,1,-1,1,-1,... A033999
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...2
3...5....3
5...12...11...5
9...26...34...24...8
First three polynomials u(n,x): 1, 1 + 2x, 3 + 5x + 3x^2.

Cf. A209770, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209769 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209770 *)

u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=(x+1)*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A209770 Triangle of coefficients of polynomials v(n,x) jointly generated with A209769; see the Formula section.

Original entry on oeis.org

1, 3, 1, 5, 4, 2, 9, 12, 10, 3, 15, 29, 33, 19, 5, 25, 64, 93, 77, 37, 8, 41, 132, 234, 251, 171, 69, 13, 67, 261, 548, 719, 629, 362, 127, 21, 109, 500, 1216, 1884, 2004, 1482, 742, 230, 34, 177, 936, 2592, 4628, 5784, 5196, 3342, 1482, 412, 55, 287

Offset: 1

Clark Kimberling, Mar 15 2012

Column 1: A001595
Row n ends with F(n), where F=A000045, the Fibonacci numbers.
Row sums: 1,4,11,34,101,304,911,2734,... A060925
Alternating row sums: 1,2,3,4,5,6,7,.... A000027
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
3....1
5....4....2
9....12...10...3
15...29...33...19...5
First three polynomials v(n,x): 1, 3 + x , 5 + 4x + 2x^2.

Cf. A209669, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209769 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209770 *)

u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=(x+1)*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A259598 Number of representations of n as u(h) + v(k), where u = A000201 (lower Wythoff numbers), v = A001950 (upper Wythoff numbers), h>=1, k>=1.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 3, 1, 2, 4, 0, 4, 4, 1, 6, 2, 4, 7, 0, 8, 4, 4, 9, 1, 8, 8, 2, 11, 4, 7, 12, 0, 12, 9, 4, 14, 4, 10, 14, 1, 16, 8, 8, 17, 2, 15, 14, 4, 19, 7, 12, 20, 0, 21, 12, 9, 22, 4, 18, 19, 4, 24, 10, 14, 25, 1, 24, 18, 8, 27, 8, 19, 26, 2, 29, 15

Offset: 1

Clark Kimberling, Jul 22 2015

Three conjectures. The numbers that are not a sum u(h) + v(k) are (1,2,4,7,12, ...) = A000071 = -1 + Fibonacci numbers. The numbers that have exactly one such representation are (3, 5, 9, 15, 25, 41, ...) = A001595. The numbers that have exactly two such representations are (6, 10, 17, 28, 46, ...) = A001610.

Cf. A259556, A000071, A001595, A001610, A295540.

r = GoldenRatio; z = 500;
u[n_] := u[n] = Floor[n*r]; v[n_] := v[n] = Floor[n*r^2];
s[m_, n_] := s[m, n] = u[m] + v[n]; t = Table[s[m, n], {m, 1, z}, {n, 1, z}];
w = Flatten[Table[Count[Flatten[t], n], {n, 1, z/5}]]  (* A259598 *)

{a(n) = my(phi = (1 + sqrt(5))/2, WL=1, WU=1);
WL = sum(m=1, floor(n/phi)+1, x^floor(m*phi) +x*O(x^n));
WU = sum(m=1, floor(n/phi^2)+1, x^floor(m*phi^2) +x*O(x^n));
polcoeff(WL*WU, n)}
for(n=1, 120, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 02 2017

G.f.: [Sum_{n>=1} x^floor(n*phi)] * [Sum_{n>=1} x^floor(n*phi^2)], where phi = (1+sqrt(5))/2. - Paul D. Hanna, Dec 02 2017

G.f.: [Sum_{n>=1} x^A000201(n)] * [Sum_{n>=1} x^A001950(n)], where A000201 and A001950 are the lower and upper Wythoff sequences, respectively. - Paul D. Hanna, Dec 02 2017

A122194 Numbers that are the sum of exactly two sets of Fibonacci numbers.

Original entry on oeis.org

3, 5, 6, 9, 10, 15, 17, 25, 28, 41, 46, 67, 75, 109, 122, 177, 198, 287, 321, 465, 520, 753, 842, 1219, 1363, 1973, 2206, 3193, 3570, 5167, 5777, 8361, 9348, 13529, 15126, 21891, 24475, 35421, 39602, 57313, 64078, 92735, 103681, 150049, 167760, 242785

Offset: 1

Ron Knott, Aug 25 2006

			a(1)=3 as 3 is the sum of just 2 Fibonacci sets {3=Fibonacci(4)} and {1=Fibonacci(2), 2=Fibonacci(3)};
a(2)=5 as 5 is sum of Fibonacci sets {5} and {2,3} only.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique Theorique, Vol. 35, No 6, 2001, pp. 491-498, in the issue dedicated to Aldo De Luca on the occasion of his 60th anniversary.
M. Bicknell-Johnson & D. C. Fielder, The number of Representations of N Using Distinct Fibonacci Numbers, Counted by Recursive Formulas, Fibonacci Quart. 37.1 (1999) pp. 47 ff.
Ron Knott Sumthing about Fibonacci Numbers
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1).

Cf. A000032, A000045, A000071, A000119, A013583, A122195.

a:= function(n)
    if n mod 2=0 then return 2*Fibonacci(Int((n+6)/2)) -1;
    else return Lucas(1,-1, Int((n+5)/2))[2] -1;
    fi;
  end;
List([1..50], n-> a(n) ); # G. C. Greubel, Jul 13 2019

f:=Floor; [(n mod 2) eq 0 select 2*Fibonacci(f((n+6)/2))-1 else Lucas(f((n+5)/2))-1: n in [1..50]]; // G. C. Greubel, Jul 13 2019

fib:= combinat[fibonacci]:
lucas:=n->fib(n-1)+fib(n+1):
a:=n -> if n mod 2 = 0 then 2 *fib(n/2+3) -1 else lucas((n+1)/2+2)-1 fi:
seq(a(n), n=1..50);

LinearRecurrence[{1, 1, -1, 1, -1}, {3, 5, 6, 9, 10, 15}, 40] (* Vincenzo Librandi, Jul 25 2017 *)
Table[If[Mod[n,2]==0, 2*Fibonacci[(n+6)/2]-1, LucasL[(n+5)/2]-1], {n,50}] (* G. C. Greubel, Jul 13 2019 *)

vector(50, n, f=fibonacci; if(n%2==0, 2*f((n+6)/2)-1, f((n+7)/2) + f((n+3)/2)-1)) \\ G. C. Greubel, Jul 13 2019

def a(n):
    if (mod(n,2)==0): return 2*fibonacci((n+6)/2) - 1
    else: return lucas_number2((n+5)/2, 1,-1) -1
[a(n) for n in (1..50)] # G. C. Greubel, Jul 13 2019

a(2n-1) = A000032(n+2) - 1,
a(2n) = 2*A000045(n+3) - 1.
a(2n-1) = A001610(n+2), a(2n) = A001595(n+2).
a(1)=3, a(2)=5, a(3)=6, a(4)=9, a(n) = a(n-2) + a(n-4) + 1, n > 4.
G.f.: (3 + 2*x - 2*x^2 + x^3 - 3*x^4)/(1-x-x^2+x^3-x^4+x^5).
a(n) = A272632(n)-1. - R. J. Mathar, Jan 13 2023

A131240 T(n,k) = 2*A046854(n,k) - I.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 2, 4, 2, 1, 2, 4, 6, 2, 1, 2, 6, 6, 8, 2, 1, 2, 6, 12, 8, 10, 2, 1, 2, 8, 12, 20, 10, 12, 2, 1, 2, 8, 20, 20, 30, 12, 14, 2, 1, 2, 10, 20, 40, 30, 42, 14, 16, 2, 1, 2, 10, 30, 40, 70, 42, 56, 16, 18, 2, 1, 2, 12, 30, 70, 70, 112, 56, 72, 18, 20, 2, 1

Offset: 0

Gary W. Adamson, Jun 21 2007

Row sums = A001595: (1, 3, 5, 9, 15, 25, 41, 67, ...).

A131241 = 3*A046854 - 2*I.

			First few rows of the triangle:
  1;
  2, 1;
  2, 2,  1;
  2, 4,  2, 1;
  2, 4,  6, 2,  1;
  2, 6,  6, 8,  2, 1;
  2, 6, 12, 8, 10, 2, 1;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A001595, A046854, A131241.

T:= function(n,k)
    if k=n then return 1;
    else return 2*Binomial(Int((n+k)/2), k);
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k)))); # G. C. Greubel, Jul 12 2019

[k eq n select 1 else 2*Binomial(Floor((n+k)/2), k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2019

Table[If[k==n, 1, 2*Binomial[Floor[(n+k)/2], k]], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 12 2019 *)

T(n,k) = if(k==n, 1, 2*binomial((n+k)\2, k));

def T(n, k):
    if (k==n): return 1
    else: return 2*binomial(floor((n+k)/2), k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jul 12 2019

T(n,k) = 2*A046854(n,k) - Identity matrix, where A046854 = Pascal's triangle with repeats by columns.

More terms added by G. C. Greubel, Jul 12 2019

cp's OEIS Frontend

A128587 Row sums of A128586.

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A382843 Length of the long leg in the unique primitive Pythagorean triple (x,y,z) such that (x-y+z)/2 = A000045(n) and its long leg and hypotenuse are consecutive natural numbers.

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A049112 2-ranks of difference sets constructed from Glynn type I hyperovals.

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A049114 2-ranks of difference sets constructed from Glynn type II hyperovals.

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A209421 Triangle of coefficients of polynomials u(n,x) jointly generated with A209422; see the Formula section.

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A209769 Triangle of coefficients of polynomials u(n,x) jointly generated with A209770; see the Formula section.

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A209770 Triangle of coefficients of polynomials v(n,x) jointly generated with A209769; see the Formula section.

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