A080097 -id:A080097 - cp's OEIS frontend

A382845 Sum of the legs of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 = A000045(n) and its long leg and hypotenuse are consecutive natural numbers.

-1, 1, 1, 7, 17, 49, 127, 337, 881, 2311, 6049, 15841, 41471, 108577, 284257, 744199, 1948337, 5100817, 13354111, 34961521, 91530449, 239629831, 627359041, 1642447297, 4299982847, 11257501249, 29472520897, 77160061447, 202007663441, 528862928881, 1384581123199

Offset: 0

Miguel-Ángel Pérez García-Ortega, Apr 06 2025

			For n=4, the short leg is A382843(2,1) = 3 and the long leg is A382843(2,2) = 4  so the sum of the legs is then a(4) = 3 + 4 = 7.

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.

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

Cf. A000045, A382843, A382844, A095122, A007598, A080097.

a=Table[Fibonacci[n],{n,0,30}];Apply[Join,Map[{2#^2-1}&,a]]

a(n) = A382843(n,1) + A382843(n,2).

a(n) = 2*Fibonacci(n)^2 - 1.

A193917 Triangular array: the self-fusion of (p(n,x)), where p(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 3, 6, 9, 3, 5, 9, 15, 24, 5, 8, 15, 24, 40, 64, 8, 13, 24, 39, 64, 104, 168, 13, 21, 39, 63, 104, 168, 273, 441, 21, 34, 63, 102, 168, 272, 441, 714, 1155, 34, 55, 102, 165, 272, 440, 714, 1155, 1870, 3025, 55, 89, 165, 267, 440, 712, 1155

Offset: 0

Clark Kimberling, Aug 09 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.  (Fusion is defined at A193822; fission, at A193742; see A202503 and A202453 for infinite-matrix representations of fusion and fission.)
First five rows of P (triangle of coefficients of polynomials p(n,x)):
1
1...1
1...1...2
1...1...2...3
1...1...2...3...5
First eight rows of A193917:
1
1...1
1...2...3
2...3...6...9
3...5...9...15...24
5...8...15..24...40...64
8...13..24..39...64...104..168
13..21..39..63...104..168..273..441
...
col 1:  A000045
col 2:  A000045
col 3:  A022086
col 4:  A022086
col 5:  A022091
col 6:  A022091
col 7:  A022355
col 8:  A022355
right edge, w(n,n):  A064831
w(n,n-1):  A001654
w(n,n-2):  A064831
w(n,n-3):  A059840
w(n,n-4):  A080097
w(n,n-5):  A080143
w(n,n-6):  A080144
Suppose n is an even positive integer and w(n+1,x) is the polynomial matched to row n+1 of A193917 as in the Mathematica program (and definition of fusion at A193722), where the first row is counted as row 0.

			First six rows:
1
1...1
1...2...3
2...3...6....9
3...5...9....15...24
5...8...15...24...40...64

Cf. A193722, A064831, A193918, A194000, A194001.

z = 12;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := p[n, x];
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193917 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193918 *)

A194000 Triangular array: the self-fission of (p(n,x)), where sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 2, 3, 3, 5, 9, 5, 8, 15, 24, 8, 13, 24, 39, 64, 13, 21, 39, 63, 104, 168, 21, 34, 63, 102, 168, 272, 441, 34, 55, 102, 165, 272, 440, 714, 1155, 55, 89, 165, 267, 440, 712, 1155, 1869, 3025, 89, 144, 267, 432, 712, 1152, 1869, 3024, 4895, 7920, 144, 233

Offset: 0

Clark Kimberling, Aug 11 2011

See A193917 for the self-fusion of the same sequence of polynomials.  (Fusion is defined at A193822; fission, at A193842; see A202503 and A202453 for infinite-matrix representations of fusion and fission.)
...
First five rows of P (triangle of coefficients of polynomials p(n,x)):
1
1...1
1...1...2
1...1...2...3
1...1...2...3...5
First eight rows of A194000:
1
2....3
3....5....9
5....8....15...24
8....13...24...39...64
13...21...29...63...104...168
21...34...63...102..168...272...441
34...55...102..165..272...440...714..1155
...
col 1:  A000045
col 2:  A000045
col 3:  A022086
col 4:  A022086
col 5:  A022091
col 6:  A022091
right edge, d(n,n):  A064831
d(n,n-1):  A059840
d(n,n-2):  A080097
d(n,n-3):  A080143
d(n,n-4):  A080144
...
Suppose n is an odd positive integer and d(n+1,x) is the polynomial matched to row n+1 of A194000 as in the Mathematica program (and definition of fission at A193842), where the first row is counted as row 0.

			First six rows:
1
2....3
3....5....9
5....8....15...24
8....13...24...39...64
13...21...29...63...104...168
...
Referring to the matrix product for fission at A193842,
the row (5,8,15,24) is the product of P(4) and QQ, where
P(4)=(p(4,4), p(4,3), p(4,2), p(4,1))=(5,3,2,1); and
QQ is the 4x4 matrix
(1..1..2..3)
(0..1..1..2)
(0..0..1..1)
(0..0..0..1).

Cf. A193842, A194001, A193917, A193918.

z = 11;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := p[n, x];
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A194000 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A194001 *)

A214729 Member m=6 of the m-family of sums b(m,n) = Sum_{k=0..n} F(k+m)*F(k), m >= 0, n >= 0, with the Fibonacci numbers F.

Original entry on oeis.org

0, 13, 34, 102, 267, 712, 1864, 4893, 12810, 33550, 87835, 229968, 602064, 1576237, 4126642, 10803702, 28284459, 74049688, 193864600, 507544125, 1328767770, 3478759198, 9107509819, 23843770272, 62423800992, 163427632717, 427859097154, 1120149658758

Offset: 0

Wolfdieter Lang, Jul 27 2012

See the comment section on A080144 for the general formula and the o.g.f. for b(m,n).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Cf. A001654, A027941, A059840(n+2), A064831, A080097, A080143 and A080144 for the m=0,1,...,5 members.

Cf. A027941.

[(9*(-1)^(n+1)-20+Lucas(2*n+7))/5: n in [0..40]]; // Vincenzo Librandi, Aug 26 2017

With[{m = 6}, Table[Sum[Fibonacci[k + m]*Fibonacci[k], {k, 0, n}], {n, 0, 25}]] (* or *)
Table[(9 (-1)^(n + 1) - 20 + LucasL[2 n + 7])/5, {n, 0, 25}] (* Michael De Vlieger, Aug 23 2017 *)
LinearRecurrence[{3,0,-3,1},{0,13,34,102},40] (* Harvey P. Dale, Jun 13 2022 *)

concat(0, Vec(x*(13 - 5*x) / ((1 - x)*(1 + x)*(1 - 3*x + x^2)) + O(x^30))) \\ Colin Barker, Aug 25 2017

[fibonacci(n+3)*fibonacci(n+4) - 2*(2+(-1)^n) for n in range(41)] # G. C. Greubel, Dec 31 2023

a(n) = b(6,n) = 4*A027941(n) + 9*A001654(n), with A027941(n) = Fibonacci(2*n+1) - 1 and A001654(n) = Fibonacci(n+1)*Fibonacci(n), n >= 0. 4 = Fibonacci(6)/2 and 9 = LucasL(6)/2.
O.g.f.: x*(13-5*x)/((1-x^2)*(1-3*x+x^2)) (see a comment above). - Wolfdieter Lang, Jul 30 2012
a(n) = (9*(-1)^(n+1) - 20 + Lucas(2*n + 7))/5. - Ehren Metcalfe, Aug 21 2017
From Colin Barker, Aug 25 2017: (Start)
a(n) = (1/10)*((29 - 13*sqrt(5))*((3 - sqrt(5))/2)^n + (29 + 13*sqrt(5))*((3 + sqrt(5))/2)^n - 2*(20 + 9*(-1)^n) ).
a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4) for n>3. (End)
a(n) = A001654(n+3) - 2*(2 + (-1)^n). - G. C. Greubel, Dec 31 2023

A281619 Integer values of (A000045^2-1)/9 where A000045(m) is the m-th Fibonacci number.

Original entry on oeis.org

0, 7, 336, 880, 6032, 15792, 741895, 34853280, 91247072, 625416736, 1637362272, 76921173511, 3613657792752, 9460678925136, 64844458022832, 169764995085840, 7975341111241735, 374671267233275712, 980902112224710592, 6723203096097857600, 17601574218852716736, 826899317018844410887

Offset: 1

Michel Marcus, Jan 25 2017

Also the integer values of A080097/9 where A080097(m) = Fibonacci(n+2)^2 - 1.

The indices of the Fibonacci numbers are 1, 2, 6, 10, and 11 mod 12. See the Wulczy link.

Robert Israel, Table of n, a(n) for n = 1..997
G. Wulczy, Unity with Fibonacci, Problem H-247 and solution, Fib. Quarter. p. 89_90, Vol 15, 1, Feb. 1977.

Cf. A000045, A007887, A080097.

seq(seq((combinat:-fibonacci(12*m+j)^2-1)/9,j=[2,6,10,11,13]),m=0..20); # Robert Israel, Mar 05 2017

Select[(#^2-1)/9&/@Fibonacci[Range[100]],IntegerQ] (* Harvey P. Dale, Feb 07 2017 *)

lista(nn)=v = [1, 2, 6, 10, 11, 13, 14, 18, 22, 23]; for (n=1, nn, j = (n % #v) +1; k = n\#v; print1((fibonacci(24*k+v[j])^2-1)/9, ", "););

a(n)=fibonacci(n\5*12+[1, 2, 6, 10, 11][n%5+1])^2\9 \\ Charles R Greathouse IV, Jan 26 2017

G.f.: x^2*(7*x^10+336*x^9+880*x^8+6032*x^7+15792*x^6+16114*x^5+15792*x^4

+6032*x^3+880*x^2+336*x+7)/(-x^15+103683*x^10-103683*x^5+1). - Robert Israel, Mar 05 2017

cp's OEIS Frontend

A382845 Sum of the legs of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 = A000045(n) and its long leg and hypotenuse are consecutive natural numbers.

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A193917 Triangular array: the self-fusion of (p(n,x)), where p(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

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A194000 Triangular array: the self-fission of (p(n,x)), where sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

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A214729 Member m=6 of the m-family of sums b(m,n) = Sum_{k=0..n} F(k+m)*F(k), m >= 0, n >= 0, with the Fibonacci numbers F.

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A281619 Integer values of (A000045^2-1)/9 where A000045(m) is the m-th Fibonacci number.

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