A055819 -id:A055819 - cp's OEIS frontend

A236331 Positive integers n such that x^2 - 18xy + y^2 + n = 0 has integer solutions.

64, 256, 320, 576, 704, 1024, 1216, 1280, 1600, 1856, 1984, 2304, 2624, 2816, 2880, 3136, 3520, 3776, 3904, 4096, 4544, 4864, 5056, 5120, 5184, 5696, 6080, 6336, 6400, 6464, 6976, 7424, 7744, 7936, 8000, 8384, 8896, 9216, 9280, 9536, 9664, 9920, 10496, 10816

Offset: 1

Colin Barker, Feb 16 2014

nonn

			64 is in the sequence because x^2 - 18xy + y^2 + 64 = 0 has integer solutions, for example (x, y) = (1, 13).

Cf. A001519 (n = 64), A052995 (n = 256), A055819 (n = 256), A005248 (n = 320), A237132 (n = 704), A237133 (n = 1216).

Cf. A031363, A084917, A237351, A237599, A237606, A237609, A237610, A236330.

A237133 Values of x in the solutions to x^2 - 3xy + y^2 + 19 = 0, where 0 < x < y.

Original entry on oeis.org

4, 5, 7, 11, 17, 28, 44, 73, 115, 191, 301, 500, 788, 1309, 2063, 3427, 5401, 8972, 14140, 23489, 37019, 61495, 96917, 160996, 253732, 421493, 664279, 1103483, 1739105, 2888956, 4553036, 7563385, 11920003, 19801199, 31206973, 51840212, 81700916, 135719437

Offset: 1

Colin Barker, Feb 04 2014

The corresponding values of y are given by a(n+2).

Positive values of x (or y) satisfying x^2 - 18xy + y^2 + 1216 = 0.

			11 is in the sequence because (x, y) = (11, 28) is a solution to x^2 - 3xy + y^2 + 19 = 0.

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

Cf. A001519, A005248, A055819, A237132, A218735.

LinearRecurrence[{0,3,0,-1},{4,5,7,11},40] (* Harvey P. Dale, Dec 15 2014 *)

Vec(-x*(x-1)*(4*x^2+9*x+4)/((x^2-x-1)*(x^2+x-1)) + O(x^100))

a(n) = 3*a(n-2)-a(n-4).
G.f.: -x*(x-1)*(4*x^2+9*x+4) / ((x^2-x-1)*(x^2+x-1)).
a(n) = (1/2) * (F(n+4) + (-1)^n*F(n-5)), n>4, with F the Fibonacci numbers (A000045). - Ralf Stephan, Feb 05 2014

A218735 Values of x in the solutions to x^2 - 3xy + y^2 + 29 = 0, where 0 < x < y.

Original entry on oeis.org

5, 6, 9, 13, 22, 33, 57, 86, 149, 225, 390, 589, 1021, 1542, 2673, 4037, 6998, 10569, 18321, 27670, 47965, 72441, 125574, 189653, 328757, 496518, 860697, 1299901, 2253334, 3403185, 5899305, 8909654, 15444581, 23325777, 40434438, 61067677, 105858733

Offset: 1

Colin Barker, Feb 05 2014

The corresponding values of y are given by a(n+2).

Positive values of x (or y) satisfying x^2 - 18xy + y^2 + 1856 = 0.

			13 is in the sequence because (x, y) = (13, 33) is a solution to x^2 - 3xy + y^2 + 29 = 0.

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

Cf. A001519, A005248, A055819, A237132, A237133.

LinearRecurrence[{0,3,0,-1},{5,6,9,13},40] (* Harvey P. Dale, Nov 30 2024 *)

Vec(-x*(x-1)*(5*x^2+11*x+5)/((x^2-x-1)*(x^2+x-1)) + O(x^100))

a(n) = 3*a(n-2)-a(n-4).

G.f.: -x*(x-1)*(5*x^2+11*x+5) / ((x^2-x-1)*(x^2+x-1)).

A105292 Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n, having leftmost column of height k.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 4, 3, 1, 13, 10, 6, 4, 1, 34, 26, 15, 8, 5, 1, 89, 68, 39, 20, 10, 6, 1, 233, 178, 102, 52, 25, 12, 7, 1, 610, 466, 267, 136, 65, 30, 14, 8, 1, 1597, 1220, 699, 356, 170, 78, 35, 16, 9, 1, 4181, 3194, 1830, 932, 445, 204, 91, 40, 18, 10, 1, 10946, 8362

Offset: 1

Emeric Deutsch, Apr 25 2005

T(n,k) is the number of nondecreasing Dyck paths of semilength n, having height of leftmost peak equal to k. Example: T(3,2)=2 because we have UUDDUD and UUDUDD, where U=(1,1) and D(1,-1). Sum of row n = fibonacci(2n-1) (A001519). T(n,1)=fibonacci(2n-3) (A001519). Column 2 yields A055819.

			Triangle begins:
  1;
  1,1;
  2,2,1;
  5,4,3,1;
  13,10,6,4,1;

Elena Barcucci, Alberto del Lungo, S. Fezzi, and Renzo Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
Emeric Deutsch and Helmut Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.
Juan B. Gil, Felix H. Xu, and William Y. Zhu, Odd-indexed Fibonacci numbers via pattern-avoiding permutations, arXiv:2506.15800 [math.CO], 2025. See p. 4.

Cf. A001519, A055819.

Maple
```
with(combinat):T:=proc(n,k) if k
				
```

Flatten[Join[{1},#]&/@Table[k*Fibonacci[2n-2k-1],{n,15},{k,n-1}]] (* Harvey P. Dale, Aug 21 2013 *)

T(n, k)=k*fibonacci(2n-2k-1) if k

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A236331 Positive integers n such that x^2 - 18xy + y^2 + n = 0 has integer solutions.

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A237133 Values of x in the solutions to x^2 - 3xy + y^2 + 19 = 0, where 0 < x < y.

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A218735 Values of x in the solutions to x^2 - 3xy + y^2 + 29 = 0, where 0 < x < y.

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A105292 Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n, having leftmost column of height k.

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