A333650 -id:A333650 - cp's OEIS frontend

A339493 a(n) = A333650(n, 3) for n >= 3; a(1) = a(2) = 0.

0, 0, 4, 11, 24, 52, 110, 228, 467, 949, 1916, 3848, 7696, 15342, 30505, 60527, 119892, 237156, 468590, 925032, 1824727, 3597289, 7088216, 13961168, 27489168, 54110618, 106489245, 209531251, 412217072, 810865812, 1594878078, 3136673996, 6168522827, 12130206941

Offset: 1

Peter Luschny, Dec 07 2020

nonn

Cf. A333650, A339494.

A339346 a(n) is the number of domino towers of height n consisting of 2n bricks, a(n) = A333650(2n, n).

Original entry on oeis.org

1, 4, 24, 168, 1257, 9778, 77994, 633056, 5204434

Offset: 1

Peter Luschny, Dec 01 2020

Cf. A333650.

A339494 T(n, k) is the number of domino towers of n bricks with height at most 3 and k bricks in the base floor. Triangle read by rows, T(n, k) for 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 5, 9, 4, 1, 3, 14, 14, 5, 1, 1, 16, 29, 20, 6, 1, 0, 12, 46, 51, 27, 7, 1, 0, 5, 52, 101, 81, 35, 8, 1, 0, 1, 41, 150, 190, 120, 44, 9, 1, 0, 0, 22, 169, 345, 323, 169, 54, 10, 1, 0, 0, 7, 143, 495, 687, 511, 229, 65, 11, 1

Offset: 1

Peter Luschny, Dec 07 2020

This is the third triangle in a sequence of triangles: The first is the unit triangle A023531; the second is the binomial triangle C(k, n-k) without the first column, triangle A030528. This triangle highlights the connection between the Pascal triangle and the Fibonacci numbers in the case m = 2. Similarly, the current triangle and its row sums generalizes this to the case m = 3 of the construction of Union(A333650(n, j), j=1..m), classified by the number of bricks in the base floor.

			Triangle starts:        n: [row] sum
                          1: [1] 1
                         2: [2, 1] 3
                       3: [5, 3, 1] 9
                     4: [5, 9, 4, 1] 19
                   5: [3, 14, 14, 5, 1] 37
                 6: [1, 16, 29, 20, 6, 1] 73
              7: [0, 12, 46, 51, 27, 7, 1] 144
            8: [0, 5, 52, 101, 81, 35, 8, 1] 283
         9: [0, 1, 41, 150, 190, 120, 44, 9, 1] 556
     10: [0, 0, 22, 169, 345, 323, 169, 54, 10, 1] 1093

Peter Luschny, Domino towers classified by the size of the base floor. Illustrating row 4 of the triangle.

Cf. A339495 (row sums), A333650, A030528, A023531.

A339029 Expansion of (1 + 4x - 20x^2 + 8x^3 + 33x^4 - 4x^5 - 33x^6)/(1 - 2*x)^4.

Original entry on oeis.org

1, 12, 52, 168, 497, 1412, 3879, 10360, 27016, 69024, 173264, 428288, 1044480, 2516992, 6001408, 14174208, 33191936, 77127680, 177967104, 408027136, 930021376, 2108424192, 4756275200, 10680270848, 23880794112, 53185871872, 118016180224, 260969594880, 575223627776

Offset: 0

Peter Luschny, Dec 01 2020

nonn

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A339252 (k=2), A339254 (k=3), A333650.

gf := (1 + 4*x - 20*x^2 + 8*x^3 + 33*x^4 - 4*x^5 - 33*x^6)/(1 - 2*x)^4:
ser := series(gf, x, 32): seq(coeff(ser, x, n), n = 0..28);

LinearRecurrence[{8, -24, 32, -16}, {1, 12, 52, 168, 497, 1412, 3879}, 30] (* Paolo Xausa, Feb 01 2024 *)

a(n) = 2^(n-7)*(588 + 367*n + 84*n^2 + 9*n^3) for n >= 3.

A339495 Row sums of A339494.

Original entry on oeis.org

1, 3, 9, 19, 37, 73, 144, 283, 556, 1093, 2149, 4225, 8306, 16329, 32102, 63111, 124073, 243921, 479536, 942743, 1853384, 3643657, 7163241, 14082561, 27685586, 54428429, 107003474, 210363291, 413563341, 813044121, 1598402656, 3142376883, 6177750292, 12145137293

Offset: 1

Peter Luschny, Dec 07 2020

nonn

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

Cf. A339494, A339493, A000045, A333650.

a(n) = A339493(n) + Fibonacci(n + 2) for n >= 2.

a(n) = [x^n] -x*(x*(x + 1)*(x*(x*(x*(x*(x + 2) + 2) + 3) + 3) + 2) + 1)/(x^5 + x^4 + x^3 + x^2 + x - 1).

cp's OEIS Frontend

A339493 a(n) = A333650(n, 3) for n >= 3; a(1) = a(2) = 0.

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A339346 a(n) is the number of domino towers of height n consisting of 2n bricks, a(n) = A333650(2n, n).

Views

Author

Keywords

Crossrefs

A339494 T(n, k) is the number of domino towers of n bricks with height at most 3 and k bricks in the base floor. Triangle read by rows, T(n, k) for 1 <= k <= n.

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A339029 Expansion of (1 + 4x - 20x^2 + 8x^3 + 33x^4 - 4x^5 - 33x^6)/(1 - 2*x)^4.

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Maple

Mathematica

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A339495 Row sums of A339494.

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Formula

cp's OEIS Frontend

A339493 a(n) = A333650(n, 3) for n >= 3; a(1) = a(2) = 0.

Views

Author

Keywords

Crossrefs

A339346 a(n) is the number of domino towers of height n consisting of 2*n bricks, a(n) = A333650(2*n, n).

Views

Author

Keywords

Crossrefs

A339494 T(n, k) is the number of domino towers of n bricks with height at most 3 and k bricks in the base floor. Triangle read by rows, T(n, k) for 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A339029 Expansion of (1 + 4*x - 20*x^2 + 8*x^3 + 33*x^4 - 4*x^5 - 33*x^6)/(1 - 2*x)^4.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A339495 Row sums of A339494.

Views

Author

Keywords

Links

Crossrefs

Formula

A339346 a(n) is the number of domino towers of height n consisting of 2n bricks, a(n) = A333650(2n, n).

A339029 Expansion of (1 + 4x - 20x^2 + 8x^3 + 33x^4 - 4x^5 - 33x^6)/(1 - 2*x)^4.