A079957 -id:A079957 - cp's OEIS frontend

A378716 Triangle read by rows: T(n,k) is the number of k-Fibonacci polyominoes with an area of n, with k > 1.

1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 2, 2, 0, 0, 1, 3, 0, 1, 0, 0, 1, 4, 2, 1, 0, 0, 0, 1, 5, 3, 1, 1, 0, 0, 0, 1, 7, 1, 1, 1, 0, 0, 0, 0, 1, 9, 5, 2, 0, 1, 0, 0, 0, 0, 1, 12, 5, 1, 1, 1, 0, 0, 0, 0, 0, 1, 16, 3, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1, 21, 10, 3, 3, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 3

Stefano Spezia, Dec 05 2024

			The triangle begins as:
   1;
   1, 1;
   1, 0, 1;
   2, 1, 0, 1;
   2, 2, 0, 0, 1;
   3, 0, 1, 0, 0, 1;
   4, 2, 1, 0, 0, 0, 1;
   5, 3, 1, 1, 0, 0, 0, 1;
   7, 1, 1, 1, 0, 0, 0, 0, 1;
   9, 5, 2, 0, 1, 0, 0, 0, 0, 1;
  12, 5, 1, 1, 1, 0, 0, 0, 0, 0, 1;
  ...

Juan F. Pulido, José L. Ramírez, and Andrés R. Vindas-Meléndez, Generating Trees and Fibonacci Polyominoes, arXiv:2411.17812 [math.CO], 2024. See page 9.

Cf. A079957 (k=3), A182097 (k=2), A378704, A378706, A378707.

T[n_, k_]:=SeriesCoefficient[1/(1-Sum[x^((k+i)(k-i+1)/2), {i, k}]), {x, 0, n}]; Table[T[n, k], {n, 2, 14}, {k, 2,n}]//Flatten

T(n, k) = [x^n] 1/(1 - Sum_{i=1..k} x^((k+i)*(k-i+1)/2) ).

A245367 Compositions of n into parts 3, 5 and 7.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 3, 3, 3, 6, 5, 8, 10, 11, 17, 18, 25, 32, 37, 52, 61, 79, 102, 123, 163, 200, 254, 326, 402, 519, 649, 819, 1045, 1305, 1664, 2096, 2643, 3358, 4220, 5352, 6759, 8527, 10806, 13622, 17237, 21785, 27501, 34802, 43934, 55544, 70209, 88672, 112131, 141644, 179018, 226274, 285860, 361358

Offset: 0

David Neil McGrath, Aug 20 2014

			a(16) = 10: the compositions are the permutations of [5533] (there are 4!/2!2!=6 of them) and the permutations of [7333] (there are 4!/3!=4).

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

Cf. A000073, A013979, A017818, A060945, A079956, A079957, A079971, A079973, A120400, A121833.

LinearRecurrence[{0,0,1,0,1,0,1},{1,0,0,1,0,1,1},70] (* Harvey P. Dale, Jan 27 2017 *)

Vec(1/(1-x^3-x^5-x^7) +O(x^66)) \\ Joerg Arndt, Aug 20 2014

G.f: 1/(1-x^3-x^5-x^7).

a(n) = a(n-3) + a(n-5) + a(n-7).

A245369 Number of compositions of n into parts 3, 5 and 8.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 0, 3, 1, 1, 5, 1, 5, 7, 2, 13, 9, 8, 25, 12, 26, 41, 22, 64, 62, 56, 130, 96, 146, 233, 174, 340, 391, 376, 703, 661, 862, 1327, 1211, 1905, 2379, 2449, 3935, 4251, 5216, 7641, 7911, 11056, 14271, 15576, 22632, 26433, 31848, 44544, 49920, 65536, 85248, 97344, 132712, 161601, 194728, 262504, 308865

Offset: 0

David Neil McGrath, Aug 23 2014

			a(19)=25. The compositions of 19 into parts 3, 5, and 8 are the permutations of (883) (these are 3!/2!=3), (8533) (these are 4!/2!=12), and (55333) (these are 5!/3!2!=10).

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

Cf. A079957, A245367.

LinearRecurrence[{0,0,1,0,1,0,0,1},{1,0,0,1,0,1,1,0},70] (* Harvey P. Dale, Sep 05 2022 *)

Vec( 1/(1-x^3-x^5-x^8) +O(x^66) ) \\ Joerg Arndt, Aug 25 2014

G.f.: 1/(1-x^3-x^5-x^8).

a(n) = a(n-3) + a(n-5) + a(n-8).

A245370 Number of compositions of n into parts 3, 5 and 9.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 0, 2, 2, 1, 3, 3, 3, 6, 5, 6, 11, 10, 13, 19, 19, 27, 35, 37, 52, 65, 74, 100, 121, 145, 192, 230, 282, 365, 440, 548, 695, 843, 1058, 1327, 1621, 2035, 2535, 3119, 3910, 4851, 5997, 7503, 9297, 11528, 14389, 17829, 22150, 27596, 34208, 42536, 52928, 65655, 81660, 101525, 126020, 156738, 194776, 241888

Offset: 0

David Neil McGrath, Aug 24 2014

			a(28)=100 The compositions of n into parts 3,5 and 9 are the permutations of (9955)(these are 4!/2!2!=6), (555553) (these are 6!/5!=6), (955333) (these are 6!/3!2!=60), (55333333) (these are 8!/6!2!=28).

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

Cf. A079957, A245367, A245369.

Vec( 1/(1-x^3-x^5-x^9) +O(x^66) ) \\ Joerg Arndt, Aug 24 2014

G.f.: 1/(1-x^3-x^5-x^9).

a(n) = a(n-3) + a(n-5) + a(n-9).

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A378716 Triangle read by rows: T(n,k) is the number of k-Fibonacci polyominoes with an area of n, with k > 1.

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A245367 Compositions of n into parts 3, 5 and 7.

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A245369 Number of compositions of n into parts 3, 5 and 8.

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A245370 Number of compositions of n into parts 3, 5 and 9.

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