A088305 -id:A088305 - cp's OEIS frontend

A372015 Product of Fibonacci and self-convolution of Fibonacci numbers: a(n) = A000045(n+1)*A001629(n+1).

0, 1, 4, 15, 50, 160, 494, 1491, 4420, 12925, 37380, 107136, 304764, 861445, 2421700, 6775755, 18879734, 52413856, 145038890, 400183575, 1101277060, 3023462521, 8282790024, 22646131200, 61805595000, 168399404425, 458128878724, 1244567262471, 3376576740410, 9149594423200

Offset: 0

Vladimir Kruchinin, Apr 15 2024

Conjecture: a(n) is the total number of pairs of adjacent parts that are the same color in all n-color compositions of n+1. - John Tyler Rascoe, Jul 30 2024

John Tyler Rascoe, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (5,-5,-5,5,-1).

Cf. A000045, A001629, A088305.

a := proc(n) option remember; if n < 3 then return n^2 fi;
-((2 - 2*n^2 + n)*a(n - 1) + (1 - 2*n^2 + 3*n)*a(n - 2) + n^2*a(n - 3))/(n - 1)^2 end: seq(a(n), n = 0..29);  # Peter Luschny, Apr 16 2024

CoefficientList[Series[x(1-x)/((1+x)*(1-3*x+x^2)^2),{x,0,29}],x] (* Stefano Spezia, Apr 16 2024 *)

A_x(N)= {my(x='x+O('x^N)); concat([0],Vec(x*(1-x)/((1+x)*(1-3*x+x^2)^2)))}
A_x(40) \\ John Tyler Rascoe, Jul 29 2024

a(n) = F(n+1)*((n+2)*F(n) + (n)*F(n+2))/5 where F(n) = A000045(n) is the Fibonacci numbers.

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

A383101 Number of compositions of n such that any part 1 can be m different colors where m is the largest part of the composition.

Original entry on oeis.org

1, 1, 2, 6, 21, 77, 294, 1178, 4978, 22191, 104146, 513385, 2653003, 14349804, 81125023, 478686413, 2943737942, 18838530436, 125268429098, 864256288435, 6177766228172, 45689641883377, 349173454108407, 2754058599745239, 22393206702946457, 187501022603071090

Offset: 0

John Tyler Rascoe, Apr 16 2025

			a(3) = 6 counts: (3), (2,1_a), (2,1_b), (1_a,2), (1_b,2), (1_a,1_a,1_a).

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A011782, A088305, A105422, A171632, A363262, A382991, A382992.

b:= proc(n, p, m) option remember; binomial(n+p, n)*
      m^n+add(b(n-j, p+1, max(m, j)), j=2..n)
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 23 2025

A_x(N) = {my(x='x+O('x^N)); Vec(1+sum(m=1,N, x^m/((1-m*x-(x^2-x^m)/(1-x))*(1-m*x-(x^2-x^(m+1))/(1-x)))))}
A_x(30)

G.f.: 1 + Sum_{m>0} x^m/((1 - m*x - (x^2 - x^m)/(1 - x)) * (1 - m*x - (x^2 - x^(m+1))/(1 - x))).

A383175 Number of compositions of n such that any fixed point k can be k different colors.

Original entry on oeis.org

1, 1, 2, 5, 10, 22, 48, 101, 213, 450, 945, 1961, 4064, 8385, 17242, 35332, 72141, 146924, 298552, 605377, 1225277, 2475912, 4995754, 10067848, 20267680, 40762951, 81916919, 164504411, 330155437, 662265817, 1327860471, 2661376529, 5332341881, 10680912173

Offset: 0

John Tyler Rascoe, Apr 18 2025

			a(3) = 5 counts: (3), (2,1), (1_a,2_a), (1_a,2_b), (1_a,1,1).

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A011782, A088305, A238349, A238350, A238351, A335713, A352512.

b:= proc(n, i) option remember; `if`(n=0, 1, add(
     `if`(n<=i+j, ceil(2^(n-j-1)), b(n-j, i+1))*
     `if`(i=j, j, 1), j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..33);  # Alois P. Heinz, Apr 18 2025

A_x(N) = {my(x='x+O('x^N)); Vec(1+sum(i=1,N, prod(j=1,i, j*x^j-x^j+x/(1-x))))}
A_x(30)

G.f.: 1 + Sum_{i>0} Product_{j=1..i} ( j*x^j - x^j + x/(1-x) ).

A141688 Triangle T(n, k) = Fibonacci(2k)T(n-1, k) + Fibonacci(2(n-k+1))T(n-1, k-1), with T(n, 1) = T(n, n) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 26, 26, 1, 1, 99, 416, 99, 1, 1, 352, 5407, 5407, 352, 1, 1, 1200, 62616, 227094, 62616, 1200, 1, 1, 3977, 673728, 8212854, 8212854, 673728, 3977, 1, 1, 12918, 6889153, 269486766, 903413940, 269486766, 6889153, 12918, 1, 1, 41338, 67863290, 8256432767, 88493861004, 88493861004, 8256432767, 67863290, 41338, 1

Offset: 1

Roger L. Bagula, Sep 09 2008

Row sums are: {1, 2, 8, 54, 616, 11520, 354728, 17781120, 1456191616, 193636396800, ...}.

			Triangle begins as:
  1;
  1,     1;
  1,     6,       1;
  1,    26,      26,         1;
  1,    99,     416,        99,         1;
  1,   352,    5407,      5407,       352,        1;
  1,  1200,   62616,    227094,     62616,     1200,       1;
  1,  3977,  673728,   8212854,   8212854,   673728,    3977,     1;
  1, 12918, 6889153, 269486766, 903413940,269486766, 6889153, 12918, 1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A088305, A186314.

function T(n,k)
  if k eq 1 or k eq n then return 1;
  else return Fibonacci(2*(n-k+1))*T(n-1, k-1) + Fibonacci(2*k)*T(n-1, k);
  end if; return T;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 29 2021

(* First program *)
b[n_]:= b[n]= If[n==0, 1, Sum[k*b[n-k], {k,n}]];
T[n_, k_]:= If[k==1 || k==n, 1, b[n-k+1]*T[n-1, k-1] + b[k]*T[n-1, k]];
Table[T[n, k], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 29 2021 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, Fibonacci[2*(n-k+1)]*T[n-1, k-1] + Fibonacci[2*k]*T[n-1, k]];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Mar 29 2021 *)

@CachedFunction
def T(n,k): return 1 if (k==1 or k==n) else fibonacci(2*(n-k+1))*T(n-1, k-1) + fibonacci(2*k)*T(n-1, k)
flatten([[T(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 29 2021

Let A088305(n) be defined by b(n) = Sum_{j=1..n} j*b(n-j), with b(0)=1, then T(n, k) = b(n-k+1)*T(n-1, k-1) + b(k)*T(n-1, k) with T(n,1) = T(n,n) = 1.
From G. C. Greubel, Mar 29 2021: (Start)
T(n, k) = Fibonacci(2*k)*T(n-1, k) + Fibonacci(2*(n-k+1))*T(n-1, k-1), with T(n, 1) = T(n, n) = 1.
T(n, 2) = A186314(n+1). (End)

Edited by G. C. Greubel, Mar 29 2021

A143929 Eigentriangle by rows, termwise products of the natural numbers decrescendo and the bisected Fibonacci series.

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 4, 3, 6, 8, 5, 4, 9, 16, 21, 6, 5, 12, 24, 42, 55, 7, 6, 15, 32, 63, 110, 144, 8, 7, 18, 40, 84, 165, 288, 377, 9, 8, 21, 48, 105, 220, 432, 754, 987, 10, 9, 24, 56, 126, 275, 576, 1131, 1974, 2584

Offset: 1

Gary W. Adamson, Sep 05 2008

Row sums = even-indexed Fibonacci terms A001906.

Sum of n-th row terms = rightmost term of next row.

			First rows of the triangle T(n, m), n >= 1, m = 1..n:
  1;
  2, 1;
  3, 2,  3;
  4, 3,  6,  8;
  5, 4,  9, 16,  21;
  6, 5, 12, 24,  42,  55;
  7, 6, 15, 32,  63, 110, 144;
  8, 7, 18, 40,  84, 165, 288, 377;
  9, 8, 21, 48, 105, 220, 432, 754, 987;
  ...
Example: row 4 = (4, 3, 6, 8) = termwise product of (4, 3, 2, 1) and (1, 1, 3, 8).

Cf. A000045, A001906, A004736, A088305.

Given A004736: (1; 2,1; 3,2,1; 4,3,2,1; ...), we apply the termwise products of the sequence {A088305(n-1)}_{n>=1} starting (1, 1, 3, 8, 21, ...).
From Wolfdieter Lang, Jan 07 2021: (Start)
T(n, m) = 0 if n < m; T(n, 1) = n, for n >= 1, and T(n, m) = F(2*(m-1))*(n-m+1) for n >= m >= 2, with F = A000045 (Fibonacci).
G.f. column m: G(1, x) = x/(1-x)^2, G(m, x) = F(2*(m-1))*x^m/(1-x)^2, for m >= 2. (End)
With offset 0: g.f. of row polynomials R(n, x) := Sum_{m=0..n} t(n, m)*x^m, that is, g.f. of triangle t(n,m) = T(n+1, m+1):
G(z, x) = (1 - x*z)^2 / ((1 - z)^2*(1 - 3*x*z + (x*z)^2)). - Wolfdieter Lang, Apr 09 2021

A259714 a(n) = Sum_{k=1..n-1}((k mod 5)*a(n-k)), a(1) = 1.

Original entry on oeis.org

1, 1, 3, 8, 21, 50, 129, 327, 827, 2089, 5290, 13386, 33868, 85693, 216836, 548660, 1388269, 3512737, 8888292, 22490049, 56906580, 143990771, 364339983, 921889753, 2332658401, 5902327520, 14934664284, 37789193522, 95618028007, 241942376384

Offset: 1

Anders Hellström, Jul 03 2015

nonn

Anders Hellström, Table of n, a(n) for n = 1..1000

Cf. A088305 (sequence obtained without mod 5 in the formula).

f[n_] := Block[{k, a = {1}}, Do[AppendTo[a, Sum[Mod[k, 5] a[[i - k]], {k, 1, i - 1}]], {i, 2, n}]; a]; f@ 30 (* Michael De Vlieger, Jul 03 2015 *)

main(size)=my(v=vector(size),n,s); v[1]=1; for(n=2, size, for(s=1, n-1, v[n] = v[n] + (s%5)*v[n-s] )); v;

Conjectures from Colin Barker, Jul 04 2015: (Start)
a(n) = a(n-1)+2*a(n-2)+3*a(n-3)+4*a(n-4)+a(n-5) for n>6.
G.f.: x*(x-1)*(x^4+x^3+x^2+x+1) / ((x+1)*(x^4+3*x^3+2*x-1)).
(End)

A296106 Square array T(n,k) n >= 1, k >= 1 read by antidiagonals: T(n, k) is the number of distinct Bojagi boards with dimensions n X k that have a unique solution.

Original entry on oeis.org

1, 3, 3, 8, 17, 8, 21, 130, 130, 21, 55, 931, 2604, 931, 55, 144, 6871, 54732, 54732, 6871, 144, 377, 50778

Offset: 1

Taotao Liu, Dec 04 2017

Bojagi is a puzzle game created by David Radcliffe.

A Bojagi board is a rectangular board with some cells empty and some cells containing positive integers. A solution for a Bojagi board partitions the board into rectangles such that each rectangle contains exactly one integer, and that integer is the area of the rectangle.

			Array begins:
======================================
n\k|   1    2     3     4    5   6
---+----------------------------------
1  |   1    3     8    21   55 144 ...
2  |   3   17   130   931 6871 ...
3  |   8  130  2604 54732 ...
4  |  21  931 54732 ...
5  |  55 6871 ...
6  | 144 ...
...
As a triangle:
    1;
    3,    3;
    8,   17,     8;
   21,  130,   130,    21;
   55,  931,  2604,   931,   55;
  144, 6871, 54732, 54732, 6871, 144;
  ...
If n=1 or k=1, any valid board (a board whose numbers add up to the area of the board) has a unique solution.
For n=2 and k=2, there are 17 boards that have a unique solution. There is 1 board in which each of the four cells has a 1.
There are 4 boards which contain two 2's. The 2's must be adjacent (not diagonally opposite) in order for the board to have a unique solution.
There are 8 boards which contain one 2 and two 1's. The 1's must be adjacent in order for the board to have a solution. The 2 can be placed in either of the remaining two cells.
There are 4 boards which contain one 4. It can be placed anywhere.

Taotao Liu, Thomas Ledbetter C# Program
David Radcliffe, Rules of puzzle game Bojagi

Cf. A088305.

T(n,1) = A088305(n), the even-indexed Fibonacci numbers.

T(n,1) = Sum_{i=1..n} i*T(n-i,1) if we take T(0,1) = 1.

A320251 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/(1 - Sum_{j>=1} j^k*x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 5, 8, 8, 1, 1, 9, 18, 21, 16, 1, 1, 17, 44, 63, 55, 32, 1, 1, 33, 114, 207, 221, 144, 64, 1, 1, 65, 308, 723, 991, 776, 377, 128, 1, 1, 129, 858, 2631, 4805, 4752, 2725, 987, 256, 1, 1, 257, 2444, 9843, 24655, 31880, 22769, 9569, 2584, 512

Offset: 0

Ilya Gutkovskiy, Oct 08 2018

A(n,k) is the invert transform of k-th powers evaluated at n.

			G.f. of column k: A_k(x) = 1 + x + (2^k + 1)*x^2 + (2^(k + 1) + 3^k + 1)*x^3 + (3*2^k + 2^(2*k + 1) + 2*3^k + 1)*x^4 + ...
Square array begins:
   1,   1,    1,    1,     1,      1,  ...
   1,   1,    1,    1,     1,      1,  ...
   2,   3,    5,    9,    17,     33,  ...
   4,   8,   18,   44,   114,    308,  ...
   8,  21,   63,  207,   723,   2631,  ...
  16,  55,  221,  991,  4805,  24655,  ...

N. J. A. Sloane, Transforms

Columns k=0..3 give A011782, A088305, A033453, A144109.
Main diagonal gives A301655.
Cf. A144048.

Table[Function[k, SeriesCoefficient[1/(1 - Sum[i^k x^i, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 - PolyLog[-k, x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

G.f. of column k: 1/(1 - PolyLog(-k,x)), where PolyLog() is the polylogarithm function.

A374925 Number of n-color compositions of n having at least one pair of adjacent parts that are the same color.

Original entry on oeis.org

0, 0, 1, 3, 10, 31, 91, 259, 726, 2007, 5489, 14888, 40122, 107574, 287239, 764405, 2028679, 5371858, 14198008, 37467982, 98749767, 259984452, 683865318, 1797500121, 4721662597, 12396308875, 32531025970, 85337831350, 223794544179, 586736215856, 1537941527011

Offset: 0

John Tyler Rascoe, Jul 24 2024

			a(4) = 10 counts: (1,1,1,1), (1,1,2_a), (1,1,2_b), (1,2_a,1), (1,3_a), (2_a,1,1), (2_a,2_a), (2_b,1,1), (2_b,2_b), (3_a,1).

Cf. A003242, A088305, A242551, A330774, A374727, A374728.

C_x(N) = {my(x='x+O('x^N), h=(sum(i=1,N,(x^(2*i))/((1-x)*(1-x+x^i)*(1-sum(j=1,N, (x^j)/(1-x+x^j))))))/(1-sum(i=1,N,(x^i)/(1-x)))); concat([0,0],Vec(h))}
C_x(40)

G.f.: Sum_{i>0} ( x^(2*i)/((1 - x)*(1 - x + x^i)*(1 - Sum_{j>0} ((x^j)/(1 - x + x^j)))) )/( 1 - Sum_{k>0} ((x^k)/(1 - x)) ).

a(n) = A088305(n) - A242551(n).

A383275 Number of compositions of n such that any part 1 can be k different colors where k is the current record having appeared in the composition.

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 134, 454, 1634, 6245, 25321, 108779, 494443, 2374288, 12024257, 64100444, 358948674, 2106756217, 12931155910, 82823317389, 552400947902, 3829070637080, 27534807426150, 205066734143893, 1579309451332366, 12559941159979791, 103013928588389695

Offset: 0

John Tyler Rascoe, Apr 21 2025

A record in a composition is a part that is greater than all parts before it, reading left to right. The first part of any nonempty composition is considered a record. A part 1 can be a record, iff it is the first part of a composition.

			a(3) = 5: (3), (1_a,2), (2,1_a), (2,1_b), (1_a,1_a,1_a).

Alois P. Heinz, Table of n, a(n) for n = 0..600

Cf. A000108, A011782, A088305, A382312, A382991, A383101, A383175.

b:= proc(n, m) option remember; `if`(n=0, 1, add(
      b(n-j, max(j, m))*`if`(j=1, m, 1), j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..26);  # Alois P. Heinz, Apr 23 2025

A_x(N) = {my(x='x+O('x^N)); Vec(prod(i=1,N,1+x^i/(1-i*x+(-x^2+x^(i+1))/(1-x))))}
A_x(30)

G.f.: Product_{i>0} 1 + x^i/(1 - i*x - (x^2 - x^(i+1))/(1-x)).

cp's OEIS Frontend

A372015 Product of Fibonacci and self-convolution of Fibonacci numbers: a(n) = A000045(n+1)*A001629(n+1).

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Maple

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A383101 Number of compositions of n such that any part 1 can be m different colors where m is the largest part of the composition.

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Maple

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A383175 Number of compositions of n such that any fixed point k can be k different colors.

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Maple

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A141688 Triangle T(n, k) = Fibonacci(2*k)*T(n-1, k) + Fibonacci(2*(n-k+1))*T(n-1, k-1), with T(n, 1) = T(n, n) = 1, read by rows.

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Magma

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A143929 Eigentriangle by rows, termwise products of the natural numbers decrescendo and the bisected Fibonacci series.

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A259714 a(n) = Sum_{k=1..n-1}((k mod 5)*a(n-k)), a(1) = 1.

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Mathematica

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A296106 Square array T(n,k) n >= 1, k >= 1 read by antidiagonals: T(n, k) is the number of distinct Bojagi boards with dimensions n X k that have a unique solution.

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A320251 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/(1 - Sum_{j>=1} j^k*x^j).

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A141688 Triangle T(n, k) = Fibonacci(2k)T(n-1, k) + Fibonacci(2(n-k+1))T(n-1, k-1), with T(n, 1) = T(n, n) = 1, read by rows.