A123231 -id:A123231 - cp's OEIS frontend

A233323 Triangle read by rows: T(n,k) = number of palindromic compositions of n in which the largest part is equal to k, 1 <= k <= n.

1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 1, 0, 1, 1, 4, 1, 1, 0, 1, 1, 2, 3, 0, 1, 0, 1, 1, 7, 3, 3, 0, 1, 0, 1, 1, 4, 6, 1, 2, 0, 1, 0, 1, 1, 12, 7, 7, 1, 2, 0, 1, 0, 1, 1, 7, 12, 3, 5, 0, 2, 0, 1, 0, 1, 1, 20, 16, 15, 3, 5, 0, 2, 0, 1, 0, 1

Offset: 1

L. Edson Jeffery, Dec 10 2013

A palindromic composition of a natural number m is an ordered partition of m into N+1 natural numbers (or parts), p_0, p_1, ..., p_N, of the form m = p_0 + p_1 + ... + p_N such that p_j = p_{N-j}, for each j in {0,...,N}. Two palindromic compositions, sum_{j=0..N} p_j and sum_{j=0..N} q_j (say), are identical if and only if p_j = q_j, j = 0,...,N; otherwise they are taken to be distinct.

			There are eight palindromic compositions of n=7, namely, {7}, {3,1,3}, {2,3,2}, {2,1,1,1,2}, {1,5,1}, {1,2,1,2,1}, {1,1,3,1,1}, {1,1,1,1,1,1,1}, and three of them have the largest part equal to 3, so T(7,3) = 3.
Triangle T(n,k) begins:
  1;
  1,  1;
  1,  0, 1,
  1,  2, 0, 1;
  1,  1, 1, 0, 1;
  1,  4, 1, 1, 0, 1;
  1,  2, 3, 0, 1, 0, 1;
  1,  7, 3, 3, 0, 1, 0, 1;
  1,  4, 6, 1, 2, 0, 1, 0, 1;
  1, 12, 7, 7, 1, 2, 0, 1, 0, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened (rows n = 1..50 from Charles R Greathouse IV)
V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

Cf. A016116 (row sums), A233324 (row partial sums).

T(n,2)+1 = A053602(n+1) = A123231(n). T(4n-2,2n) = A011782(n-1). - Alois P. Heinz, Dec 11 2013

b:= proc(n, k) option remember; `if`(n<=k, 1, 0)+
      add(b(n-2*j, k), j=1..min(k, iquo(n, 2)))
    end:
T:= (n, k)-> b(n, k) -b(n, k-1):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Dec 11 2013

b[n_, k_] := b[n, k] = If[n <= k, 1, 0] + Sum[b[n-2*j, k], { j, 1, Min[k, Quotient[n, 2]]}]; t[n_, k_] := b[n, k] - b[n, k-1]; Table[Table[t[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Alois P. Heinz's Maple code *)

T(n,k,ok=0)={
    if(n<1,return(n==0 && ok));
    if(k>n/2 && !ok,
        n-=k;
        if(n<0||n%2, return(0));
        return(2^max(n/2-1,0))
    );
    sum(i=1,k,
        T(n-2*i,k,ok||i==k)
    )+(n==k || (ok && nCharles R Greathouse IV, Dec 11 2013

A233324 Triangle read by rows: T(n,k) = number of palindromic compositions of n in which no part exceeds k, 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 3, 3, 4, 1, 2, 3, 3, 4, 1, 5, 6, 7, 7, 8, 1, 3, 6, 6, 7, 7, 8, 1, 8, 11, 14, 14, 15, 15, 16, 1, 5, 11, 12, 14, 14, 15, 15, 16, 1, 13, 20, 27, 28, 30, 30, 31, 31, 32, 1, 8, 20, 23, 28, 28, 30, 30, 31, 31, 32, 1, 21, 37, 52, 55, 60, 60, 62, 62, 63, 63, 64

Offset: 1

L. Edson Jeffery, Dec 11 2013

A palindromic composition of a natural number m is an ordered partition of m into N+1 natural numbers (or parts), p_0, p_1, ..., p_N, of the form m = p_0 + p_1 + ... + p_N such that p_j = p_{N-j}, for each j in {0,...,N}. Two palindromic compositions, sum_{j=0..N} p_j and sum_{j=0..N} q_j (say), are identical if and only if p_j = q_j, j = 0,...,N; otherwise they are taken to be distinct.
Partial sums of rows of A233323.
T(n,k) is defined for n,k >= 0.  T(n,k) = T(n,n) = A016116(n) for k>= 0. - Alois P. Heinz, Dec 11 2013

			Triangle T(n,k) begins:
1;
1,  2;
1,  1,  2;
1,  3,  3,  4;
1,  2,  3,  3,  4;
1,  5,  6,  7,  7,  8;
1,  3,  6,  6,  7,  7,  8;
1,  8, 11, 14, 14, 15, 15, 16;
1,  5, 11, 12, 14, 14, 15, 15, 16;
1, 13, 20, 27, 28, 30, 30, 31, 31, 32;

Alois P. Heinz, Rows n = 1..141, flattened
V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

Cf. A233323.

T(n,2) = A053602(n+1) = A123231(n). T(2n,3) = A001590(n+3). T(2n,4) = A001631(n+4). - Alois P. Heinz, Dec 11 2013

T:= proc(n, k) option remember; `if`(n<=k, 1, 0)+
      add(T(n-2*j, k), j=1..min(k, iquo(n, 2)))
    end:
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Dec 11 2013

T[n_, k_] := T[n, k] = If[n <= k, 1, 0] + Sum[T[n-2*j, k], {j, 1, Min[k, Quotient[ n, 2]]}]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

T(n,k)=if(n<1,return(n==0));sum(i=1,k,T(n-2*i,k))+(n<=k) \\ Charles R Greathouse IV, Dec 11 2013

A272912 Difference sequence of the sequence A116470 of all distinct Fibonacci numbers and Lucas numbers (A000032).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 5, 3, 8, 5, 13, 8, 21, 13, 34, 21, 55, 34, 89, 55, 144, 89, 233, 144, 377, 233, 610, 377, 987, 610, 1597, 987, 2584, 1597, 4181, 2584, 6765, 4181, 10946, 6765, 17711, 10946, 28657, 17711, 46368, 28657, 75025, 46368, 121393, 75025

Offset: 1

Clark Kimberling, May 10 2016

Every term is a Fibonacci number (A000045).

			A116470 = (1, 2, 3, 4, 5, 7, 8, 11, 13, 18, 21, 29, 34, 47, 55, 76,...), so that (a(n)) = (1,1,1,1,2,1,3,2,5,3,8,5,13,8,12,...).

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Cf. A000045, A000032, A116470, A053602, A123231.

u = Table[Fibonacci[n], {n, 1, 200}]; v = Table[LucasL[n], {n, 1, 200}];
Take[Differences[Union[u, v]], 100]

Vec(x*(1+x-x^5)/(1-x^2-x^4) + O(x^50)) \\ Colin Barker, May 10 2016

From Colin Barker, May 10 2016: (Start)
a(n) = a(n-2)+a(n-4) for n>4.
G.f.: x*(1+x-x^5) / (1-x^2-x^4).
(End)
a(n) = A053602(n-2), n>2. - R. J. Mathar, May 20 2016
a(n) = A123231(n-3), n>3. - Georg Fischer, Oct 23 2018

A382478 Number of palindromic binary strings of length n having no 4-runs of 1's.

Original entry on oeis.org

1, 2, 2, 4, 3, 7, 6, 14, 12, 27, 23, 52, 44, 100, 85, 193, 164, 372, 316, 717, 609, 1382, 1174, 2664, 2263, 5135, 4362, 9898, 8408, 19079, 16207, 36776, 31240, 70888, 60217, 136641, 116072, 263384, 223736, 507689, 431265, 978602, 831290, 1886316, 1602363, 3635991, 3088654, 7008598, 5953572

Offset: 0

R. J. Mathar, Mar 28 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
M. A. Nyblom, Counting Palindromic Binary Strings Without r-Runs of Ones, J. Int. Seq. 16 (2013) #13.8.7, P_4(n).
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,1,0,1).

Cf. A001630 (bisection), A001631 (bisection).

Cf. A123231 (2-runs), A001590 (3-runs), A251653 (5-runs), A382479 (6-runs).

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(-(x^2+1)*(x^5+2*x+1) / (-1+x^2+x^4+x^6+x^8))); // Vincenzo Librandi, May 19 2025

LinearRecurrence[{0,1,0,1,0,1,0,1},{1,2,2,4,3,7,6,14},50] (* Vincenzo Librandi, May 19 2025 *)

G.f.: -(x^2+1)*(x^5+2*x+1)/(-1+x^2+x^4+x^6+x^8).

A382479 Number of palindromic binary strings of length n having no 6-runs of 1's.

Original entry on oeis.org

1, 2, 2, 4, 4, 8, 7, 15, 14, 30, 28, 60, 56, 119, 111, 236, 220, 468, 436, 928, 865, 1841, 1716, 3652, 3404, 7244, 6752, 14369, 13393, 28502, 26566, 56536, 52696, 112144, 104527, 222447, 207338, 441242, 411272, 875240, 815792, 1736111, 1618191, 3443720, 3209816, 6830904, 6366936

Offset: 0

R. J. Mathar, Mar 28 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
M. A. Nyblom, Counting Palindromic Binary Strings Without r-Runs of Ones, J. Int. Seq. 16 (2013) #13.8.7, P_6(n).
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,1,0,1,0,1,0,1).

Cf. A251707 (bisection), A251708 (bisection).

Cf. A123231 (2-runs), A001590 (3-runs), A382478 (4-runs), A251653 (5-runs).

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(-(1+x+x^2)*(x^2-x+1)*(x^7+2*x+1)/(-1+x^2+x^4+x^6+x^8+x^10+x^12))); // Vincenzo Librandi, May 20 2025

LinearRecurrence[{0,1,0,1,0,1,0,1,0,1,0,1},{1,2,2,4,4,8,7,15,14,30,28,60},50] (* Vincenzo Librandi, May 20 2025 *)

G.f.: -(1+x+x^2)*(x^2-x+1)*(x^7+2*x+1)/(-1+x^2+x^4+x^6+x^8+x^10+x^12).

A380696 a(n) = A007598(floor(n/2) - (-1)^n).

Original entry on oeis.org

1, 1, 0, 1, 1, 4, 1, 9, 4, 25, 9, 64, 25, 169, 64, 441, 169, 1156, 441, 3025, 1156, 7921, 3025, 20736, 7921, 54289, 20736, 142129, 54289, 372100, 142129, 974169, 372100, 2550409, 974169, 6677056, 2550409, 17480761, 6677056, 45765225, 17480761, 119814916

Offset: 0

Benjamin G. Brunsden, Jan 30 2025

The Fibonacci spiral is produced by creating a quarter circle of radius 1, then adding successive quarter circles such that the radius of the new quarter circle is the sum of the radii of the previous two quarter circles, and that the circumference of the new quarter circle continues where the previous quarter circle ended. When the center of the first quarter circle is at 0,0 the circumference turns clockwise from -1,0, and terms after n=1 are given signs - + + - repeating, these are the x coordinates where the circumferences meet. The y coordinates are the golden rectangle numbers (A001654) with the same pattern of alternation (x,a,b,x), and the same pattern of signs shifted backward one.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Benjamin G. Brunsden, Sequence shown as coordinates on the Fibonacci spiral
Index entries for linear recurrences with constant coefficients, signature (0,2,0,2,0,-1).

Cf. A000045, A007598, A053602, A069921, A123231, A272912, A001654.

A380696[n_] := Fibonacci[Floor[n/2] - (-1)^n]^2; Array[A380696, 50, 0] (* or *)
LinearRecurrence[{0, 2, 0, 2, 0, -1}, {1, 1, 0, 1, 1, 4}, 50] (* Paolo Xausa, Mar 27 2025 *)

from sympy import fibonacci
def A380696(n): return fibonacci(n+1>>1 if n&1 else (n>>1)-1)**2 # Chai Wah Wu, Mar 26 2025

a(n) = Fibonacci(floor(n/2)-(-1)^n)^2.
a(n) = A053602(n-2)^2 for n >= 2.
a(n) = A272912(n)^2 for n >= 3.
G.f. ( 1+x-x^3-x^4-2*x^2 ) / ( (1+x^2)*(x^2-x-1)*(x^2+x-1) ).
a(2*n) + a(2*n+1) = A069921(n-1) for n>=1.

cp's OEIS Frontend

A233323 Triangle read by rows: T(n,k) = number of palindromic compositions of n in which the largest part is equal to k, 1 <= k <= n.

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A233324 Triangle read by rows: T(n,k) = number of palindromic compositions of n in which no part exceeds k, 1 <= k <= n.

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A272912 Difference sequence of the sequence A116470 of all distinct Fibonacci numbers and Lucas numbers (A000032).

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A382478 Number of palindromic binary strings of length n having no 4-runs of 1's.

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A382479 Number of palindromic binary strings of length n having no 6-runs of 1's.

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A380696 a(n) = A007598(floor(n/2) - (-1)^n).

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