A011794 -id:A011794 - cp's OEIS frontend

A055801 Triangle T read by rows: T(i,0)=T(i,i)=1, T(i,j) = Sum_{k=1..floor(n/2)} T(i-2k, j-2k+1) for 1<=j<=i-1, where T(m,n) := 0 if m<0 or n<0.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 3, 3, 1, 1, 1, 1, 2, 3, 4, 3, 1, 1, 1, 1, 2, 3, 5, 6, 4, 1, 1, 1, 1, 2, 3, 5, 7, 7, 4, 1, 1, 1, 1, 2, 3, 5, 8, 11, 10, 5, 1, 1, 1, 1, 2, 3, 5, 8, 12, 14, 11, 5, 1, 1, 1, 1, 2, 3, 5, 8, 13, 19, 21, 15, 6, 1

Offset: 0

Clark Kimberling, May 28 2000

T(i+j,j) is the number of strings (s(1),...,s(m)) of nonnegative integers s(k) such that m<=i+1, s(m)=j and s(k)-s(k-1) is an odd positive integer for k=2,3,...,m.

T(i+j,j) is the number of compositions of numbers <=j using up to i parts, each an odd positive integer.

			Rows:
  1
  1  1
  1  1  1
  1  1  1  1
  1  1  1  2  1
  1  1  1  2  2  1
  1  1  1  2  3  3  1
  1  1  1  2  3  4  3  1
  1  1  1  2  3  5  6  4  1
  1  1  1  2  3  5  7  7  4  1
  1  1  1  2  3  5  8 11 10  5  1
  1  1  1  2  3  5  8 12 14 11  5  1
  1  1  1  2  3  5  8 13 19 21 15  6  1
  1  1  1  2  3  5  8 13 20 26 25 16  6  1
  1  1  1  2  3  5  8 13 21 32 40 36 21  7  1
  1  1  1  2  3  5  8 13 21 33 46 51 41 22  7  1
T(9,6) counts the strings 3456, 1236, 1256, 1456, 036, 016, 056.
T(9,6) counts the compositions 111, 113, 131, 311, 33, 15, 51.

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Clark Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 2B.

Infinitely many of the columns are (1, 1, 1, 2, 3, 5, 8, ..., Fibonacci numbers)
Essentially a reflected version of A011794.
Cf. A055802, A055803, A055804, A055805, A055806.

T:= function(n,k)
    if n<0 or k<0 then return 0;
    elif k=0 or k=n then return 1;
    else return Sum([1..Int(n/2)], j-> T(n-2*j, k-2*j+1));
    fi; end;
Flat(List([0..15], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Jan 23 2020

function T(n,k)
  if n lt 0 or k lt 0 then return 0;
  elif k eq 0 or k eq n then return 1;
  else return (&+[T(n-2*j, k-2*j+1): j in [1..Floor(n/2)]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jan 23 2020

A055801 := proc(i,j) option remember;
    if j =0 or j = i then 1;
    elif i < 0 or j < 0 then 0;
    else add(procname(i-2*k,j-2*k+1),k=1..floor(i/2)) ;
    end if;
end proc:
seq(seq(A055801(n,k), k=0..n),n=0..20); # R. J. Mathar, Feb 11 2018

T[n_, k_]:= T[n, k]= If[n<0 || k<0, 0, If[k==0 || k==n, 1, Sum[T[n-2*j, k-2*j+1 ], {j, Floor[n/2]}]]]; Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 23 2020 *)

T(n,k) = if(n<0 || k<0, 0, if(k==0 || k==n, 1, sum(j=1, n\2, T(n-2*j, k-2*j+1))));
for(n=0,15, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jan 23 2020

@CachedFunction
def T(n, k):
    if (n<0 or k<0): return 0
    elif (k==0 or k==n): return 1
    else: return sum(T(n-2*j, k-2*j+1) for j in (1..floor(n/2)))
[[T(n, k) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Jan 23 2020

A054469 a(n) = a(n-1) + a(n-2) + (n+2)*binomial(n+3, 3)/2, with a(0) = 1, a(1) = 7.

Original entry on oeis.org

1, 7, 28, 85, 218, 499, 1053, 2092, 3970, 7272, 12958, 22596, 38739, 65535, 109714, 182185, 300620, 493635, 807555, 1317360, 2144396, 3485032, 5657028, 9174560, 14869613, 24088399, 39009168, 63156437, 102233030, 165466347, 267786673

Offset: 0

Barry E. Williams, Mar 31 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. F. Horadam, Special Properties of the Sequence W(n){a,b; p,q}, Fib. Quart., Vol. 5, No. 5 (1967), pp. 424-434.
A. K. Whitford, Binet's Formula Generalized, Fibonacci Quarterly, Vol. 15, No. 1, 1979, pp. 21, 24, 29.
Index entries for linear recurrences with constant coefficients, signature (6,-14,15,-5,-4,4,-1).

Cf. A000045, A001891, A001911.

Right-hand column 11 of triangle A011794.

A054469:= func< n | Fibonacci(n+12) -(1/12)*(1716 +802*n +173*n^2 +20*n^3 +n^4) >;
[A054469(n): n in [0..40]]; // G. C. Greubel, Oct 21 2024

RecurrenceTable[{a[0]==1,a[1]==7,a[n]==a[n-1]+a[n-2]+(n+2) Binomial[ n+3,3]/2},a,{n,30}] (* Harvey P. Dale, Sep 22 2013 *)
CoefficientList[Series[(1+x)/((1-x)^5*(1-x-x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 23 2013 *)

a(n) = sum(i=1,(n+2)\2,binomial(n+5-i,n+2-2*i))+2*sum(i=1,(n+1)\2,binomial(n+5-i,n+1-2*i)) \\ Jason Yuen, Aug 13 2024

def A054469(n): return fibonacci(n+12) - (1716 + 802*n + 173*n^2 + 20*n^3 + n^4)//12
[A054469(n) for n in range(41)] # G. C. Greubel, Oct 21 2024

a(n) = a(n-1) + a(n-2) + (n+1)*(n+2)^2*(n+3)/12.
a(-n) = 0.
a(n) = (Sum_{i=1..floor((n+2)/2)} binomial(n+5-i, n+2-2*i)) + 2*(Sum_{i=1..floor((n+1)/2)} binomial(n+5-i, n+1-2*i)).
G.f.: (1+x) / ((1-x)^5*(1-x-x^2)). - Colin Barker, Jun 11 2013
From G. C. Greubel, Oct 21 2024: (Start)
a(n) = Fibonacci(n+12) - Sum_{j=0..4} Fibonacci(11-2*j) * binomial(n+j, j).
a(n) = Fibonacci(n+12) - (1/12)*(1716 + 802*n + 173*n^2 + 20*n^3 + n^4). (End)

A053809 Second partial sums of A001891.

Original entry on oeis.org

1, 6, 21, 57, 133, 281, 554, 1039, 1878, 3302, 5686, 9638, 16143, 26796, 44179, 72471, 118435, 193015, 313920, 509805, 827036, 1340636, 2171996, 3517532, 5695053, 9218786, 14920769, 24147269, 39076593, 63233317, 102320326

Offset: 0

Barry E. Williams, Mar 27 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,6,1,-3,1).

Cf. A001911, A001891, A053808.

Right-hand column 9 of triangle A011794. Pairwise sums of A014166.

List([0..40], n-> Fibonacci(n+10) - (2*n^3 + 27*n^2 + 145*n + 324)/6) # G. C. Greubel, Jul 06 2019

[Fibonacci(n+10) - (2*n^3 + 27*n^2 + 145*n + 324)/6: n in [0..40]]; // G. C. Greubel, Jul 06 2019

Table[Fibonacci[n+10] - (2*n^3+27*n^2+145*n+324)/6, {n,0,40}] (* G. C. Greubel, Jul 06 2019 *)

vector(40, n, n--; fibonacci(n+10) - (2*n^3 + 27*n^2 + 145*n + 324)/6) \\ G. C. Greubel, Jul 06 2019

[fibonacci(n+10) - (2*n^3 + 27*n^2 + 145*n + 324)/6 for n in (0..40)] # G. C. Greubel, Jul 06 2019

a(n) = a(n-1) + a(n-2) + (2*n+3)*C(n+2, 2)/3; a(-x)=0.
a(n) = Fibonacci(n+10) - (2*n^3 + 27*n^2 + 145*n + 324)/6.
G.f.: (1+x)/((1-x)^4*(1-x-x^2)).
a(n) = 5*a(n-1) - 9*a(n-2) + 6*a(n-3) + a(n-4) - 3*a(n-5) + a(n-6). - Wesley Ivan Hurt, Apr 21 2021

A054470 Partial sums of A054469.

Original entry on oeis.org

1, 8, 36, 121, 339, 838, 1891, 3983, 7953, 15225, 28183, 50779, 89518, 155053, 264767, 446952, 747572, 1241207, 2048762, 3366122, 5510518, 8995550, 14652578, 23827138, 38696751, 62785150, 101794318, 164950755, 267183785, 432650132

Offset: 0

Barry E. Williams, Mar 31 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. F. Horadam, Special Properties of the Sequence W(n){a,b; p,q}, Fib. Quart., Vol. 5, No. 5 (1967), pp. 424-434.
A. K. Whitford, Binet's Formula Generalized, Fibonacci Quarterly, Vol. 15, No. 1, 1979, pp. 21, 24, 29.
Index entries for linear recurrences with constant coefficients, signature (7,-20,29,-20,1,8,-5,1).

Cf. A000045, A001891, A001911, A054469.

Right-hand column 13 of triangle A011794.

A054470:= func< n | Fibonacci(n+14) - (45120 +21458*n +4925*n^2 +680*n^3 +55*n^4 +2*n^5)/120 >;
[A054470(n): n in [0..40]]; // G. C. Greubel, Oct 21 2024

Accumulate[RecurrenceTable[{a[0]==1,a[1]==7,a[n]==a[n-1]+a[n-2]+(n+2) Binomial[n+3,3]/2},a,{n,40}]] (* Harvey P. Dale, Sep 22 2013 *)
CoefficientList[Series[(1+x)/((1-x)^6*(1-x-x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 23 2013 *)

def A054470(n): return fibonacci(n+14) -(45120 +21458*n +4925*n^2 +680*n^3 +55*n^4 +2*n^5)//120
[A054470(n) for n in range(41)] # G. C. Greubel, Oct 21 2024

a(n) = a(n-1) + a(n-2) + (2*n+5)*C(n+4, 4)/5, with a(-n) = 0.
a(n) = Sum_{j=1..[(n+2)/2]} binomial(n+6-j, n+2-2*j) + 2*Sum_{j=1..[(n+1)/2]} binomial(n+6-j, n+1-2*j), where [x]=greatest integer in x.
G.f.: (1+x) / ((1-x)^6*(1-x-x^2)). - Colin Barker, Jun 11 2013
From G. C. Greubel, Oct 21 2024: (Start)
a(n) = Fibonacci(n+14) - Sum_{j=0..5} Fibonacci(13-2*j)*binomial(n+j,j).
a(n) = Fibonacci(n+14) - (1/120)*(45120 + 21458*n + 4925*n^2 + 680*n^3 + 55*n^4 + 2*n^5). (End)

cp's OEIS Frontend

A055801 Triangle T read by rows: T(i,0)=T(i,i)=1, T(i,j) = Sum_{k=1..floor(n/2)} T(i-2k, j-2k+1) for 1<=j<=i-1, where T(m,n) := 0 if m<0 or n<0.

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A054469 a(n) = a(n-1) + a(n-2) + (n+2)*binomial(n+3, 3)/2, with a(0) = 1, a(1) = 7.

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A053809 Second partial sums of A001891.

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A054470 Partial sums of A054469.

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