A055819 -id:A055819 - cp's OEIS frontend

A055818 Triangle T read by rows: T(i,j) = R(i-j,j), where R(i,0) = R(0,i) = 1 for i >= 0, R(i,j) = Sum_{h=0..i-1} Sum_{m=0..j} R(h,m) for i >= 1, j >= 1.

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 11, 9, 4, 1, 1, 23, 24, 14, 5, 1, 1, 47, 60, 43, 20, 6, 1, 1, 95, 144, 122, 69, 27, 7, 1, 1, 191, 336, 328, 217, 103, 35, 8, 1, 1, 383, 768, 848, 640, 354, 146, 44, 9, 1, 1, 767, 1728, 2128, 1800, 1131, 543, 199, 54, 10, 1

Offset: 0

Clark Kimberling, May 28 2000

			Rows begins as:
  1;
  1,  1;
  1,  2, 1;
  1,  5, 3, 1;
  1, 11, 9, 4, 1;
  ...

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 3B.

Cf. A055819, A055820, A055821, A055822, A055823, A055824, A055825, A055826, A055827, A055828, A055829.

T:= function(i,j)
    if i=0 or j=0 then return 1;
    else return Sum([0..i-1], h-> Sum([0..j], m-> T(h,m) ));
    fi; end;
Flat(List([0..12], n-> List([0..n], k-> T(n-k,k) ))); # G. C. Greubel, Jan 21 2020

function T(i,j)
  if i eq 0 or j eq 0 then return 1;
  else return (&+[(&+[T(h,m): m in [0..j]]): h in [0..i-1]]);
  end if; return T; end function;
[T(n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 21 2020

T:= proc(i, j) option remember;
      if i=0 or j=0 then 1
    else add(add(T(h,m), m=0..j), h=0..i-1)
      fi; end:
seq(seq(T(n-k, k), k=0..n), n=0..12); # G. C. Greubel, Jan 21 2020

T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n-k, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 21 2020 *)

T(i,j) = if(i==0 || j==0, 1, sum(h=0,i-1, sum(m=0,j, T(h,m) )));
for(n=0,12, for(k=0, n, print1(T(n-k,k), ", "))) \\ G. C. Greubel, Jan 21 2020

@CachedFunction
def T(i, j):
    if (i==0 or j==0): return 1
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[[T(n-k, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 21 2020

A052995 Expansion of 2x(1 - x)/(1 - 3*x + x^2).

Original entry on oeis.org

0, 2, 4, 10, 26, 68, 178, 466, 1220, 3194, 8362, 21892, 57314, 150050, 392836, 1028458, 2692538, 7049156, 18454930, 48315634, 126491972, 331160282, 866988874, 2269806340, 5942430146, 15557484098, 40730022148, 106632582346, 279167724890, 730870592324

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Terms >=4 give solutions x to floor(phi^2*x^2) - floor(phi*x)^2 = 5, where phi =(1 + sqrt(5))/2. - Benoit Cloitre, Mar 16 2003
Except for the first term, positive values of x (or y) satisfying x^2 - 18*x*y + y^2 + 256 = 0. - Colin Barker, Feb 14 2014
a(n+1) is the square of the distance AB, where A is the point (F(n), F(n+1)), B is the 90-degree rotation of A about the origin, and F(n)=A000045(n) are the Fibonacci numbers. - Burak Muslu, Mar 24 2021

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 30.
B. Muslu, Sayılar ve Bağlantılar 2, Luna, 2021, pages 60-61.

Colin Barker, Table of n, a(n) for n = 0..1000
Younseok Choo, Some Results on the Infinite Sums of Reciprocal Generalized Fibonacci Numbers, International Journal of Mathematical Analysis, 12(12) (2018), 621-629.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1072.
Youngwoo Kwon, Binomial transforms of the modified k-Fibonacci-like sequence, arXiv:1804.08119 [math.NT], 2018.
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Bisection of A006355.
First differences of A025169.
Cf. A000032, A000045, A055819, A006355, A025169.

spec := [S, S=Prod(Sequence(Union(Prod(Sequence(Z),Z),Z)),Union(Z,Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);

LinearRecurrence[{3, -1}, {0, 2, 4}, 30] (* or *)
Nest[Append[#, 3 #[[-1]] - #[[-2]]] &, {0, 2, 4}, 27] (* or *)
CoefficientList[Series[-2 x (-1 + x)/(1 - 3 x + x^2), {x, 0, 29}], x] (* Michael De Vlieger, Jul 18 2018 *)

concat(0, Vec(2*x*(1-x)/(1-3*x+x^2) + O(x^50))) \\ Colin Barker, Mar 30 2016

a(n) = fibonacci(max(0,2*n-1))<<1; \\ Kevin Ryde, Mar 25 2021

G.f.: -2*x*(-1 + x)/(1 - 3*x + x^2).
a(0) = 0, a(1) = 2, a(2) = 4; for n > 0, a(n) - 3*a(n+1) + a(n+2) = 0.
a(n) = A069403(n-1)+1.
a(n) = Sum(2/5*(-1 + 4*_alpha)*_alpha^(-1-n), _alpha = RootOf(_Z^2 - 3*_Z + 1)).
a(n) = 2*Fibonacci(2*n-1) = 2*A001519(n) for n > 0. - Vladeta Jovovic, Mar 19 2003
a(n+2) = F(n)^2 + F(n+3)^2 = 2*F(n+1)^2 + 2*F(n+2)^2, where F = A000045. -  N. J. A. Sloane, Feb 20 2005
a(n) = 1/2*(F(2*n+8) mod F(2*n+2)) for n > 2. - Gary Detlefs, Nov 22 2010
a(n) = F(n-3)*F(n-1) + F(n)*F(n+2) for n > 0, F(-2) = -1, F(-1) = 1. - Bruno Berselli, Nov 03 2015
a(n) = (2^(-n)*((3 - sqrt(5))^n*(1 + sqrt(5)) + (-1 + sqrt(5))*(3 + sqrt(5))^n))/sqrt(5) for n > 0. - Colin Barker, Mar 30 2016
a(n) = Fibonacci(2*n-2) + Lucas(2*n-2) for n > 0. - Bruno Berselli, Oct 13 2017
a(n) = Lucas(2*n) - Fibonacci(2*n) for n > 0. - Diego Rattaggi, Mar 08 2023

More terms from James Sellers, Jun 05 2000

A055823 a(n) = T(n,n-6), array T as in A055818.

Original entry on oeis.org

1, 95, 336, 848, 1800, 3422, 6017, 9974, 15782, 24045, 35498, 51024, 71672, 98676, 133475, 177734, 233366, 302555, 387780, 491840, 617880, 769418, 950373, 1165094, 1418390, 1715561, 2062430, 2465376, 2931368, 3468000, 4083527, 4786902, 5587814, 6496727, 7524920

Offset: 6

Clark Kimberling, May 28 2000

nonn

Vincenzo Librandi, Table of n, a(n) for n = 6..5000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A055818, A055819, A055820, A055821, A055822, A055824, A055825, A055826, A055827, A055828, A055829.

Concatenation([1], List([7..50], n-> (n^6 +15*n^5 -65*n^4 -795*n^3 +1864*n^2 +6180*n -7200)/720 )); # G. C. Greubel, Jan 22 2020

I:=[1,95,336,848,1800,3422,6017,9974,15782,24045, 35498,51024,71672]; [n le 13 select I[n] else 7*Self(n-1)- 21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6)+Self(n-7): n in [1..50]]; // Vincenzo Librandi, Dec 30 2016

seq( `if`(n=6, 1, (n^6 +15*n^5 -65*n^4 -795*n^3 +1864*n^2 +6180*n -7200)/720), n=6..50); # G. C. Greubel, Jan 22 2020

Join[{1, 95, 336, 848, 1800, 3422}, LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {6017, 9974, 15782, 24045, 35498, 51024, 71672}, 50]] (* Vincenzo Librandi, Dec 30 2016 *)
Table[If[n==6, 1, (n^6 +15*n^5 -65*n^4 -795*n^3 +1864*n^2 +6180*n -7200)/720], {n, 6, 50}] (* G. C. Greubel, Jan 22 2020 *)

vector(45, n, my(m=n+5); if(m==6, 1, (m^6 +15*m^5 -65*m^4 -795*m^3 +1864*m^2 +6180*m -7200)/720)) \\ G. C. Greubel, Jan 22 2020

[1]+[(n^6 +15*n^5 -65*n^4 -795*n^3 +1864*n^2 +6180*n -7200)/720 for n in (7..50)] # G. C. Greubel, Jan 22 2020

From Chai Wah Wu, Dec 29 2016: (Start)
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 13.
G.f.: x^6*(1 + 88*x - 308*x^2 + 456*x^3 - 370*x^4 + 174*x^5 - 45*x^6 + 5*x^7)/(1-x)^7. (End)
From G. C. Greubel, Jan 22 2020: (Start)
a(n) = (n^6 + 15*n^5 - 65*n^4 - 795*n^3 + 1864*n^2 + 6180*n -7200)/720, for n > 6, with a(6) = 1.
E.g.f.: (7200 - 2880*x^2 - 960*x^3 + 30*x^4 + 60*x^5 - 5*x^6 + (-7200 + 7200*x - 720*x^2 - 720*x^3 + 150*x^4 + 30*x^5 + x^6)*exp(x))/720. (End)

A055824 a(n) = T(2*n,n), array T as in A055818.

Original entry on oeis.org

1, 2, 9, 43, 217, 1131, 6017, 32467, 177009, 972691, 5378425, 29889531, 166795977, 934039419, 5246059761, 29540072355, 166708076001, 942651407907, 5339465635049, 30291114653131, 172081678284729, 978807205953931

Offset: 0

Clark Kimberling, May 28 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A055818, A055819, A055820, A055821, A055822, A055823, A055825, A055826, A055827, A055828, A055829.

T:= proc(i, j) option remember;
      if i=0 or j=0 then 1
    else add(add(T(h,m), m=0..j), h=0..i-1)
fi; end: seq(T(n, n), n=0..30); # G. C. Greubel, Jan 22 2020

T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n, n], {n,0,25}] (* G. C. Greubel, Jan 22 2020 *)

@CachedFunction
def T(i, j):
    if (i==0 or j==0): return 1
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[T(n, n) for n in (0..30)] # G. C. Greubel, Jan 22 2020

A055825 a(n) = T(2n+1,n), array T as in A055818.

Original entry on oeis.org

1, 5, 24, 122, 640, 3422, 18536, 101362, 558336, 3093302, 17218168, 96214890, 539415552, 3032659086, 17091411912, 96527966178, 546184965120, 3095613086822, 17571039730136, 99868193737306, 568303494617472

Offset: 0

Clark Kimberling, May 28 2000

nonn

Also main diagonal of array: A(i,1)=i, A(1,j)=j; A(i,j) = 2*A(i,j-1) - A(i-1,j). - Benoit Cloitre, Feb 26 2003

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A055818, A055819, A055820, A055821, A055822, A055823, A055824, A055826, A055827, A055828, A055829.

T:= proc(i, j) option remember;
      if i=0 or j=0 then 1
    else add(add(T(h,m), m=0..j), h=0..i-1)
  fi; end:
seq(T(n+1, n), n=0..30); # G. C. Greubel, Jan 22 2020

T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n+1, n], {n,0,25}] (* G. C. Greubel, Jan 22 2020 *)

@CachedFunction
def T(i, j):
    if (i==0 or j==0): return 1
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[T(n+1, n) for n in (0..30)] # G. C. Greubel, Jan 22 2020

A055826 a(n) = T(2n+2,n), array T as in A055818.

Original entry on oeis.org

1, 11, 60, 328, 1800, 9924, 54964, 305680, 1706256, 9554620, 53653996, 302038488, 1703995800, 9631951476, 54539233380, 309296779296, 1756495236128, 9987721546860, 56857004161180, 324008331785320, 1848182861702184

Offset: 0

Clark Kimberling, May 28 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A055818, A055819, A055820, A055821, A055822, A055823, A055824, A055825, A055827, A055828, A055829.

T:= proc(i, j) option remember;
      if i=0 or j=0 then 1
    else add(add(T(h,m), m=0..j), h=0..i-1)
  fi; end:
seq(T(n+2, n), n=0..30); # G. C. Greubel, Jan 22 2020

T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n+2, n], {n,0,25}] (* G. C. Greubel, Jan 22 2020 *)

@CachedFunction
def T(i, j):
    if (i==0 or j==0): return 1
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[T(n+2, n) for n in (0..30)] # G. C. Greubel, Jan 22 2020

A055827 a(n) = T(2n+3,n), array T as in A055818.

Original entry on oeis.org

1, 23, 144, 848, 4880, 27816, 157920, 895264, 5074272, 28772280, 163262704, 927203184, 5270629104, 29988361032, 170780080320, 973422085184, 5552990609344, 31702646247768, 181128948471888, 1035584204252560

Offset: 0

Clark Kimberling, May 28 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A055818, A055819, A055820, A055821, A055822, A055823, A055824, A055825, A055826, A055828, A055829.

T:= proc(i, j) option remember;
      if i=0 or j=0 then 1
    else add(add(T(h,m), m=0..j), h=0..i-1)
  fi; end:
seq(T(n+3, n), n=0..30); # G. C. Greubel, Jan 22 2020

T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n+3, n], {n,0,25}] (* G. C. Greubel, Jan 22 2020 *)

@CachedFunction
def T(i, j):
    if (i==0 or j==0): return 1
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[T(n+3, n) for n in (0..30)] # G. C. Greubel, Jan 22 2020

A055828 a(n) = T(2n+4,n), array T as in A055818.

Original entry on oeis.org

1, 47, 336, 2128, 12848, 75808, 441824, 2557024, 14737312, 84726992, 486388912, 2789840688, 15995087184, 91689974592, 525608606912, 3013422222272, 17280193763392, 99117586397296, 568696489153808, 3263973649089808

Offset: 0

Clark Kimberling, May 28 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A055818, A055819, A055820, A055821, A055822, A055823, A055824, A055825, A055826, A055827, A055829.

T:= proc(i, j) option remember;
      if i=0 or j=0 then 1
    else add(add(T(h,m), m=0..j), h=0..i-1)
  fi; end:
seq(T(n+4, n), n=0..30); # G. C. Greubel, Jan 22 2020

T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n+4, n], {n,0,25}] (* G. C. Greubel, Jan 22 2020 *)

@CachedFunction
def T(i, j):
    if (i==0 or j==0): return 1
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[T(n+4, n) for n in (0..30)] # G. C. Greubel, Jan 22 2020

A055829 a(n) = T(2n+5,n), array T as in A055818.

Original entry on oeis.org

1, 95, 768, 5216, 33024, 201792, 1208320, 7145792, 41919744, 244590496, 1421823232, 8243669664, 47708339712, 275738420864, 1592186658816, 9187634766976, 52992487665152, 305556178607328, 1761501729738496

Offset: 0

Clark Kimberling, May 28 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A055818, A055819, A055820, A055821, A055822, A055823, A055824, A055825, A055826, A055827, A055828.

T:= proc(i, j) option remember;
      if i=0 or j=0 then 1
    else add(add(T(h,m), m=0..j), h=0..i-1)
  fi; end:
seq(T(n+5, n), n=0..30); # G. C. Greubel, Jan 23 2020

T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n+5, n], {n,0,30}] (* G. C. Greubel, Jan 23 2020 *)

@CachedFunction
def T(i, j):
    if (i==0 or j==0): return 1
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[T(n+5, n) for n in (0..30)] # G. C. Greubel, Jan 23 2020

A237132 Values of x in the solutions to x^2 - 3xy + y^2 + 11 = 0, where 0 < x < y.

Original entry on oeis.org

3, 4, 5, 9, 12, 23, 31, 60, 81, 157, 212, 411, 555, 1076, 1453, 2817, 3804, 7375, 9959, 19308, 26073, 50549, 68260, 132339, 178707, 346468, 467861, 907065, 1224876, 2374727, 3206767, 6217116, 8395425, 16276621, 21979508, 42612747, 57543099, 111561620

Offset: 1

Colin Barker, Feb 04 2014

The corresponding values of y are given by a(n+2).

Positive values of x (or y) satisfying x^2 - 18xy + y^2 + 704 = 0.

			9 is in the sequence because (x, y) = (9, 23) is a solution to x^2 - 3xy + y^2 + 11 = 0.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Cf. A001519, A005248, A055819, A237133, A218735.

Vec(-x*(x-1)*(3*x^2+7*x+3)/((x^2-x-1)*(x^2+x-1)) + O(x^100))

a(n) = 3*a(n-2)-a(n-4).
G.f.: -x*(x-1)*(3*x^2+7*x+3) / ((x^2-x-1)*(x^2+x-1)).
a(n) = F(n+2) + (-1)^n*F(n-3), n>1, with F the Fibonacci numbers (A000045). - Ralf Stephan, Feb 05 2014
Let h(n) = hypergeom([(1 - n)/2, n mod 2 - n/2], [1 - n], -4) then a(n) = h(n-1) + h(n) for n > 3. - Peter Luschny, Sep 03 2019

cp's OEIS Frontend

A055818 Triangle T read by rows: T(i,j) = R(i-j,j), where R(i,0) = R(0,i) = 1 for i >= 0, R(i,j) = Sum_{h=0..i-1} Sum_{m=0..j} R(h,m) for i >= 1, j >= 1.

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A052995 Expansion of 2*x*(1 - x)/(1 - 3*x + x^2).

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A055823 a(n) = T(n,n-6), array T as in A055818.

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A055824 a(n) = T(2*n,n), array T as in A055818.

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A055825 a(n) = T(2n+1,n), array T as in A055818.

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Sage

A055826 a(n) = T(2n+2,n), array T as in A055818.

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Sage

A055827 a(n) = T(2n+3,n), array T as in A055818.

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A052995 Expansion of 2x(1 - x)/(1 - 3*x + x^2).