A006498 -id:A006498 - cp's OEIS frontend

A273461 Number of physically stable n X n placements of water source-blocks in Minecraft.

1, 2, 9, 40, 484, 9717, 338724, 21624680, 2504301849, 520443847520, 195145309791364, 131850659243316222, 160668896658179472676, 352891729183598844656996, 1397187513066371784602204416, 9972288382286063615850619475640

Offset: 0

Gus Wiseman, May 23 2016

nonn

In Minecraft worlds, a source block of water can be reacted with another source block, two blocks away. This reaction creates a third "infinite" source block in the unoccupied intermediate block, so called because if the intermediate water source is destroyed or picked up by a player using a bucket, it will immediately regenerate itself.
A placement of water at several positions in an n X n board is said to be *stable* if no infinite water physics can in fact occur (under otherwise optimal conditions). This means that the total quantity of water in the system is held constant.
In short, no two source blocks can be graph-distance 2 from each other. - Gus Wiseman, Nov 27 2019
Often incorrectly described as cellular automata, the observed behaviors of liquids within a board are inseparable in certain ways from states of affair outside of the board and events outside of the system. This aspect of Minecraft is poorly understood.

			a(2) = 9: {{}, {(2,2)}, {(2,1)}, {(2,1),(2,2)}, {(1,2)}, {(1,2),(2,2)}, {(1,1)}, {(1,1),(2,1)}, {(1,1),(1,2)}}.

EthosLab, Minecraft - Tutorial: Water
Wikipedia, Distance (graph theory)
Gus Wiseman, Example of a physically stable arrangement of water source-blocks (n=11)
Christopher Cormier, C# Program

The one-dimensional version is A006498.
Dominated by A329871.
Cf. A002416, A005251, A027624, A114901.

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
allflows[n_]:=stableSets[Join@@Array[List,{n,n}],Function[{v,w},Plus@@Abs/@(w-v)===2]];
Table[Length[allflows[i]],{i,6}] (* Gus Wiseman, May 23 2016 *)

a(7) from Tae Lim Kook, May 25 2016
a(8) from Tae Lim Kook, May 29 2016
a(7)-a(8) corrected by Christopher Cormier, Dec 17 2019
a(9)-a(15) from Christopher Cormier, Dec 19 2019

A329871 Number of static n X n placements of water source-blocks in Minecraft.

Original entry on oeis.org

1, 2, 10, 55, 754, 18853, 82931, 70143802, 11087020614, 3243227117597, 1772826333285009, 1806938280429412270, 3430002591378184399879, 12137184871791092506807847, 80047171080361800628780500638, 983838070049011459232146327319193

Offset: 0

Gus Wiseman, Nov 26 2019

nonn

In Minecraft worlds, a source block of water can be reacted with another source block, two blocks away, linearly or diagonally. This reaction creates a third "infinite" source block in the unoccupied intermediate block or blocks, so called because if the intermediate water source is destroyed or picked up by a player using a bucket, it will immediately regenerate itself.

A placement of water at several positions in an n X n board is said to be static if no infinite water sources are created that are not already present. In particular, the total quantity of water in the system is held constant.

EthosLab, Minecraft - Tutorial: Water
Gus Wiseman, The a(3) = 55 static placements of water source-blocks (black = water).
Christopher Cormier, C# Program

Dominates A273461.
The one-dimensional case is A005251.
Cf. A002416, A006498, A027624, A114901.

vdist[v_,w_]:=Total[Abs[v-w]];
flowdown[prs_]:=Union[prs,With[{ovs=Select[Subsets[prs,{2}],vdist@@#==2&]},Union@@Function[{v,w},Select[Tuples[{Range[Min@@Union[First/@prs],Max@@Union[First/@prs]],Range[Min@@Union[Last/@prs],Max@@Union[Last/@prs]]}],vdist[v,#]==1&&vdist[w,#]==1&]]@@@ovs]];
Table[Length[Select[Subsets[Tuples[Range[n],2]],flowdown[#]==#&]],{n,0,3}]

a(5)-a(6) from Christopher Cormier, Dec 10 2019

a(7)-a(15) from Christopher Cormier, Dec 19 2019

A376743 Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-2,-1,4} for all i=1,...,n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 5, 5, 6, 8, 11, 15, 25, 35, 46, 61, 85, 125, 175, 245, 341, 470, 650, 925, 1300, 1810, 2521, 3520, 4915, 6880, 9640, 13476, 18801, 26251, 36721, 51346, 71776, 100335, 140210, 195886, 273813, 382821, 535105, 747850, 1045220

Offset: 0

Michael A. Allen, Oct 03 2024

Other sequences related to strongly restricted permutations pi(i) of i in {1,..,n} along with the sets of allowed p(i)-i (containing at least 3 elements): A000045 {-1,0,1}, A189593 {-1,0,2,3,4,5,6}, A189600 {-1,0,2,3,4,5,6,7}, A006498 {-2,0,2}, A080013 {-2,1,2}, A080014 {-2,0,1,2}, A033305 {-2,-1,1,2}, A002524 {-2,-1,0,1,2}, A080000 {-2,0,3}, A080001 {-2,1,3}, A080004 {-2,0,1,3}, A080002 {-2,2,3}, A080005 {-2,0,2,3}, A080008 {-2,1,2,3}, A080011 {-2,0,1,2,3}, A079999 {-2,-1,3}, A080003 {-2,-1,0,3}, A080006 {-2,-1,1,3}, A080009 {-2,-1,0,1,3}, A080007 {-2,-1,2,3}, A080010 {-2,-1,0,2,3}, A080012 {-2,-1,1,2,3}, A072827 {-2,-1,0,1,2,3}, A224809 {-2,0,4}, A189585 {-2,0,1,3,4}, A189581 {-2,-1,0,3,4}, A072850 {-2,-1,0,1,2,3,4}, A189587 {-2,0,1,3,4,5}, A189588 {-2,-1,0,3,4,5}, A189594 {-2,0,1,3,4,5,6}, A189595 {-2,-1,0,3,4,5,6}, A189601 {-2,0,1,3,4,5,6,7}, A189602 {-2,-1,0,3,4,5,6,7}, A224811 {-2,0,8}, A224812 {-2,0,10}, A224813 {-2,0,12}, A006500 {-3,0,3}, A079981 {-3,1,3}, A079983 {-3,0,1,3}, A079982 {-3,2,3}, A079984 {-3,0,2,3}, A079988 {-3,1,2,3}, A079989 {-3,0,1,2,3}, A079986 {-3,-1,1,3}, A079992 {-3,-1,0,1,3}, A079987 {-3,-1,2,3}, A079990 {-3,-1,0,2,3}, A079993 {-3,-1,1,2,3}, A079985 {-3,-2,2,3}, A079991 {-3,-2,0,2,3}, A079996 {-3,-2,0,1,2,3}, A079994 {-3,-2,1,2,3}, A079997 {-3,-2, -1,1,2,3}, A002526 {-3,-2,-1,0,1,2,3}, A189586 {-3,0,1,2,4}, A189583 {-3,-1,0,2,4}, A189582 {-3,-2,0,1,4}, A189584 {-3,-2,-1,0,4}, A189589 {-3,0,1,2,4,5}, A189590 {-3,-1,0,2,4,5}, A189591 {-3,-2,1,4,5}, A189592 {-3,-2,-1,0,4,5}, A224810 {-3,0,6}, A189596 {-3,0,1,2,4,5,6}, A189597 {-3,-1,0,2,4,5,6}, A189598 {-3,-2,0,1,4,5,6}, A189599 {-3,-2,-1,0,4,5,6}, A224814 {-3,0,9}, A031923 {-4,0,4}, A072856 {-4,-3, -2,-1,0,1,2,3,4}, A224815 {-4,0,8}, A154654 {-5,-4,-3,-2,-1,0,1,2,3,4,5}, A154655 {-6,-5,-4,-3, -2,-1,0,1,2,3,4,5,6}.

[Keyword "less", because this comment should be moved to the Index to the OEIS, it is not appropriate here. - N. J. A. Sloane, Oct 25 2024]

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), North-Holland, Amsterdam, 1970, pp. 755-770.

Michael A. Allen and Kenneth Edwards, Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices, Lin. Multilin. Alg. 72:13 (2024) 2091-2103.
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics, 4(1) (2010), 119-135.
Kenneth Edwards and Michael A. Allen, Strongly restricted permutations and tiling with fences, Discrete Applied Mathematics, 187 (2015), 82-90.
Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,1,2,1,0,-2,-2,0,-1,0,0,1).

See comments for other sequences related to strongly restricted permutations.

CoefficientList[Series[(1 - x^3 - x^4 - x^6 + x^9)/(1 - x^3 - x^4 - x^5 - 2*x^6 - x^7 + 2*x^9 + 2*x^10 + x^12 - x^15),{x,0,49}],x]
LinearRecurrence[{0, 0, 1, 1, 1, 2, 1, 0, -2, -2, 0, -1, 0, 0, 1}, {1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 5, 5, 6, 8}, 50]

a(n) = a(n-3) + a(n-4) + a(n-5) + 2*a(n-6) + a(n-7) - 2*a(n-9) - 2*a(n-10) - a(n-12) + a(n-15).

G.f.: (1 - x^3 - x^4 - x^6 + x^9)/(1 - x^3 - x^4 - x^5 - 2*x^6 - x^7 + 2*x^9 + 2*x^10 + x^12 - x^15).

A116698 Expansion of (1-x+3x^2+x^3) / ((1-x-x^2)(1+2*x^2)).

Original entry on oeis.org

1, 0, 2, 5, 5, 4, 13, 29, 34, 39, 89, 176, 233, 313, 610, 1115, 1597, 2328, 4181, 7277, 10946, 16687, 28657, 48416, 75025, 117297, 196418, 326003, 514229, 815656, 1346269, 2211077, 3524578, 5637351, 9227465

Offset: 0

Creighton Dement, Feb 23 2006

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-1,2,2).

Cf. A000045, A001519, A006498, A115008, A116697, A116699.

A116698:= func< n | Fibonacci(n+1) -((n mod 2) -2*0^((n+1) mod 4))*2^Floor(n/2) >;
[A116898(n): n in [0..50]]; // G. C. Greubel, Aug 24 2025

CoefficientList[Series[(1-x+3x^2+x^3)/((1-x-x^2)(1+2x^2)),{x,0,100}],x] (* or *) LinearRecurrence[{1,-1,2,2},{1,0,2,5},100] (* Harvey P. Dale, May 14 2022 *)
Table[Fibonacci[n+1] -I^(n-1)*Mod[n,2]*2^Floor[n/2], {n,0,50}] (* G. C. Greubel, Aug 24 2025 *)

Vec((1-x +3*x^2 +x^3)/((1-x-x^2)*(1+2*x^2)) + O(x^40)) \\ Colin Barker, May 18 2019

def A116898(n): return fibonacci(n+1) - (-1)**((n-1)//2)*(n%2)*2**(n//2)
print([A116898(n) for n in range(51)]) # G. C. Greubel, Aug 24 2025

a(2*n) = A000045(2*n+1) = A001519(n).
a(n) = a(n-1) - a(n-2) + 2*a(n-3) + 2*a(n-4) for n > 3. - Colin Barker, May 18 2019
From G. C. Greubel, Aug 24 2025: (Start)
a(n) = A000045(n+1) - (-1)^floor((n-1)/2) * (n mod 2) * 2^floor(n/2).
E.g.f.: exp(x/2)*(cosh(sqrt(5)*x/2) + (1/sqrt(5))*sinh(sqrt(5)*x/2))  - sin(sqrt(2)*x)/sqrt(2). (End)

A208743 Number of subsets of the set {1,2,...,n} which do not contain two elements whose difference is 6.

Original entry on oeis.org

2, 4, 8, 16, 32, 64, 96, 144, 216, 324, 486, 729, 1215, 2025, 3375, 5625, 9375, 15625, 25000, 40000, 64000, 102400, 163840, 262144, 425984, 692224, 1124864, 1827904, 2970344, 4826809, 7797153, 12595401, 20346417, 32867289, 53093313, 85766121, 138859434

Offset: 1

David Nacin, Mar 01 2012

			If n=7 then we must count all subsets not containing both 1 and 7.  There are 2^5 subsets containing 1 and 7, giving us 2^7 - 2^5 = 48.  Thus a(7) = 96.

M. El-Mikkawy, T. Sogabe, A new family of k-Fibonacci numbers, Appl. Math. Comput. 215 (2010) 4456-4461 doi:10.1016/j.amc.2009.12.069, Table 1 k=6.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Katharine A. Ahrens, Combinatorial Applications of the k-Fibonacci Numbers: A Cryptographically Motivated Analysis, Ph. D. thesis, North Carolina State University (2020).
M. Tetiva, Subsets that make no difference d, Mathematics Magazine 84 (2011), no. 4, 300-301.
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, 0, -5, 5, 5, 0, 0, 0, 15, -15, -15, 0, 0, 0, 15, -15, -15, 0, 0, 0, -5, 5, 5, 0, 0, 0, -1, 1, 1).

Cf. A006498, A006500, A208741, A208742.

Table[Fibonacci[Floor[n/6] + 3]^Mod[n, 6] * Fibonacci[Floor[n/6] + 2]^(6 - Mod[n, 6]), {n, 1, 80}]
LinearRecurrence[{1, 1, 0, 0, 0, -5, 5, 5, 0, 0, 0, 15, -15, -15, 0, 0, 0, 15, -15, -15, 0, 0, 0, -5, 5, 5, 0, 0, 0, -1, 1, 1}, {2, 4, 8, 16, 32, 64, 96, 144, 216, 324, 486, 729, 1215, 2025, 3375, 5625, 9375, 15625, 25000, 40000, 64000, 102400, 163840, 262144, 425984, 692224, 1124864, 1827904, 2970344, 4826809, 7797153, 12595401}, 80]

Vec(x*(2 + 2*x + 2*x^2 + 4*x^3 + 8*x^4 + 16*x^5 + 10*x^6 - 6*x^7 - 14*x^8 - 16*x^9 - 14*x^10 - x^11 - 30*x^12 - 29*x^13 - 15*x^14 - 15*x^15 - 15*x^16 - 20*x^17 - 30*x^18 - 10*x^19 + 5*x^20 + 5*x^21 + 5*x^22 + 4*x^23 + 10*x^24 + 6*x^25 + x^26 + x^27 + x^28 + x^29 + 2*x^30 + x^31) / ((1 + x^2)*(1 - x - x^2)*(1 - x^2 + x^4)*(1 + x^3 - x^6)*(1 - x^3 - x^6)*(1 + 7*x^6 + x^12)) + O(x^30)) \\ Colin Barker, Feb 23 2017

a(n) = F(floor(n/6) + 3)^(n mod 6)*F(floor(n/6) + 2)^(6 - (n mod 6)) where F(n) is the n-th Fibonacci number.
a(n) = a(n-1) + a(n-2) - 5*a(n-6) + 5*a(n-7) + 5*a(n-8) + 15*a(n-12) - 15*a(n-13) - 15*a(n-14) + 15*a(n-18) - 15*a(n-19) - 15*a(n-20) - 5*a(n-24) + 5*a(n-25) + 5*a(n-26) - a(n-30) + a(n-31) + a(n-32).
G.f.: x*(2 + 2*x + 2*x^2 + 4*x^3 + 8*x^4 + 16*x^5 + 10*x^6 - 6*x^7 - 14*x^8 - 16*x^9 - 14*x^10 - x^11 - 30*x^12 - 29*x^13 - 15*x^14 - 15*x^15 - 15*x^16 - 20*x^17 - 30*x^18 - 10*x^19 + 5*x^20 + 5*x^21 + 5*x^22 + 4*x^23 + 10*x^24 + 6*x^25 + x^26 + x^27 + x^28 + x^29 + 2*x^30 + x^31) / ((1 + x^2)*(1 - x - x^2)*(1 - x^2 + x^4)*(1 + x^3 - x^6)*(1 - x^3 - x^6)*(1 + 7*x^6 + x^12)). - Colin Barker, Feb 23 2017

A246690 Number A(n,k) of compositions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 1, 0, 1, 1, 5, 0, 1, 0, 1, 1, 0, 2, 0, 8, 1, 1, 0, 1, 0, 1, 0, 3, 0, 13, 0, 1, 0, 1, 0, 1, 1, 1, 4, 1, 21, 1, 1, 0, 1, 1, 0, 1, 2, 0, 6, 0, 34, 0, 1, 0, 1, 1, 2, 0, 1, 3, 0, 9, 0, 55, 1, 1, 0

Offset: 0

Alois P. Heinz, Sep 01 2014

The first lists of distinct parts in the order given by A246688 are: 0:[], 1:[1], 2:[2], 3:[1,2], 4:[3], 5:[1,3], 6:[4], 7:[1,4], 8:[2,3], 9:[5], 10:[1,2,3], 11:[1,5], 12:[2,4], 13:[6], 14:[1,2,4], 15:[1,6], 16:[2,5], 17:[3,4], 18:[7], 19:[1,2,5], 20:[1,3,4], ... .

			Square array A(n,k) begins:
  1, 1, 1,  1, 1,  1, 1,  1, 1, 1,   1, 1, 1, 1,   1, ...
  0, 1, 0,  1, 0,  1, 0,  1, 0, 0,   1, 1, 0, 0,   1, ...
  0, 1, 1,  2, 0,  1, 0,  1, 1, 0,   2, 1, 1, 0,   2, ...
  0, 1, 0,  3, 1,  2, 0,  1, 1, 0,   4, 1, 0, 0,   3, ...
  0, 1, 1,  5, 0,  3, 1,  2, 1, 0,   7, 1, 2, 0,   6, ...
  0, 1, 0,  8, 0,  4, 0,  3, 2, 1,  13, 2, 0, 0,  10, ...
  0, 1, 1, 13, 1,  6, 0,  4, 2, 0,  24, 3, 3, 1,  18, ...
  0, 1, 0, 21, 0,  9, 0,  5, 3, 0,  44, 4, 0, 0,  31, ...
  0, 1, 1, 34, 0, 13, 1,  7, 4, 0,  81, 5, 5, 0,  55, ...
  0, 1, 0, 55, 1, 19, 0, 10, 5, 0, 149, 6, 0, 0,  96, ...
  0, 1, 1, 89, 0, 28, 0, 14, 7, 1, 274, 8, 8, 0, 169, ...

Alois P. Heinz, Antidiagonals n = 0..140

Columns k=0-21, 23, 25-28 give: A000007, A000012, A059841, A000045(n+1), A079978, A000930, A121262, A003269(n+1), A182097, A079998, A000073(n+2), A003520, A079977, A079979, A060945, A005708, A001687(n+1), A017817, A082784, A079971, A006498, A005709, A052920, A120400, A060961, A005710, A013979.
Main diagonal gives A246691.
Cf. A246688, A246720 (the same for partitions).

b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
      [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
    end:
f:= proc() local i, l; i, l:=0, [];
      proc(n) while n>=nops(l)
        do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]
      end
    end():
g:= proc(n, l) option remember; `if`(n=0, 1,
      add(`if`(i>n, 0, g(n-i, l)), i=l))
    end:
A:= (n, k)-> g(n, f(k)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i>n, {}, Join[Prepend[#, i]& /@ b[n - i, i + 1], b[n, i + 1]]]];
f = Module[{i = 0, l = {}}, Function[n, While[n >= Length[l], l = Join[l, b[i, 1]]; i++]; l[[n + 1]]]];
g[n_, l_] := g[n, l] = If[n==0, 1, Sum[If[i>n, 0, g[n - i, l]], {i, l}]];
A[n_, k_] := g[n, f[k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

A171645 Partial products of Product_{n=1..inf.} (p(n)/p(n-1)p(n)/p(n-1)), = 222(3/2)(3/2)(5/3)(5/3)(7/5)(7/5)(11/7)(11/7)...; p = primes, A000040, a(1) = 2.

Original entry on oeis.org

2, 4, 8, 12, 18, 30, 50, 70, 98, 154, 242, 286, 338, 442, 578, 646, 722, 874, 1058, 1334, 1682, 1798, 1922, 2294, 2738

Offset: 1

Gary W. Adamson, Dec 13 2009

nonn

Analogous formulas using A000041 terms = A171646; Fibonacci numbers, A006498; factorials, A010551.

			a(10) = 154 = 2*2*2*(3/2)*(3/2)*(5/3)*(5/3)*(7/5)*(7/5)*(11/7).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A171646, A006498, A010551.

FoldList[Times,Join[{2,2,2},Flatten[{#[[2]]/#[[1]],#[[2]]/#[[1]]}&/@Partition[Prime[Range[20]],2,1]]]] (* Harvey P. Dale, Oct 02 2024 *)

Partial products of Product_{n=1..inf.} (p(n)/p(n-1)*p(n)/p(n-1)), =
2*2*2*(3/2)*(3/2)*(5/3)*(5/3)*(7/5)*(7/5)*(11/7)*(11/7)*...; p = primes,
A000040, a(1) = 2.
a(n)=2*A057602(n-1). [From R. J. Mathar, Dec 15 2009]

A171646 a(1) = 1, then partial products of Product_{n>=1} (p(n)/p(n-1)p(n)/p(n-1)) = 11*1(2)(2)(3/2)(3/2)(5/3)(5/3)(7/5)(7/5)...; p = partition numbers, A000041 starting (1, 2, 3, 5, ...).

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 9, 15, 25, 35, 49, 77, 121, 165, 225, 330, 484, 660, 900, 1260, 1764, 2352, 3136, 4312, 5929, 7777, 10201, 13635, 18225, 23760, 30976, 40656, 53361, 68607, 88209, 114345, 148225, 188650, 240100, 307230

Offset: 1

Gary W. Adamson, Dec 13 2009

nonn

A006498 = analogous sequence using the Fibonacci numbers.
A171645 = .............................Primes, analogous formula.
A010551 = .............................Factorial numbers, analogous formula.

			a(12) = 77 = 1*1*1*2*2*(3/2)*(3/2)*(5/3)*(5/3)*(7/5)*(7/5)*(11/7).

Cf. A000041, A171645, A006498, A010551.

A171646t := proc(n)
    local nh;
    nh := floor(n/2) ;
    combinat[numbpart](nh)/combinat[numbpart](nh-1) ;
end proc:
A171646 := proc(n)
    mul(A171646t(i),i=2..n) ;
end proc:
1,seq(A171646(n),n=2..40) ; # R. J. Mathar, Jul 21 2015

a(1) = 1, then partial products of Product_{n>=1} (p(n)/p(n-1)*p(n)/p(n-1)) = 1*1*1*(2)*(2)*(3/2)*(3/2)*(5/3)*(5/3)*(7/5)*(7/5)*...; p = partition numbers, A000041 starting (1, 2, 3, 5, ...).

Corrected by R. J. Mathar, Jul 21 2015

A178534 Triangle T(n,k) read by rows. T(n,1) = A000045(n+1), k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) - (Sum_{i=1..k-1} T(n-i,k)).

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 5, 2, 1, 1, 8, 3, 1, 1, 1, 13, 5, 3, 1, 1, 1, 21, 8, 4, 2, 1, 1, 1, 34, 13, 6, 4, 2, 1, 1, 1, 55, 21, 11, 6, 3, 2, 1, 1, 1, 89, 34, 17, 9, 6, 3, 2, 1, 1, 1, 144, 55, 27, 15, 9, 5, 3, 2, 1, 1, 1, 233, 89, 45, 25, 14, 9, 5, 3, 2, 1, 1, 1, 377, 144, 72, 40, 23, 14, 8, 5, 3, 2, 1, 1, 1

Offset: 1

Mats Granvik, May 29 2010

			Table begins:
   1;
   2,  1;
   3,  1,  1;
   5,  2,  1,  1;
   8,  3,  1,  1,  1;
  13,  5,  3,  1,  1,  1;
  21,  8,  4,  2,  1,  1,  1;
  34, 13,  6,  4,  2,  1,  1,  1;
  55, 21, 11,  6,  3,  2,  1,  1,  1;
  89, 34, 17,  9,  6,  3,  2,  1,  1,  1;

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Cf. 1st column=A000045(n+1), 2nd=A000045, 3rd=A093040, 4th=A006498. Matrix inverse of A178535.

Cf. A051731, A129713.

A178534 := proc(n, k)
    option remember;
    if k= 1 then
        combinat[fibonacci](n+1) ;
    elif k > n then
        0 ;
    else
        add(procname(n-i, k-1), i=1..k-1)-add(procname(n-i, k), i=1..k-1) ;
    end if;
end proc:
seq(seq(A178534(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Oct 28 2010

T[n_, 1] := Fibonacci[n+1];
T[n_, k_] := T[n, k] = If[k > n, 0, Sum[T[n-i, k-1], {i, 1, k-1}] - Sum[T[n-i, k], {i, 1, k-1}]];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 23 2024 *)

T(n,k)=(n % k==0) + sum(j=1,n\k,fibonacci(n-j*k)) \\ Andrew Howroyd, Feb 23 2024

from sympy.core.cache import cacheit
from sympy import fibonacci
@cacheit
def A(n, k): return fibonacci(n + 1) if k==1 else 0 if k>n else sum([A(n - i, k - 1) for i in range(1, k)]) - sum([A(n - i, k) for i in range(1, k)])
for n in range(1, 13): print([A(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, Sep 15 2017

T(n,1) = A000045(n+1), k>1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1) - Sum_{i=1..k-1} T(n-i,k).
T(n,k) = A129713*A051731. - Mats Granvik, Oct 22 2010
From R. J. Mathar, Sep 16 2017: (Start)
G.f. 3rd column: x^3*(1+x)/((1-x-x^2)*(1+x+x^2)).
G.f. 4th column: x^4/((1-x-x^2)*(1+x^2)) =x^4*(1+x)/((1-x-x^2)*(1+x+x^2+x^3)).
G.f. 5th column: x^5*(1+x)/((1-x-x^2)*(1+x+x^2+x^3+x^4)).
G.f. 6th column: x^6/((1-x-x^2)*(1+x+x^2)*(1-x+x^2)) = x^6*(1+x)/((1-x-x^2)*(1+x+x^2+x^3+x^4+x^5)).
G.f. 7th column: x^7*(1+x)/((1-x-x^2)*(1+x+x^2+x^3+x^4+x^5+x^6)).
G.f. 8th column: x^8/((1-x-x^2)*(1+x^2)*(1+x^4)) = x^8*(1+x)/((1-x-x^2)*(1+x+x^2+x^3+x^4+x^5+x^6+x^7)).
Conjecture (by extrapolating): G.f. k-th column: x^k*(1-x^2)/((1-x-x^2)*(1-x^k)).
G.f.: (1-x^2)/(1-x-x^2)*Sum_{i>=1} (x*y)^i/(1-x^i) = (1-x^2)/(1-x-x^2)*A051731(x,y). (End)
T(n,k) = A051731(n,k) + Sum_{j=1..floor(n/k)} Fibonacci(n-j*k). - Andrew Howroyd, Feb 23 2024

A191373 Sum of binomial coefficients C(i+j,i) modulo 2 over all pairs (i,j) of positive integers satisfying 5i+j=n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 2, 4, 1, 2, 2, 4, 2, 3, 3, 5, 1, 3, 2, 5, 2, 3, 4, 6, 1, 3, 2, 6, 2, 3, 4, 6, 2, 4, 3, 7, 3, 5, 5, 8, 1, 4, 3, 8, 2, 3, 5, 8, 2, 4, 3, 8, 4, 6, 6, 9, 1, 5, 3, 9, 2, 3, 6, 9, 2, 4, 3, 9

Offset: 0

Johannes W. Meijer, Jun 05 2011

nonn

The Le1{1,5} and Le2{5,1} triangle sums of Sierpinski’s triangle A047999 equal this sequence; see the formulas for their definitions. The Le1{1,5} and Le2{5,1} triangle sums are similar to the Kn11 and Kn21 sums, the Ca1 and Ca2 sums, and the Gi1 and Gi2 sums, see A180662.

Some A191373(2^n-p) sequences, 0<=p<=32, lead to known sequences, see the crossrefs.

Sam Northshield, Sums across Pascal's triangle modulo 2, Congressus Numerantium, 200, pp. 35-52, 2010.

Cf. A001316 (1,1), A002487 (2,1), A120562 (3,1), A112970 (4,1), A191373 (5,1)

Cf. A000012 (p=0), A006498 (p=1, p=2, p=4, p=8, p=16, p=32), A070550 (p=3, p=6, p=12, p=24), A000071 (p=15, p=30), A115008 (p=23).

A191373:=proc(n) option remember; if n <0 then A191373(n):=0 fi: if (n=0 or n=1) then 1 elif n mod 2 = 0 then A191373(n/2) else A191373((n-1)/2) + A191373(((n-1)/2)-2); fi; end: seq(A191373(n),n=0..75);

a(2*n) = a(n) and a(2*n+1) = a(n) + a(n-2) with a(0) = 1, a(1) = 1 and a(n)=0 for n<=-1.
a(n) = Le1{1,5}(n) = add(T(n-4*k,k),k=0..floor(n/5))
a(n) = Le1{1,5}(n) = sum(binomial(i + j, i) mod 2 | (i + 5*j) = n)
a(n) = Le2{5,1}(n) = add(T(n-4*k,n-5*k),k=0..floor(n/5))
a(n) = Le2{5,1}(n) = sum(binomial(i + j, i) mod 2 | (5*i + j) = n)
G.f.: Product_{n>=0} (1+x^(2^n)+x^(5*2^n)).
G.f. A(x) satisfies: A(x) = (1 + x + x^5) * A(x^2). - Ilya Gutkovskiy, Jul 09 2019

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