0,3

Table A125790 is related to partitions into powers of 2, with A002577 in column 1 of A125790; further, column k of A125790 equals row sums of matrix power A078121^k, where triangle A078121 shifts left one column under matrix square.

Also number of distinct instances of the one-in-three monotone 3SAT problem for n variables. - Paul Tarau (paul.tarau(AT)gmail.com), Jan 25 2008

a(n)=2^(C(3+n,n)), n>=-2. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 16 2007

a(n) is a pyramidal power of 2; exponents of 2 in a(n) begin:

[0, 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, ..., n(n-1)(n-2)/6, ...].

seq(2^(binomial(3+n, n)), n=-2..8); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 16 2007

(PARI) a(n)=if(n<1, 0, 2^(n*(n-1)*(n-2)/6)) (PARI) /* As determinant of n X n matrix: */ {a(n)=local(q=2, A=Mat(1), B); for(m=1, n, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i|j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(matdet(matrix(n, n, r, c, (A^c)[r, 1])))}

(Prolog program from Paul Tarau (paul.tarau(AT)gmail.com), Jan 25 2008) This generates all 3SAT problem instances

test:-test(4).

test(Max):-

between(1, Max, N),

nl,

one_in_three_monotone_3sat(N, Pss),

write(N:Pss), nl,

fail

; nl.

% generates all one-in-three monotone 3SAT problems involving N variables

one_in_three_monotone_3sat(N, Pss):-

ints(1, N, Is),

findall(Xs, ksubset(3, Is, Xs), Xss),

subset_of(Xss, Pss).

% subset generator

subset_of([], []).

subset_of([X|Xs], Zs):-

subset_of(Xs, Ys),

add_element(X, Ys, Zs).

add_element(_, Ys, Ys).

add_element(X, Ys, [X|Ys]).

% subsets of K elements

ksubset(0, _, []).

ksubset(K, [X|Xs], [X|Rs]):-K&gt; 0, K1 is K-1, ksubset(K1, Xs, Rs).

ksubset(K, [_|Xs], Rs):-K&gt; 0, ksubset(K, Xs, Rs).

% list of integers in [From..To]

ints(From, To, Is):-findall(I, between(From, To, I), Is).

Cf. A125790, A078121; A002577, A000292 (pyramidal numbers).

Sequence in context: A013178 A084595 A002543 this_sequence A102103 A060597 A091479

Adjacent sequences: A125788 A125789 A125790 this_sequence A125792 A125793 A125794

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 10 2006