A391682 - OEIS

%I #17 Feb 15 2026 08:58:19

%S 1,2,4,7,13,22,39,65,112,185,313,514,859,1405,2328,3797,6253,10178,

%T 16687,27121,44320,71953,117297,190274,309619,501941,815656,1321693,

%U 2145541,3475426,5637351,9129161,14799280,23961209,38826025,62852770,101809867,164793709

%N Number of integer compositions of n that are the first sums of more than one nonnegative sequence.

%C The first sums of a nonempty sequence (a, b, c, d, ...) are (a+b, b+c, c+d, ...).

%C A nonnegative sequence is not unique in having its first sums iff every other of its terms is at least 1. - _Christian Sievers_, Jan 09 2026

%H Christian Sievers, <a href="/A391682/b391682.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f.: (x+x^2-x^3-x^4)/(1-x-3*x^2+2*x^3+2*x^4). - _Christian Sievers_, Jan 11 2026

%e The composition (1,2,1) is the first sums of (0,1,1,0) only, so (1,2,1) is not counted under a(4).

%e The composition (2,1,2) is the first sums of both (1,1,0,2) and (2,0,1,1), so (2,1,2) is counted under a(5).

%e The a(1) = 1 through a(5) = 13 compositions:

%e   (1)  (2)    (3)      (4)        (5)

%e        (1,1)  (1,2)    (1,3)      (1,4)

%e               (2,1)    (2,2)      (2,3)

%e               (1,1,1)  (3,1)      (3,2)

%e                        (1,1,2)    (4,1)

%e                        (2,1,1)    (1,1,3)

%e                        (1,1,1,1)  (1,2,2)

%e                                   (2,1,2)

%e                                   (2,2,1)

%e                                   (3,1,1)

%e                                   (1,1,1,2)

%e                                   (2,1,1,1)

%e                                   (1,1,1,1,1)

%t pas[y_,k_]:=Table[(-1)^j*k+Sum[(-1)^(i+j)*y[[i]],{i,j}],{j,0,Length[y]}];

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Length[Select[Table[pas[#,b],{b,0,Max[#]}],Min@@#>=0&]]>1&]],{n,1,10}]

%o (PARI) a(n)=polcoef((x+x^2-x^3-x^4)/(1-x-3*x^2+2*x^3+2*x^4)+O(x*x^n),n) \\ _Christian Sievers_, Jan 11 2026

%Y These compositions are ranked by A391623.

%Y For a unique choice we have A391643, ranks A391622.

%Y For no choices we have A391645 (for all parts > 1 A391679), ranks A390747.

%Y For compositions we have:

%Y - more than one choice: A391628, ranks A391627

%Y - unique choice: A391644, ranks A390745

%Y - no choices: A391680, ranks A390677

%Y - at least one choice: A391683, ranks A390676 (union of A390568)

%Y A011782 counts compositions.

%Y A357213 counts compositions by sum of first sums.

%Y A390432 lists first sums of standard compositions.

%Y A390673 ranks compositions with distinct first sums, counted by A390567.

%Y A391621 counts nonnegative sequences with standard first sums.

%Y Cf. A008965, A070211, A342527, A390448, A390675, A390678, A391235, A391641, A391642, A391983.

%K nonn,easy

%O 1,2

%A _Gus Wiseman_, Jan 08 2026

%E a(21) onward from _Christian Sievers_, Jan 11 2026