Rob Bunce has authored 3 sequences.
A376325
A Catalan-like sequence formed by summing the truncation of the terms of a Catalan Triangle A009766 where the number of row terms are truncated to ceiling((n+3)*log(3)/log(2)) - (n+4).
Original entry on oeis.org
1, 2, 5, 14, 28, 76, 151, 412, 1239, 2689, 7724, 16351, 46607, 98276, 280035, 871218, 1967577, 5819850, 12749014, 37260985, 118163637, 272787542, 819934670, 1829959304, 5422130623, 11963162678, 35243160809, 112614062317, 262572197079, 795710438547, 1794155974237
Offset: 1
When n=6, number of terms is restricted to 5 dropping 1 term; ceiling((6+3)*log(3)/log(2)) - (6+4) = 5.
When n=10, number of terms is restricted to 7 dropping 3 terms; ceiling((10+3)*log(3)/log(2)) - (10+4) = 7.
Truncating A009766 at the point indicated by - and summing the remaining triangle terms in the normal way results in:
row sum truncated triangle terms
1 1 = 1;
2 2 = 1, 1;
3 5 = 1, 2, 2;
4 14 = 1, 3, 5, 5;
5 28 = 1, 4, 9, 14, -;
6 76 = 1, 5, 14, 28, 28, -;
7 151 = 1, 6, 20, 48, 76, -, -;
8 412 = 1, 7, 27, 75, 151, 151, -, -;
9 1239 = 1, 8, 35, 110, 261, 412, 412, -, -;
10 2689 = 1, 9, 44, 154, 415, 827, 1239, -, -, -;
...
-
lista(nn) = {
my(terms(j)=ceil((j+3)*log(3)/log(2)) - (j+4));
my(T=vector(nn));
my(S=vector(nn));
for(y=1, nn,
if(y==1,
T[1]=[1];
S[1]=1
,
my(k=terms(y));
T[y]=vector(k);
for(i=1, k, if(i==1,T[y][i]=1,if(i<=length(T[y-1]),T[y][i]=T[y-1][i]+T[y][i-1],T[y][i]=T[y][i-1])));
S[y]=vecsum(T[y])
);
);
S;
}
A376319
A Catalan-like sequence formed by summing the truncation of the terms of the fourth convolution of the Catalan Triangle where the number of row terms are truncated to ceiling((n+4)*log(3)/log(2)) - (n+4).
Original entry on oeis.org
1, 4, 14, 34, 103, 228, 665, 2096, 4787, 14239, 31330, 91728, 199328, 580128, 1834665, 4223092, 12667903, 28207395, 83435822, 267154051, 623837740, 1891453021, 4265101202, 12735718304, 28359351604, 84126071303, 270338873771, 634653510356, 1933488496208
Offset: 1
When n=6, number of terms is restricted to 6, dropping 2 terms from the standard triangle; ceiling((6+4)*log(3)/log(2)) - (6+4) = 6.
When n=9, number of terms is restricted to 8, dropping 3 terms; ceiling((9+4)*log(3)/log(2)) - (9+4) = 8.
etc.
Truncating A002057 at this point, with dropped terms indicated by - and summing the remaining triangle terms in the normal way results in:
n sum truncated triangle terms
1 1 = 1;
2 4 = 1, 1, 1, 1;
3 14 = 1, 2, 3, 4, 4;
4 34 = 1, 3, 6, 10, 14, -;
5 103 = 1, 4, 10, 20, 34, 34, -;
6 228 = 1, 5, 15, 35, 69, 103, -, -;
7 665 = 1, 6, 21, 56, 125, 228, 228, -, -;
8 2096 = 1, 7, 28, 84, 209, 437, 665, 665, -, -;
9 4787 = 1, 8, 36, 120, 329, 766, 1431, 2096, -, -, -;
10 14239 = 1, 9, 45, 165, 494, 1260, 2691, 4787, 4787, -, -, -;
...
-
lista(nn) = {
my(terms(j)=ceil((j+4)*log(3)/log(2)) - (j+4));
my(T=vector(nn));
my(S=vector(nn));
for(y=1, nn,
if(y==1,
T[1]=[1];
S[1]=1
,
my(k=terms(y));
T[y]=vector(k);
for(i=1, k, if(i==1,T[y][i]=1,if(i<=length(T[y-1]),T[y][i]=T[y-1][i]+T[y][i-1],T[y][i]=T[y][i-1])));
S[y]=vecsum(T[y])
);
);
S;
}
A374244
A Catalan-like sequence formed from the row sums of a Catalan-like triangle where row n is truncated to have ceiling((n+4)*log(3)/log(2)) - (n + 6) terms.
Original entry on oeis.org
1, 2, 5, 9, 23, 43, 113, 331, 698, 1966, 4072, 11433, 23701, 66734, 205712, 459632, 1348864, 2927822, 8499580, 26809375, 61495590, 183946295, 408179706, 1204202538, 2643267587, 7756962475, 24708004563, 57390010121, 173405214133, 389606249120, 1160606285961, 3738436950162
Offset: 1
Standard Catalan:
n Sum Triangle terms
1 1 = 1;
2 2 = 1, 1;
3 5 = 1, 2, 2;
4 14 = 1, 3, 5; /5
5 42 = 1, 4, 9, 14; /14
6 132 = 1, 5, 14, 28; /42; 14
7 429 = 1, 6, 20, 48, 90; /132; 132
...
When n=4, number of terms is restricted to 3 dropping 1 term; ceiling((4+4)*log(3)/log(2)) - (4 + 6) = 3.
When n=6, number of terms is restricted to 4 dropping 2 terms; ceiling((6+4)*log(3)/log(2)) - (6 + 6) = 4.
etc.
Truncating at the point indicated by / and summing the remaining triangle terms in the normal way results in:
n Sum Truncated Triangle terms
1 1 = 1;
2 2 = 1, 1;
3 5 = 1, 2, 2;
4 9 = 1, 3, 5;
5 23 = 1, 4, 9, 9;
6 43 = 1, 5, 14, 23;
7 113 = 1, 6, 20, 43, 43;
8 331 = 1, 7, 27, 70, 113, 113;
9 698 = 1, 8, 35, 105, 218, 331;
10 1966 = 1, 9, 44, 149, 367, 698, 698;
11 4072 = 1, 10, 54, 203, 570, 1268, 1966;
12 11433 = 1, 11, 65, 268, 838, 2106, 4072, 4072;
13 23701 = 1, 12, 77, 345, 1183, 3289, 7361, 11433;
...
-
lista(N) = {
my(T=vector(N, n, vector(logint(3^(n+4), 2)-n-5)));
for(n=1, #T
, for(k=1, #T[n]
, T[n][k]= if(1==k, 1, k<=#T[n-1], T[n][k-1]+T[n-1][k], T[n][k-1])
);
);
vector(#T, n, vecsum(T[n]));
}
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