Displaying 31-40 of 63 results found.
Total number of odd coefficients in (1+x+x^2)^k for k=0,...,n.
+10
7
1, 4, 7, 12, 15, 24, 29, 40, 43, 52, 61, 76, 81, 96, 107, 128, 131, 140, 149, 164, 173, 200, 215, 248, 253, 268, 283, 308, 319, 352, 373, 416, 419, 428, 437, 452, 461, 488, 503, 536, 545, 572, 599, 644, 659, 704, 737, 800, 805, 820, 835, 860, 875, 920, 945, 1000
MATHEMATICA
Sum[PolynomialMod[(1+x+x^2)^k, 2] /. x->1, {k, 0, n-1}]
Number of triangles after n-th stage in a cellular automaton based in isosceles triangles of two sizes (see Comments lines for precise definition).
+10
7
0, 1, 3, 7, 11, 15, 23, 33, 41, 45, 53, 65, 81, 91, 111, 133, 149, 153, 161, 173, 189, 201, 225, 253, 285, 295, 315, 343, 383, 405, 449, 495, 527, 531, 539, 551, 567, 579, 603, 631, 663, 675, 699, 731, 779, 807, 863, 923, 987, 997, 1017, 1045, 1085, 1113, 1169, 1233, 1313, 1335, 1379, 1439, 1527, 1573, 1665, 1759, 1823
COMMENTS
On the semi-infinite square grid the structure of this C.A. contains "black" triangles and "gray" triangles (see the Links section). Both types of triangles have two sides of length 5^(1/2). Every black triangle has a base of length 2 hence its height is 2 and its area is 2. Every gray triangle has a base of length 2^(1/2) hence its height is 3/(2^(1/2)) and its area is 3/2. Both types of triangles are arranged in the same way as the triangles of Sierpinski gasket (see A047999 and A006046). The black triangles are arranged in vertical direction. On the other hand the gray triangles are arranged in diagonal direction in the holes of the structure formed by the black triangles. Note that the vertices of all triangles coincide with the grid points. The sequence gives the total number of triangles (black and gray) in the structure after n-th stage. A231349 (the first differences) gives the number of triangles added at n-th stage. For a more complex structure see A233780.
EXAMPLE
We start at stage 0 with no triangles, so a(0) = 0.
At stage 1 we add a black triangle, so a(1) = 1.
At stage 2 we add two black triangles, so a(2) = 1+2 = 3.
At stage 3 we add two black triangles and two gray triangles from the vertices of the master triangle, so a(3) = 3+2+2 = 7.
At stage 4 we add four black triangles, so a(4) = 7+4 = 11.
At stage 5 we add two black triangles and two gray triangles from the vertices of the master triangle, so a(5) = 11+2+2 = 15.
At stage 6 we add four black triangles and four gray triangles, so a(6) = 15+4+4 = 23.
At stage 7 we add four black triangles and six gray triangles, so a(7) = 23+4+6 = 33.
At stage 8 we add eight black triangles, so a(8) = 33+8 = 41.
And so on.
Note that always we add both black triangles and gray triangles except if n is a power of 2. In this case at stage 2^k we add only 2^k black triangles, for k >= 0.
CROSSREFS
Cf. A047999, A006046, A139250, A151566, A160406, A173530, A182634, A194444, A220524, A220478, A230981, A231349, A233780.
If n = binomial(b,2)+binomial(c,1), b>c>=0 then a(n) = binomial(b+1,3)+binomial(c+1,2).
+10
5
1, 2, 4, 5, 7, 10, 11, 13, 16, 20, 21, 23, 26, 30, 35, 36, 38, 41, 45, 50, 56, 57, 59, 62, 66, 71, 77, 84, 85, 87, 90, 94, 99, 105, 112, 120, 121, 123, 126, 130, 135, 141, 148, 156, 165, 166, 168, 171, 175, 180, 186, 193, 201, 210, 220, 221, 223, 226, 230, 235, 241
COMMENTS
Triangle-tree numbers: a(n) = sum(b(m), m = 1..n), b(m) = 1,1,2,1,2,3,1,2,3,4,... = A002260. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
REFERENCES
W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge, 1993, p. 159.
FORMULA
a(n*(n+1)/2+m)=n*(n+1)*(n+2)/6 + m*(m+1)/2= A000292(n)+ A000217(m), m=0...n+1, n=1, 2, 3.. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de) a(n) = a(n-1)+ A002260(n). As a triangle with n >= k >= 1: a(n, k) =a(n-1, k)+(n-1)*n/2 =a(n, k-1)+k =(n^3-n+3k^2+3k)/6. - Henry Bottomley, Nov 14 2001 a(n) = b(n) * (b(n) + 1) * (b(n) + 2) / 6 + c(n) * (c(n) + 1) / 2, where b(n) = [sqrt(2 * n) - 1/2] and c(n) = n - b(n) * (b(n) + 1) / 2 - Robert A. Stump (bee_ess107(AT)msn.com), Sep 20 2002
EXAMPLE
The triangle starts:
1
2 4
5 7 10
11 13 16 20
21 23 26 30 35
MAPLE
a := 0: for i from 1 to 15 do for j from 1 to i do a := a+j: printf(`%d, `, a); od:od:
a(n) = Sum_{k=1..n} 2^b(k) where b(k) denotes the number of 1's in the binary representation of k.
+10
5
2, 4, 8, 10, 14, 18, 26, 28, 32, 36, 44, 48, 56, 64, 80, 82, 86, 90, 98, 102, 110, 118, 134, 138, 146, 154, 170, 178, 194, 210, 242, 244, 248, 252, 260, 264, 272, 280, 296, 300, 308, 316, 332, 340, 356, 372, 404, 408, 416, 424, 440, 448, 464, 480, 512, 520, 536
FORMULA
a(0)=0, a(2n) = 2a(n-1) + a(n) + 2, a(2n+1) = 3a(n) + 2.
G.f.: (1/(1-x)) * Product_{k>=0} (1 + 2x^2^k). (End)
MAPLE
f:= proc(n) option remember; if n::even then 2*procname(n/2-1)+procname(n/2)+2
else 3*procname((n-1)/2)+2
fi
end proc:
f(0):= 0:
MATHEMATICA
b[n_] := IntegerDigits[n, 2] // Total;
a[n_] := 2^(b /@ Range[n]) // Total;
PROG
(PARI) a(n)=sum(i=1, n, 2^sum(k=1, length(binary(i)), component(binary(i), k)))
a(0)=1, a(1)=1, a(n) = 11*a(n/2) for even n, and a(n) = 10*a((n-1)/2) + a((n+1)/2) for odd n >= 3.
+10
5
0, 1, 11, 21, 121, 131, 231, 331, 1331, 1341, 1441, 1541, 2541, 2641, 3641, 4641, 14641, 14651, 14751, 14851, 15851, 15951, 16951, 17951, 27951, 28051, 29051, 30051, 40051, 41051, 51051, 61051, 161051, 161061, 161161, 161261, 162261, 162361, 163361, 164361
COMMENTS
Let M =
1, 0, 0, 0, 0, ...
11, 0, 0, 0, 0, ...
10, 1, 0, 0, 0, ...
0, 11, 0, 0, 0, ...
0, 10, 1, 0, 0, ...
0, 0, 11, 0, 0, ...
0, 0, 10, 1, 0, ...
...
Then lim_{k->infinity} M^k converges to a single nonzero column giving the sequence.
The sequence divided by its aerated variant is (1, 11, 10, 0, 0, 0, ...). (End)
FORMULA
Let r(x) = (1 + 11x + 10x^2). The sequence is r(x) * r(x^2) * r(x^4) * r(x^8) * ... - Gary W. Adamson, Aug 30 2016
MAPLE
a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 11*a(n/2) else 10*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n), n=0..42);
MATHEMATICA
b[0] := 0; b[1] := 1; b[n_?EvenQ] := b[n] = 11*b[n/2]; b[n_?OddQ] := b[n] = 10*b[(n - 1)/2] + b[(n + 1)/2]; a = Table[b[n], {n, 1, 25}]
a(0)=1, a(1)=1, a(n) = 9*a(n/2) for even n >= 2, and a(n) = 8*a((n-1)/2) + a((n+1)/2) for odd n >= 3.
+10
5
0, 1, 9, 17, 81, 89, 153, 217, 729, 737, 801, 865, 1377, 1441, 1953, 2465, 6561, 6569, 6633, 6697, 7209, 7273, 7785, 8297, 12393, 12457, 12969, 13481, 17577, 18089, 22185, 26281, 59049, 59057, 59121, 59185, 59697, 59761, 60273, 60785, 64881, 64945, 65457, 65969
COMMENTS
The interest this one has is in the prime form of even odd 2^n+1, 2^n.
Let M =
1, 0, 0, 0, 0, ...
9, 0, 0, 0, 0, ...
8, 1, 0, 0, 0, ...
0, 9, 0, 0, 0, ...
0, 8, 1, 0, 0, ...
0, 0, 9, 0, 0, ...
0, 0, 8, 1, 0, ...
...
Then M^k converges to a single nonzero column giving the sequence.
The sequence divided by its aerated variant is (1, 9, 8, 0, 0, 0, ...). (End)
MAPLE
a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 9*a(n/2) else 8*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n), n=0..45);
MATHEMATICA
b[0] := 0; b[1] := 1; b[n_?EvenQ] := b[n] = 9*b[n/2]; b[n_?OddQ] := b[n] = 8*b[(n - 1)/2] + b[(n + 1)/2]; a = Table[b[n], {n, 1, 25}]
Number of entries in the n-th row of Pascal's triangle not divisible by 5.
+10
5
1, 2, 3, 4, 5, 2, 4, 6, 8, 10, 3, 6, 9, 12, 15, 4, 8, 12, 16, 20, 5, 10, 15, 20, 25, 2, 4, 6, 8, 10, 4, 8, 12, 16, 20, 6, 12, 18, 24, 30, 8, 16, 24, 32, 40, 10, 20, 30, 40, 50, 3, 6, 9, 12, 15, 6, 12, 18, 24, 30, 9, 18, 27, 36, 45, 12, 24, 36, 48, 60, 15, 30
COMMENTS
Pascal triangles modulo p with p prime have the dimension D = log(p*(p+1)/2)/log(p). [Corrected by Connor Lane, Nov 28 2022]
FORMULA
a(n) = Product_{d=1..4} (d+1)^b(n,d) with b(n,d) = number of digits d in base 5 expansion of n. The formula generalizes to other prime bases p.
EXAMPLE
n = 32 = 112|_5: b(32,1) = 2, b(32,2) = 1, thus a(32) = 2^2 * 3^1 = 12.
MAPLE
a:= proc(n) local l, m, t;
m:= n;
l:= [0$5];
while m>0 do t:= irem(m, 5, 'm')+1; l[t]:=l[t]+1 od;
mul(r^l[r], r=2..5)
end:
seq(a(n), n=0..100);
PROG
(Haskell)
a194459 = sum . map (signum . flip mod 5) . a007318_row
1, 5, 9, 13, 17, 33, 37, 45, 49, 65, 81, 85, 89, 105, 113, 129, 133, 149, 165, 181, 197, 261, 265, 273, 277, 293, 309, 317, 325, 357, 373, 405, 409, 425, 441, 457, 473, 537, 553, 585, 601, 665, 729, 733, 737, 753, 761, 777, 781, 797, 813, 829, 845, 909, 917, 933, 941, 973, 1005, 1021, 1037, 1101, 1133, 1197
1, 4, 7, 14, 17, 26, 33, 46, 49, 58, 67, 86, 93, 114, 127, 154, 157, 166, 175, 196, 205, 232, 251, 286, 293, 314, 335, 376, 389, 428, 455, 510, 513, 522, 531, 552, 561, 588, 609, 648, 657, 684, 711, 766, 785, 842, 877, 950, 957, 978, 999, 1048, 1069, 1132, 1173, 1248, 1261, 1300, 1339, 1418, 1445, 1526, 1581, 1690
1, 4, 7, 16, 19, 26, 35, 52, 55, 64, 71, 92, 101, 118, 135, 168, 171, 180, 189, 216, 223, 240, 261, 304, 313, 340, 357, 408, 425, 460, 493, 560, 563, 572, 581, 608, 617, 638, 665, 716, 723, 744, 761, 812, 833, 874, 917, 1000, 1009, 1036, 1063, 1144, 1161, 1204, 1255, 1368, 1385, 1436, 1471, 1576, 1609, 1676, 1743, 1880
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