A253145
Triangular numbers (A000217) omitting the term 1.
Original entry on oeis.org
0, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275
Offset: 0
Cf.
A000217,
A179865,
A001045,
A062510,
A002378,
A046092,
A033996,
A122803,
A007283,
A091629,
A020714,
A110286,
A161680,
A175805,
A005010,
A194880,
A001840,
A022003,
A255935.
-
Concatenation([0],List([1..50],n->(n+1)*(n+2)/2)); # Muniru A Asiru, Oct 31 2018
-
Prepend[Table[(n + 1) (n + 2)/2, {n, 49}], 0] (* Michael De Vlieger, Mar 23 2015 *)
-
a(n)=if(n,(n+1)*(n+2)/2,0) \\ Charles R Greathouse IV, Mar 23 2015
A321483
a(n) = 7*2^n + (-1)^n.
Original entry on oeis.org
8, 13, 29, 55, 113, 223, 449, 895, 1793, 3583, 7169, 14335, 28673, 57343, 114689, 229375, 458753, 917503, 1835009, 3670015, 7340033, 14680063, 29360129, 58720255, 117440513, 234881023, 469762049, 939524095, 1879048193, 3758096383, 7516192769, 15032385535
Offset: 0
Cf.
A000079,
A001045,
A005029,
A010710,
A014551,
A062092,
A062510,
A070366,
A175805,
A199207,
A206372.
-
a[n_] := 7*2^n + (-1)^n ; Array[a, 32, 0] (* Amiram Eldar, Nov 12 2018 *)
CoefficientList[Series[E^-x + 7 E^(2 x), {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* Stefano Spezia, Nov 12 2018 *)
LinearRecurrence[{1,2},{8,13},40] (* Harvey P. Dale, Mar 18 2022 *)
-
Vec((8 + 5*x) / ((1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Nov 11 2018
Two terms corrected, and more terms added by
Colin Barker, Nov 11 2018
A159256
a(0)=131; for n > 0, a(n) = a(n-1) + floor(sqrt(a(n-1))).
Original entry on oeis.org
131, 142, 153, 165, 177, 190, 203, 217, 231, 246, 261, 277, 293, 310, 327, 345, 363, 382, 401, 421, 441, 462, 483, 504, 526, 548, 571, 594, 618, 642, 667, 692, 718, 744, 771, 798, 826, 854, 883, 912, 942, 972, 1003, 1034, 1066, 1098, 1131, 1164, 1198
Offset: 0
-
RecurrenceTable[{a[0]==131,a[n]==a[n-1]+Floor[Sqrt[a[n-1]]]},a,{n,50}] (* Harvey P. Dale, Apr 17 2013 *)
NestList[#+Floor[Sqrt[#]]&,131,50] (* Harvey P. Dale, May 11 2019 *)
Original entry on oeis.org
1, -3, -1, 5, -6, 3, -11, 10, -12, -5, 21, -22, 20, -24, 11, -43, 42, -44, 40, -48, -21, 85, -86, 84, -88, 80, -96, 43, -171, 170, -172, 168, -176, 160, -192, -85, 341, -342, 340, -344, 336, -352, 320, -384, 171, -683, 682, -684, 680, -688
Offset: 0
Cf.
A000096,
A000217,
A005015,
A001045,
A007283,
A020988,
A077925,
A078008,
A140503,
A146523,
A151575,
A175805,
A084247.
-
T[0, 0] = 0; T[1, 0] = T[0, 1] = 1; T[0, n_] := T[0, n] = T[0, n - 1] + 2*T[0, n - 2]; T[d_, d_] = 0; T[d_, n_] := T[d, n] = T[d - 1, n + 1] - T[d - 1, n]; A140944 = Table[T[d, n], {d, 0, 10}, {n, 0, d}] // Flatten; a[n_] := A140944[[n + 2]] - 3*A140944[[n + 1]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 18 2014 *)
A322417
a(n) - 2*a(n-1) = period 2: repeat [3, 0] for n > 0, a(0)=5, a(1)=13.
Original entry on oeis.org
5, 13, 26, 55, 110, 223, 446, 895, 1790, 3583, 7166, 14335, 28670, 57343, 114686, 229375, 458750, 917503, 1835006, 3670015, 7340030, 14680063, 29360126, 58720255, 117440510, 234881023, 469762046, 939524095, 1879048190, 3758096383, 7516192766
Offset: 0
-
a:=[13,26];; for n in [3..30] do a[n]:=a[n-2]+21*2^(n-2); od; Concatenation([5],a); # Muniru A Asiru, Dec 07 2018
-
a[0] = 5; a[1] = 13; a[n_] := a[n] = a[n - 2] + 21*2^(n - 2); Array[a, 30, 0] (* Amiram Eldar, Dec 07 2018 *)
LinearRecurrence[{2, 1, -2}, {5, 13, 26}, 31] (* Jean-François Alcover, Jan 28 2019 *)
-
Vec((5 + 3*x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Dec 07 2018
A320227
Assuming the truth of the Collatz conjecture, let T(n,i), i = 1..k be the initial k elements of the Collatz trajectory of n, up to when the first 1 appears, but excluding the 1. a(n) is the number of ordered pairs T(n,i) < T(n,j) such that gcd(T(n,i), T(n,j)) = 1.
Original entry on oeis.org
0, 0, 10, 0, 4, 11, 58, 0, 84, 4, 40, 12, 12, 62, 47, 0, 25, 89, 89, 4, 6, 43, 36, 13, 117, 13, 3395, 66, 66, 49, 3064, 0, 148, 27, 21, 94, 94, 94, 286, 4, 3246, 6, 184, 46, 42, 39, 2924, 14, 122, 122, 120, 14, 14, 3435, 3374, 70, 231, 70, 247, 51, 63, 3101
Offset: 1
a(3) = 10 because the Collatz trajectory T(3,i) of 3 up to the number 1 is 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 and gcd(T(i), T(j)) = 1 for the 10 following pairs of elements of T: (2, 3), (2, 5), (3, 4), (3, 5), (3, 8), (3, 10), (3, 16), (4, 5), (5, 8) and (5, 16). 28
In the general case, a(n) = 10 for n in the set {3} union {341, 682, 1364, 2728, ...,((4^5 - 1)/3)*2^k, ...} with k = 0, 1, 2, ...
a(6) = 11 because the Collatz trajectory T(6,i) of 6 up to the number 1 is 6 -> 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 and gcd(T(i), T(j)) = 1 for the 11 following pairs of elements of T: (2, 3), (2, 5), (3, 4), (3, 5), (3, 8), (3, 10), (3, 16), (4, 5), (5, 6), (5, 8) and (5, 16).
-
nn:=1000:
for n from 1 to 200 do:
m:=n:lst:={}:
for i from 1 to nn while(m<>1) do:
if irem(m, 2)=0
then
lst:=lst union {m}:m:=m/2:
else
lst:=lst union {m}:m:=3*m+1:
fi:
od:
n0:=nops(lst):it:=0:
for j from 1 to n0-1 do:
for k from j+1 to n0 do:
if gcd(lst[j],lst[k])=1
then
it:=it+1:
else fi:
od:
od:
printf(`%d, `,it):
od:
A323097
Numbers m such that all elements of the Collatz trajectory occur in the divisors of m.
Original entry on oeis.org
1, 2, 4, 8, 16, 32, 64, 80, 128, 160, 256, 320, 512, 640, 1024, 1280, 1344, 2048, 2560, 2688, 4096, 5120, 5376, 8192, 10240, 10752, 16384, 20480, 21504, 21760, 32768, 40960, 43008, 43520, 65536, 81920, 86016, 87040, 131072, 163840, 172032, 174080, 262144, 327680
Offset: 1
1344 is in the sequence because the set of the divisors {1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 32, 42, 48, 56, 64, 84, 96, 112, 168, 192, 224, 336, 448, 672, 1344} contains the set of the elements of the Collatz trajectory 1344 -> 672 -> 336 -> 168 -> 84 -> 42 -> 21 -> 64 -> 32 -> 16 -> 8 -> 4 -> 2 -> 1
-
with(numtheory):nn:=250000:
for n from 1 to nn do:
m:=n:it:=0:lst:={n}:
for i from 1 to nn while(m<>1) do:
if irem(m, 2)=0
then
m:=m/2:
else
m:=3*m+1:
fi:
it:=it+1:lst:=lst union {m}:
od:
x:=divisors(n):n0:=nops(x):lst1:={op(x), x[n0]}:
lst2:=lst intersect lst1:n1:=nops(lst2):
if lst2=lst
then
printf(`%d, `,n):
else fi:
od:
-
aQ[n_] := n == LCM @@ NestWhileList[If[OddQ[#], 3 # + 1, #/2] &, n, # > 1 &]; Select[Range[330000], aQ] (* Amiram Eldar, Aug 31 2019 *)
Original entry on oeis.org
0, 1, 2, 4, 8, 16, 21, 32, 42, 64, 84, 128, 168, 256, 336, 512, 672, 1024, 1344, 2048, 2231, 2688, 4096, 4462, 5376, 8192, 9324, 10752, 16384, 18648, 21504, 32768, 37296, 43008, 65536, 74592, 86016, 131072, 149184, 172032, 262144, 298368, 344064, 524288, 596736, 688128, 1048576
Offset: 1
The least positive multiple of 42 that when written in base 10 uses only 0's and 1's is 101010 = 2405*42. If we regard 101010 as binary number and converts to base 10, we get 42; hence, 42 is a term.
Successive operations for first primitive terms:
1 --> A004290(1) = 1_{10} --> 1_{2} = 1_{10},
21 --> A004290(21) = 10101_{10} --> 10101_{2} = 21_{10},
2231 --> A004290(2231) = 100010110111_{10} --> 100010110111_{2} = 2231_{10}.
-
f(n) = {if( n==0, return (0)); my(m = n); while (vecmax(digits(m)) != 1, m+=n); m; } \\ A004290
isok(m) = fromdigits(digits(f(m), 10), 2) == m; \\ Michel Marcus, May 29 2020
A352692
a(n) + a(n+1) = 2^n for n >= 0 with a(0) = 4.
Original entry on oeis.org
4, -3, 5, -1, 9, 7, 25, 39, 89, 167, 345, 679, 1369, 2727, 5465, 10919, 21849, 43687, 87385, 174759, 349529, 699047, 1398105, 2796199, 5592409, 11184807, 22369625, 44739239, 89478489, 178956967, 357913945, 715827879, 1431655769, 2863311527, 5726623065, 11453246119, 22906492249
Offset: 0
-
a := proc(n) option remember; ifelse(n = 0, 4, 2^(n-1) - a(n-1)) end: # Peter Luschny, Mar 29 2022
A352691 := proc(n)
(11*(-1)^n + 2^n)/3
end proc: # R. J. Mathar, Apr 26 2022
-
LinearRecurrence[{1, 2}, {4, -3}, 40] (* Amiram Eldar, Mar 29 2022 *)
-
a(n) = (11*(-1)^n + 2^n)/3; \\ Thomas Scheuerle, Mar 29 2022
Warning: The DATA is correct, but there may be errors in the COMMENTS, which should be rechecked. - Editors of OEIS, Apr 26 2022
Comments