A210728
a(n) = a(n-1) + a(n-2) + n + 2 with n>1, a(0)=1, a(1)=2.
Original entry on oeis.org
1, 2, 7, 14, 27, 48, 83, 140, 233, 384, 629, 1026, 1669, 2710, 4395, 7122, 11535, 18676, 30231, 48928, 79181, 128132, 207337, 335494, 542857, 878378, 1421263, 2299670, 3720963, 6020664, 9741659, 15762356, 25504049, 41266440, 66770525, 108037002, 174807565
Offset: 0
Cf.
A065220: a(n)=a(n-1)+a(n-2)+n-5, a(0)=1,a(1)=2 (except first 2 terms).
Cf.
A168043: a(n)=a(n-1)+a(n-2)+n-3, a(0)=1,a(1)=2 (except first 2 terms).
Cf.
A131269: a(n)=a(n-1)+a(n-2)+n-2, a(0)=1,a(1)=2.
Cf.
A000126: a(n)=a(n-1)+a(n-2)+n-1, a(0)=1,a(1)=2.
Cf.
A104161: a(n)=a(n-1)+a(n-2)+n, a(0)=1,a(1)=2 (except the first term).
Cf.
A192969: a(n)=a(n-1)+a(n-2)+n+1, a(0)=1,a(1)=2.
Cf.
A210729: a(n)=a(n-1)+a(n-2)+n+3, a(0)=1,a(1)=2.
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RecurrenceTable[{a[0] == 1, a[1] == 2, a[n] == a[n - 1] + a[n - 2] + n + 2}, a, {n, 36}] (* Bruno Berselli, Jun 27 2012 *)
nxt[{n_,a_,b_}]:={n+1,b,a+b+n+3}; NestList[nxt,{1,1,2},40][[;;,2]] (* Harvey P. Dale, Aug 26 2024 *)
A210729
a(n) = a(n-1) + a(n-2) + n + 3 with n>1, a(0)=1, a(1)=2.
Original entry on oeis.org
1, 2, 8, 16, 31, 55, 95, 160, 266, 438, 717, 1169, 1901, 3086, 5004, 8108, 13131, 21259, 34411, 55692, 90126, 145842, 235993, 381861, 617881, 999770, 1617680, 2617480, 4235191, 6852703, 11087927, 17940664, 29028626, 46969326, 75997989, 122967353
Offset: 0
Cf.
A065220: a(n)=a(n-1)+a(n-2)+n-5, a(0)=1,a(1)=2 (except first 2 terms).
Cf.
A168043: a(n)=a(n-1)+a(n-2)+n-3, a(0)=1,a(1)=2 (except first 2 terms).
Cf.
A131269: a(n)=a(n-1)+a(n-2)+n-2, a(0)=1,a(1)=2.
Cf.
A000126: a(n)=a(n-1)+a(n-2)+n-1, a(0)=1,a(1)=2.
Cf.
A104161: a(n)=a(n-1)+a(n-2)+n, a(0)=1,a(1)=2 (except the first term).
Cf.
A192969: a(n)=a(n-1)+a(n-2)+n+1, a(0)=1,a(1)=2.
Cf.
A210728: a(n)=a(n-1)+a(n-2)+n+2, a(0)=1,a(1)=2.
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F:=Fibonacci;; List([0..40], n-> 2*F(n+3)+3*F(n+1)-n-6); # G. C. Greubel, Jul 09 2019
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[3*Fibonacci(n+1)+2*Fibonacci(n+3)-n-6: n in [0..40]]; // Vincenzo Librandi, Jul 18 2013
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Table[3*Fibonacci[n+1]+2*Fibonacci[n+3]-n-6,{n,0,40}] (* Vaclav Kotesovec, May 13 2012 *)
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vector(40, n, n--; f=fibonacci; 2*f(n+3)+3*f(n+1)-n-6) \\ G. C. Greubel, Jul 09 2019
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prpr, prev = 1,2
for n in range(2, 99):
current = prev+prpr+n+3
print(prpr, end=',')
prpr = prev
prev = current
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f=fibonacci; [2*f(n+3)+3*f(n+1)-n-6 for n in (0..40)] # G. C. Greubel, Jul 09 2019
A123247
Let S(1)={1} and, for n>1 let S(n) be the smallest set containing x, x+1, 2x and 3x for each element x in S(n-1). a(n) is the number of elements in S(n).
Original entry on oeis.org
1, 3, 6, 13, 27, 54, 107, 213, 423, 845, 1685, 3371, 6735, 13468, 26937, 53900, 107873, 216035, 432787, 867313, 1738728, 3486464, 6993111, 14029776, 28153533, 56507114, 113435141, 227755613, 457358671, 918562597
Offset: 1
Under the indicated set mapping we have {1} -> {1,2,3} -> {1,2,3,4,6,9} -> {1,2,3,4,5,6,7,8,9,10,12,18,27}, ..., so a(2)=3, a(3)=6, a(4)=13, etc.
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lista(nn) = {my(k, v=[1]); print1(1); for(n=2, nn, v=Set(vector(4*#v, i, if(k=i%4, k*v[(3+i)\4], v[i/4]+1))); print1(", ", #v)); } \\ Jinyuan Wang, Apr 14 2020
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from itertools import chain, islice
def A123247_gen(): # generator of terms
s = {1}
while True:
yield len(s)
s = set(chain.from_iterable((x,x+1,2*x,3*x) for x in s))
A123247_list = list(islice(A123247_gen(),20)) # Chai Wah Wu, Jan 12 2022
Showing 1-3 of 3 results.
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