A157005
A Somos-4 variant.
Original entry on oeis.org
1, 2, 8, 24, 112, 736, 3776, 40192, 391424, 4203008, 85312512, 1270368256, 32235102208, 1038278549504, 27640704385024, 1549962593927168, 73624753456480256, 4273828146025070592, 435765959975516766208
Offset: 0
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a:=[1,2,8,24];; for n in [5..20] do a[n]:=(a[n-1]*a[n-3] + a[n-2]^2)/a[n-4]; od; a; # G. C. Greubel, Feb 23 2019
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I:=[1,2,8,24]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + Self(n-2)^2)/Self(n-4): n in [1..20]]; // G. C. Greubel, Feb 23 2019
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RecurrenceTable[{a[0]==1,a[1]==2,a[2]==8,a[3]==24,a[n]==(a[n-1] a[n-3]+a[n-2]^2)/a[n-4]},a,{n,20}] (* Harvey P. Dale, Apr 30 2011 *)
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m=20; v=concat([1,2,8,24], vector(m-4)); for(n=5, m, v[n] = (v[n-1]*v[n-3] +v[n-2]^2)/v[n-4]); v \\ G. C. Greubel, Feb 23 2019
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def a(n):
if (n==0): return 1
elif (n==1): return 2
elif (n==2): return 8
elif (n==3): return 24
else: return (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4)
[a(n) for n in (0..20)] # G. C. Greubel, Feb 23 2019
A162546
A Somos-4 variant: a(n) = (36*a(n-1)*a(n-3) - 68*a(n-2)^2)/a(n-4).
Original entry on oeis.org
1, 1, -16, -644, -40592, -4821056, 17059328, 2492895195136, 10659285907800064, 86296767700623425536, 1081586547380924161458176, -36649408809924048998874742784, -18144416387824430577315746611724288
Offset: 0
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a:=[1,1,-16,-644];; for n in [5..20] do a[n]:=(36*a[n-1]*a[n-3] - 68*a[n-2]^2)/a[n-4]; od; a; # G. C. Greubel, Feb 23 2019
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I:=[1,1,-16,-644]; [n le 4 select I[n] else (36*Self(n-1) *Self(n-3) - 68*Self(n-2)^2)/Self(n-4): n in [1..20]]; // G. C. Greubel, Feb 23 2019
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RecurrenceTable[{a[n]==(36*a[n-1]*a[n-3] - 68*a[n-2]^2)/a[n-4], a[0]==1, a[1]==1, a[2]==-16, a[3]==-644}, a, {n,20}] (* G. C. Greubel, Feb 23 2019 *)
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m=20; v=concat([1,1,-16,-644], vector(m-4)); for(n=5, m, v[n] = (36*v[n-1]*v[n-3] -68*v[n-2]^2)/v[n-4]); v \\ G. C. Greubel, Feb 23 2019
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def a(n):
if (n==0): return 1
elif (n==1): return 1
elif (n==2): return -16
elif (n==3): return -644
else: return (36*a(n-1)*a(n-3) - 68*a(n-2)^2)/a(n-4)
[a(n) for n in (1..20)] # G. C. Greubel, Feb 23 2019
A157100
Transform of Catalan numbers whose Hankel transform satisfies the Somos-4 recurrence.
Original entry on oeis.org
1, 2, 3, 6, 14, 37, 105, 312, 956, 2996, 9554, 30897, 101083, 333947, 1112497, 3732956, 12605030, 42800318, 146046820, 500555448, 1722402304, 5948047170, 20607691518, 71610355541, 249520257107, 871614139397, 3051737703527
Offset: 0
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a[n_]:= Sum[(-1)^Binomial[k, 2]*Binomial[n-k, Floor[k/2]]*CatalanNumber[n-k], {k,0,n}];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Jan 11 2022 *)
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def A157100(n): return sum((-1)^binomial(k,2)*binomial(n-k, k//2)*catalan_number(n-k) for k in (0..n))
[A157100(n) for n in (0..40)] # G. C. Greubel, Jan 11 2022
Showing 1-3 of 3 results.
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