A103142
a(n) = 2*a(n-1) + a(n-2) + a(n-3) + a(n-4), with a(0)=1, a(1)=2, a(3)=5, a(4)=13.
Original entry on oeis.org
1, 2, 5, 13, 34, 88, 228, 591, 1532, 3971, 10293, 26680, 69156, 179256, 464641, 1204374, 3121801, 8091873, 20974562, 54367172, 140922580, 365278767, 946821848, 2454212215, 6361447625, 16489208080, 42740897848, 110786663616, 287164880785, 744346531114
Offset: 0
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a:=[1,2,5,13];; for n in [5..40] do a[n]:=2*a[n-1]+a[n-2]+a[n-3]+a[n-4]; od; a; # G. C. Greubel, Feb 12 2020
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I:=[1,2,5,13]; [n le 4 select I[n] else 2*Self(n-1)+Self(n-2)+Self(n-3) +Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 05 2012
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m:=40; S:=series(1/(1-2*x-x^2-x^3-x^4), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 12 2020
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LinearRecurrence[{2,1,1,1}, {1,2,5,13}, 40] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)
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Vec(1/(1-2*x-x^2-x^3-x^4)+O(x^40)) \\ Charles R Greathouse IV, Jun 20 2011
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def A103142_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P( 1/(1-2*x-x^2-x^3-x^4) ).list()
A103142_list(40) # G. C. Greubel, Feb 12 2020
Deleted certain dangerous or potentially dangerous links. -
N. J. A. Sloane, Jan 30 2021
A102036
Triangle, read by rows, where the terms are generated by the rule: T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-3,k-1), with T(0,0)=1.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 9, 15, 7, 1, 1, 12, 33, 28, 9, 1, 1, 15, 60, 81, 45, 11, 1, 1, 18, 96, 189, 161, 66, 13, 1, 1, 21, 141, 378, 459, 281, 91, 15, 1, 1, 24, 195, 675, 1107, 946, 449, 120, 17, 1, 1, 27, 258, 1107, 2349, 2673, 1742, 673, 153, 19, 1
Offset: 0
Generated by adding preceding terms in the triangle at positions that form the letter 'L':
T(n,k) =
T(n-3,k-1) +
T(n-2,k-1) +
T(n-1,k-1) + T(n-1,k).
Rows begin:
[1],
[1, 1],
[1, 3, 1],
[1, 6, 5, 1],
[1, 9, 15, 7, 1],
[1, 12, 33, 28, 9, 1],
[1, 15, 60, 81, 45, 11, 1],
[1, 18, 96, 189, 161, 66, 13, 1],
[1, 21, 141, 378, 459, 281, 91, 15, 1], ...
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
- Kuhapatanakul, Kantaphon; Anantakitpaisal, Pornpawee The k-nacci triangle and applications. Cogent Math. 4, Article ID 1333293, 13 p. (2017).
- J. L. Ramírez, V. F. Sirvent, A Generalization of the k-Bonacci Sequence from Riordan Arrays, The Electronic Journal of Combinatorics, 22(1) (2015), #P1.38.
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[[(&+[Binomial(n-m,k)*(&+[Binomial(j,m-j)*Binomial(k,j):j in [0..k]]): m in [0..n-k]]): k in [0..n]]: n in [0..15]]; // G. C. Greubel, Dec 11 2018
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T:=(n,k)->add(add((binomial(j,m-j)*binomial(k,j))*binomial(n-m,k),j=0..k),m=0..n-k): seq(seq(T(n,k),k=0..n),n=0..10); # Muniru A Asiru, Dec 11 2018
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T[n_, k_] := If[n < k || k < 0, 0, If[n == 0, 1, T[n - 1, k] + T[n - 1, k - 1] + T[n - 2, k - 1] + T[n - 3, k - 1]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Dec 07 2018 *)
Table[Sum[Binomial[n-m, k]*Sum[Binomial[j, m-j]*Binomial[k, j], {j, 0, k}], {m, 0, n-k}], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 11 2018 *)
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T(n,k):=sum((sum(binomial(j,m-j)*binomial(k,j),j,0,k))*binomial(n-m,k),m,0,n-k); /* Vladimir Kruchinin, Apr 21 2015 */
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{T(n,k)=if(n
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[[sum(binomial(n-m,k)*sum(binomial(j,m-j)*binomial(k,j) for j in (0..k)) for m in (0..n-k)) for k in (0..n)] for n in range(15)] # G. C. Greubel, Dec 11 2018
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