A153518 Triangular T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1), read by rows.
2, 5, 5, 2, 46, 2, 2, 123, 123, 2, 2, 135, 476, 135, 2, 2, 147, 1226, 1226, 147, 2, 2, 159, 2048, 4832, 2048, 159, 2, 2, 171, 2942, 13010, 13010, 2942, 171, 2, 2, 183, 3908, 26192, 50180, 26192, 3908, 183, 2, 2, 195, 4946, 44810, 141422, 141422, 44810, 4946, 195, 2
Offset: 1
Examples
Triangle begins as: 2; 5, 5; 2, 46, 2; 2, 123, 123, 2; 2, 135, 476, 135, 2; 2, 147, 1226, 1226, 147, 2; 2, 159, 2048, 4832, 2048, 159, 2; 2, 171, 2942, 13010, 13010, 2942, 171, 2; 2, 183, 3908, 26192, 50180, 26192, 3908, 183, 2; 2, 195, 4946, 44810, 141422, 141422, 44810, 4946, 195, 2;
Links
- G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Crossrefs
Programs
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Magma
f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >; function T(n,k,p,q,j) if n eq 2 then return NthPrime(j); elif (n eq 3 and k eq 2 or n eq 4 and k eq 2 or n eq 4 and k eq 3) then return f(n,j); elif (k eq 1 or k eq n) then return 2; else return T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*NthPrime(j)*T(n-2,k-1,p,q,j); end if; return T; end function; [T(n,k,0,1,3): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 04 2021
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Maple
A153518 := proc(n,k) option remember ; if n =1 then 2; elif n = 2 then 5; elif k=1 or k=n then 2; elif n = 3 then 46 ; elif n = 4 then 123 ; else procname(n-1,k-1)+procname(n-1,k)+5*procname(n-2,k-1) ; end: end: for n from 1 to 13 do for k from 1 to n do printf("%d,",A153518(n,k)) ; od: od: # R. J. Mathar, Jan 22 2009
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Mathematica
T[n_, k_, p_, q_, j_]:= T[n,k,p,q,j]= If[n==2, Prime[j], If[n==3 && k==2 || n==4 && 2<=k<=3, ((3-(-1)^n)/2)*Prime[j]^(n-1) -2^((3-(-1)^n)/2), If[k==1 || k==n, 2, T[n-1,k,p,q,j] + T[n-1,k-1,p,q,j] + (p*j+q)*Prime[j]*T[n-2,k-1,p,q,j] ]]]; Table[T[n,k,0,1,3], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 04 2021 *) -
Sage
@CachedFunction def f(n,j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2) def T(n,k,p,q,j): if (n==2): return nth_prime(j) elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n,j) elif (k==1 or k==n): return 2 else: return T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*nth_prime(j)*T(n-2,k-1,p,q,j) flatten([[T(n,k,0,1,3) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 04 2021
Formula
T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1).
Recurrence row sums: s(n) = 2*s(n-1) + 5*s(n-2), n > 4, with s(1) = 2, s(2) = 10, s(3) = 50, s(4) = 250. - R. J. Mathar, Jan 22 2009
From G. C. Greubel, Mar 04 2021: (Start)
T(n,k,p,q,j) = T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*prime(j)*T(n-2,k-1,p,q,j) with T(2,k,p,q,j) = prime(j), T(3,2,p,q,j) = 2*prime(j)^2 -4, T(4,2,p,q, j) = T(4,3,p,q,j) = prime(j)^2 -2, T(n,1,p,q,j) = T(n,n,p,q,j) = 2 and (p, q, j) = (0,1,3).
Sum_{k=0..n} T(n,k,0,1,3) = 4*(-5)^n*[n<2] + 50*(i*sqrt(5))^(n-2)*(ChebyshevU(n-2, -i/sqrt(5)) - (3*i/sqrt(5))*ChebyshevU(n-3, -i/sqrt(5))) = 4*(-5)^n*[n<2] + 50*A123011(n-2). (End)
Extensions
More terms from R. J. Mathar, Jan 22 2009
Edited by G. C. Greubel, Mar 04 2021
Comments