cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A153655 Triangle T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 10, read by rows.

Original entry on oeis.org

2, 29, 29, 2, 1678, 2, 2, 24387, 24387, 2, 2, 25607, 1070676, 25607, 2, 2, 26827, 15947966, 15947966, 26827, 2, 2, 28047, 31569456, 683937616, 31569456, 28047, 2, 2, 29267, 47935146, 10427818366, 10427818366, 47935146, 29267, 2, 2, 30487, 65045036, 29701552216, 437373644876, 29701552216, 65045036, 30487, 2
Offset: 1

Views

Author

Roger L. Bagula, Dec 30 2008

Keywords

Examples

			Triangle begins as:
   2;
  29,    29;
   2,  1678,        2;
   2, 24387,    24387,           2;
   2, 25607,  1070676,       25607,            2;
   2, 26827, 15947966,    15947966,        26827,           2;
   2, 28047, 31569456,   683937616,     31569456,       28047,        2;
   2, 29267, 47935146, 10427818366,  10427818366,    47935146,    29267,     2;
   2, 30487, 65045036, 29701552216, 437373644876, 29701552216, 65045036, 30487, 2;

Links

Crossrefs

Cf. A153652 (j=7), A153653 (j=8), A153654 (j=9), this sequence (j=10).
Cf. A153516, A153518, A153520, A153521, A153648, A153649, A153650, A153651, A153656, A153657.

Programs

  • Magma
    f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
function T(n,k,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,j) + T(n-1,k-1,j) + (2*j+1)*NthPrime(j)*T(n-2,k-1,j);
  end if; return T;
end function;
[T(n,k,10): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 03 2021
  • Mathematica
    T[n_, k_, j_]:= T[n,k,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,j] + T[n-1,k-1,j] + (2*j+1)*Prime[j]*T[n-2,k-1,j] ]]];
Table[T[n,k,10], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 03 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,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,j) + T(n-1,k-1,j) + (2*j+1)*nth_prime(j)*T(n-2,k-1,j)
flatten([[T(n,k,10) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 03 2021

Formula

T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j), T(3, 2, j) = 2*prime(j)^2 - 4, T(4, 2, j) = T(4, 3, j) = prime(j)^2 - 2, T(n, 1, j) = T(n, n, j) = 2 and j = 10.
From G. C. Greubel, Mar 04 2021: (Start)
Sum_{k=0..n} T(n, k, 10) = -(76/147)*[n=0] - (116/7)*[n=1] + 1682*(i*sqrt(609))^(n-2)*(ChebyshevU(n-2, -i/sqrt(609)) - (27*i/sqrt(609))*ChebyshevU(n-3, -i/sqrt(609) )).
Row sums satisfy the recurrence S(n) = 2*S(n-1) + 609*S(n-2) for n>4 with S(0) = 2, S(1) = 58, S(2) = 1682, S(3) = 48778. (End)

Extensions

Edited by G. C. Greubel, Mar 03 2021

A153656 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (2*j +3)*prime(j)*T(n-2, k-1) with j=9, read by rows.

Original entry on oeis.org

2, 23, 23, 2, 1054, 2, 2, 12165, 12165, 2, 2, 13133, 533412, 13133, 2, 2, 14101, 6422240, 6422240, 14101, 2, 2, 15069, 12779580, 270482476, 12779580, 15069, 2, 2, 16037, 19605432, 3385203976, 3385203976, 19605432, 16037, 2, 2, 17005, 26899796, 9577346548, 137413443860, 9577346548, 26899796, 17005, 2
Offset: 1

Views

Author

Roger L. Bagula, Dec 30 2008

Keywords

Examples

			Triangle begins as:
   2;
  23,    23;
   2,  1054,        2;
   2, 12165,    12165,          2;
   2, 13133,   533412,      13133,            2;
   2, 14101,  6422240,    6422240,        14101,          2;
   2, 15069, 12779580,  270482476,     12779580,      15069,        2;
   2, 16037, 19605432, 3385203976,   3385203976,   19605432,    16037,     2;
   2, 17005, 26899796, 9577346548, 137413443860, 9577346548, 26899796, 17005, 2;

Links

Crossrefs

Sequences with variable (p,q,j): A153516 (0,1,2), A153518 (0,1,3), A153520 (0,1,4), A153521 (0,1,5), A153648 (1,0,3), A153649 (1,1,4), A153650 (1,4,5), A153651 (1,5,6), A153652 (2,1,7), A153653 (2,1,8), A153654 (2,1,9), A153655 (2,1,10), this sequence (2,3,9), A153657 (2,7,10).
Cf. A009967 (powers of 23).

Programs

  • 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,2,3,9): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 06 2021
  • 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,2,3,9], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 06 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,2,3,9) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 06 2021

Formula

T(n, k) = T(n-1, k) + T(n-1, k-1) + (2*j +3)*prime(j)*T(n-2, k-1) with j=9.
From G. C. Greubel, Mar 06 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) = (2,3,9).
Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1), for (p,q,j)=(2,3,9), = 2*A009967(n-1). (End)

Extensions

Edited by G. C. Greubel, Mar 06 2021

A153657 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (2*j +7)*prime(j)*T(n-2, k-1) with j=10, read by rows.

Original entry on oeis.org

2, 29, 29, 2, 1678, 2, 2, 24387, 24387, 2, 2, 25955, 1362648, 25955, 2, 2, 27523, 20483624, 20483624, 27523, 2, 2, 29091, 40833912, 1107920632, 40833912, 29091, 2, 2, 30659, 62413512, 17187432136, 17187432136, 62413512, 30659, 2, 2, 32227, 85222424, 49222798744, 901876719128, 49222798744, 85222424, 32227, 2
Offset: 1

Views

Author

Roger L. Bagula, Dec 30 2008

Keywords

Examples

			Triangle begins as:
   2;
  29,    29;
   2,  1678,        2;
   2, 24387,    24387,           2;
   2, 25955,  1362648,       25955,            2;
   2, 27523, 20483624,    20483624,        27523,           2;
   2, 29091, 40833912,  1107920632,     40833912,       29091,       2;
   2, 30659, 62413512, 17187432136,  17187432136,    62413512,    30659,     2;
   2, 32227, 85222424, 49222798744, 901876719128, 49222798744, 85222424, 32227, 2;

Links

Crossrefs

Sequences with variable (p,q,j): A153516 (0,1,2), A153518 (0,1,3), A153520 (0,1,4), A153521 (0,1,5), A153648 (1,0,3), A153649 (1,1,4), A153650 (1,4,5), A153651 (1,5,6), A153652 (2,1,7), A153653 (2,1,8), A153654 (2,1,9), A153655 (2,1,10), A153656 (2,3,9), this sequence (2,7,10).
Cf. A009973 (powers of 29).

Programs

  • 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,2,7,10): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 06 2021
  • 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,2,7,10], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 06 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,2,7,10) + 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,p,q,j) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 06 2021

Formula

T(n, k) = T(n-1, k) + T(n-1, k-1) + (2*j +7)*prime(j)*T(n-2, k-1) with j=10.
From G. C. Greubel, Mar 06 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) = (2,7,10).
Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1), for (p,q,j)=(2,7,10), = 2*A009973(n-1). (End)

Extensions

Edited by G. C. Greubel, Mar 06 2021
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