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.

Showing 1-10 of 13 results. Next

A153521 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + 11*T(n-2, k-1), read by rows.

Original entry on oeis.org

2, 11, 11, 2, 238, 2, 2, 1329, 1329, 2, 2, 1353, 5276, 1353, 2, 2, 1377, 21248, 21248, 1377, 2, 2, 1401, 37508, 100532, 37508, 1401, 2, 2, 1425, 54056, 371768, 371768, 54056, 1425, 2, 2, 1449, 70892, 838412, 1849388, 838412, 70892, 1449, 2, 2, 1473, 88016, 1503920, 6777248, 6777248, 1503920, 88016, 1473, 2
Offset: 1

Views

Author

Roger L. Bagula, Dec 28 2008

Keywords

Examples

			Triangle begins as:
   2;
  11,   11;
   2,  238,     2;
   2, 1329,  1329,       2;
   2, 1353,  5276,    1353,       2;
   2, 1377, 21248,   21248,    1377,       2;
   2, 1401, 37508,  100532,   37508,    1401,       2;
   2, 1425, 54056,  371768,  371768,   54056,    1425,     2;
   2, 1449, 70892,  838412, 1849388,  838412,   70892,  1449,    2;
   2, 1473, 88016, 1503920, 6777248, 6777248, 1503920, 88016, 1473, 2;

Links

Crossrefs

Sequences with variable (p,q,j): A153516 (0,1,2), A153518 (0,1,3), A153520 (0,1,4), this sequence (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), A153657 (2,7,10).
Cf. A151617 (row sums).

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,0,1,5): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 04 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,0,1,5], {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,0,1,5) 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) + 11*T(n-2, k-1).
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) = (0,1,5).
Sum_{k=1..n} T(n,k,p,q,j) = 2*(prime(j)-3)*[n=1] -2*prime(j)*(prime(j)-3)*[n=2] +2*prime(j)^2*(i*sqrt(prime(j)))^(n-3)*(ChebyshevU(n-3, -i/Sqrt(prime(j))) -((prime(j) -2)*i/sqrt(prime(j)))*ChebyshevU(n-4, -i/sqrt(prime(j)))) for (p,q,j)=(0,1,5) = A151617(n).
Row sums satisfy the recurrence relation S(n) = 2*S(n-1) + prime(j)*S(n-2), for n > 4, with S(1) = 2, S(2) = 2*prime(j), S(3) = 2*prime(j)^2, S(4) = 2*prime(j)^3 for j=5. (End)

Extensions

Edited by G. C. Greubel, Mar 04 2021

A153516 Triangle 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) and (p,q,j) = (0,1,2), read by rows.

Original entry on oeis.org

2, 3, 3, 2, 14, 2, 2, 25, 25, 2, 2, 33, 92, 33, 2, 2, 41, 200, 200, 41, 2, 2, 49, 340, 676, 340, 49, 2, 2, 57, 512, 1616, 1616, 512, 57, 2, 2, 65, 716, 3148, 5260, 3148, 716, 65, 2, 2, 73, 952, 5400, 13256, 13256, 5400, 952, 73, 2
Offset: 1

Views

Author

Roger L. Bagula, Dec 28 2008

Keywords

Examples

			Triangle begins as:
  2;
  3,  3;
  2, 14,   2;
  2, 25,  25,    2;
  2, 33,  92,   33,     2;
  2, 41, 200,  200,    41,     2;
  2, 49, 340,  676,   340,    49,    2;
  2, 57, 512, 1616,  1616,   512,   57,   2;
  2, 65, 716, 3148,  5260,  3148,  716,  65,  2;
  2, 73, 952, 5400, 13256, 13256, 5400, 952, 73, 2;

Links

Crossrefs

Sequences with variable (p,q,j): this sequence (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), A153657 (2,7,10).
Cf. A000244.

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,0,1,2): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 03 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,0,1,2], {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,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,0,1,2) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 03 2021

Formula

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) = (0,1,2).
Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1) for (p,q,j) = (0,1,2) = 2*A000244(n-1).

Extensions

Edited by G. C. Greubel, Mar 03 2021

A153518 Triangular T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1), read by rows.

Original entry on oeis.org

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

Views

Author

Roger L. Bagula, Dec 28 2008

Keywords

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

Crossrefs

Sequences with variable (p,q,j): A153516 (0,1,2), this sequences (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), A153657 (2,7,10).
Cf. A123011.

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,0,1,3): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 04 2021
  • 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
  • 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

A153520 Triangle T(n,k) = T(n-1, k) + T(n-1, k-1) + 7*T(n-2, k-1), read by rows.

Original entry on oeis.org

2, 7, 7, 2, 94, 2, 2, 341, 341, 2, 2, 357, 1340, 357, 2, 2, 373, 4084, 4084, 373, 2, 2, 389, 6956, 17548, 6956, 389, 2, 2, 405, 9956, 53092, 53092, 9956, 405, 2, 2, 421, 13084, 111740, 229020, 111740, 13084, 421, 2, 2, 437, 16340, 194516, 712404, 712404, 194516, 16340, 437, 2
Offset: 1

Views

Author

Roger L. Bagula, Dec 28 2008

Keywords

Examples

			Triangle begins as:
  2;
  7,   7;
  2,  94,     2;
  2, 341,   341,      2;
  2, 357,  1340,    357,      2;
  2, 373,  4084,   4084,    373,      2;
  2, 389,  6956,  17548,   6956,    389,      2;
  2, 405,  9956,  53092,  53092,   9956,    405,     2;
  2, 421, 13084, 111740, 229020, 111740,  13084,   421,   2;
  2, 437, 16340, 194516, 712404, 712404, 194516, 16340, 437, 2;

Links

Crossrefs

Sequences with variable (p,q,j): A153516 (0,1,2), A153518 (0,1,3), this sequence (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), A153657 (2,7,10).

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,0,1,4): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 04 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,0,1,4], {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,0,1,4) 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) + 7*T(n-2, k-1).
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) = (0,1,4).
Sum_{k=1..n} T(n,k,p,q,j) = 2*(prime(j)-3)*[n=1] -2*prime(j)*(prime(j)-3)*[n=2] +2*prime(j)^2*(i*sqrt(prime(j)))^(n-3)*(ChebyshevU(n-3, -i/Sqrt(prime(j))) -((prime(j) -2)*i/sqrt(prime(j)))*ChebyshevU(n-4, -i/sqrt(prime(j)))) for (p,q,j)=(0,1,4).
Row sums satisfy the recurrence relation S(n) = 2*S(n-1) + prime(j)*S(n-2), for n > 4, with S(1) = 2, S(2) = 2*prime(j), S(3) = 2*prime(j)^2, S(4) = 2*prime(j)^3 with j=4. (End)

Extensions

Edited by G. C. Greubel, Mar 04 2021

A153648 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + j*prime(j)*T(n-2, k-1) with j=3, read by rows.

Original entry on oeis.org

2, 5, 5, 2, 46, 2, 2, 123, 123, 2, 2, 155, 936, 155, 2, 2, 187, 2936, 2936, 187, 2, 2, 219, 5448, 19912, 5448, 219, 2, 2, 251, 8472, 69400, 69400, 8472, 251, 2, 2, 283, 12008, 159592, 437480, 159592, 12008, 283, 2, 2, 315, 16056, 298680, 1638072, 1638072, 298680, 16056, 315, 2
Offset: 1

Views

Author

Roger L. Bagula, Dec 30 2008

Keywords

Examples

			Triangle begins as:
  2;
  5,   5;
  2,  46,     2;
  2, 123,   123,      2;
  2, 155,   936,    155,       2;
  2, 187,  2936,   2936,     187,       2;
  2, 219,  5448,  19912,    5448,     219,      2;
  2, 251,  8472,  69400,   69400,    8472,    251,     2;
  2, 283, 12008, 159592,  437480,  159592,  12008,   283,   2;
  2, 315, 16056, 298680, 1638072, 1638072, 298680, 16056, 315, 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), this sequence (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), A153657 (2,7,10).
Cf. A000351 (powers of 5).

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,1,0,3): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 04 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,1,0,3], {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,1,0,3) 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) + j*prime(j)*T(n-2, k-1) with j=3.
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) = (1,0,3).
Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1), for (p,q,j) = (1,0,3), = 2*A000351(n-1). (End)

Extensions

Edited by G. C. Greubel, Mar 04 2021

A153649 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+1)*prime(j)*T(n-2, k-1) with j=4, read by rows.

Original entry on oeis.org

2, 7, 7, 2, 94, 2, 2, 341, 341, 2, 2, 413, 3972, 413, 2, 2, 485, 16320, 16320, 485, 2, 2, 557, 31260, 171660, 31260, 557, 2, 2, 629, 48792, 774120, 774120, 48792, 629, 2, 2, 701, 68916, 1917012, 7556340, 1917012, 68916, 701, 2, 2, 773, 91632, 3693648, 36567552, 36567552, 3693648, 91632, 773, 2
Offset: 1

Views

Author

Roger L. Bagula, Dec 30 2008

Keywords

Examples

			Triangle begins as:
  2;
  7,   7;
  2,  94,     2;
  2, 341,   341,       2;
  2, 413,  3972,     413,        2;
  2, 485, 16320,   16320,      485,        2;
  2, 557, 31260,  171660,    31260,      557,       2;
  2, 629, 48792,  774120,   774120,    48792,     629,     2;
  2, 701, 68916, 1917012,  7556340,  1917012,   68916,   701,   2;
  2, 773, 91632, 3693648, 36567552, 36567552, 3693648, 91632, 773, 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), this sequence (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), A153657 (2,7,10).
Cf. A000420 (powers of 7).

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,1,1,4): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 04 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,1,1,4], {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,1,1,4) 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) + (j+1)*prime(j)*T(n-2, k-1) with j=4.
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) = (1,1,4).
Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1), for (p,q,j)=(1,1,4), = 2*A000420(n-1). (End)

Extensions

Edited by G. C. Greubel, Mar 04 2021

A153651 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+5)*prime(j)*T(n-2, k-1) with j=6, read by rows.

Original entry on oeis.org

2, 13, 13, 2, 334, 2, 2, 2195, 2195, 2, 2, 2483, 52152, 2483, 2, 2, 2771, 368520, 368520, 2771, 2, 2, 3059, 726360, 8194776, 726360, 3059, 2, 2, 3347, 1125672, 61619496, 61619496, 1125672, 3347, 2, 2, 3635, 1566456, 166614648, 1295091960, 166614648, 1566456, 3635, 2
Offset: 1

Views

Author

Roger L. Bagula, Dec 30 2008

Keywords

Examples

			Triangle begins as:
   2;
  13,   13;
   2,  334,       2;
   2, 2195,    2195,         2;
   2, 2483,   52152,      2483,          2;
   2, 2771,  368520,    368520,       2771,         2;
   2, 3059,  726360,   8194776,     726360,      3059,       2;
   2, 3347, 1125672,  61619496,   61619496,   1125672,    3347,    2;
   2, 3635, 1566456, 166614648, 1295091960, 166614648, 1566456, 3635, 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), this sequence (1,5,6), A153652 (2,1,7), A153653 (2,1,8), A153654 (2,1,9), A153655 (2,1,10), A153656 (2,3,9), A153657 (2,7,10).
Cf. A001022 (powers of 13).

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,1,5,6): 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,1,5,6], {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,1,5,6) 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) + (j+5)*prime(j)*T(n-2, k-1) with j=6.
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) = (1,5,6).
Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1) for j=6, = 2*A001022(n-1). (End)

Extensions

Edited by G. C. Greubel, Mar 06 2021

A153652 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 = 7, read by rows.

Original entry on oeis.org

2, 17, 17, 2, 574, 2, 2, 4911, 4911, 2, 2, 5423, 156192, 5423, 2, 2, 5935, 1413920, 1413920, 5935, 2, 2, 6447, 2802720, 42656800, 2802720, 6447, 2, 2, 6959, 4322592, 406009120, 406009120, 4322592, 6959, 2, 2, 7471, 5973536, 1125025312, 11689502240, 1125025312, 5973536, 7471, 2
Offset: 1

Views

Author

Roger L. Bagula, Dec 30 2008

Keywords

Examples

			Triangle begins as:
   2;
  17,   17;
   2,  574,       2;
   2, 4911,    4911,          2;
   2, 5423,  156192,       5423,           2;
   2, 5935, 1413920,    1413920,        5935,          2;
   2, 6447, 2802720,   42656800,     2802720,       6447,       2;
   2, 6959, 4322592,  406009120,   406009120,    4322592,    6959,    2;
   2, 7471, 5973536, 1125025312, 11689502240, 1125025312, 5973536, 7471, 2;

Links

Crossrefs

Cf. this sequence (j=7), A153653 (j=8), A153654 (j=9), A153655 (j=10).
Cf. A153516, A153518, A153520, A153521, A153648, A153649, A153650, A153651, A153656, A153657.
Cf. A001026 (powers of 17).

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,7): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 02 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,7], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 02 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,7) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 02 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 = 7.
Sum_{k=0..n} T(n, k, j) = 2*prime(j)^(n-1) for j=7 = 2*A001026(n-1).

Extensions

Edited by G. C. Greubel, Mar 02 2021

A153653 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 = 8, read by rows.

Original entry on oeis.org

2, 19, 19, 2, 718, 2, 2, 6857, 6857, 2, 2, 7505, 245628, 7505, 2, 2, 8153, 2467944, 2467944, 8153, 2, 2, 8801, 4900212, 84273732, 4900212, 8801, 2, 2, 9449, 7542432, 886319856, 886319856, 7542432, 9449, 2, 2, 10097, 10394604, 2476630764, 28993055148, 2476630764, 10394604, 10097, 2
Offset: 1

Views

Author

Roger L. Bagula, Dec 30 2008

Keywords

Examples

			Triangle begins as:
   2;
  19,    19;
   2,   718,        2;
   2,  6857,     6857,          2;
   2,  7505,   245628,       7505,           2;
   2,  8153,  2467944,    2467944,        8153,          2;
   2,  8801,  4900212,   84273732,     4900212,       8801,        2;
   2,  9449,  7542432,  886319856,   886319856,    7542432,     9449,     2;
   2, 10097, 10394604, 2476630764, 28993055148, 2476630764, 10394604, 10097, 2;

Links

Crossrefs

Cf. A153652 (j=7), this sequence (j=8), A153654 (j=9), A153655 (j=10).
Cf. A153516, A153518, A153520, A153521, A153648, A153649, A153650, A153651, A153656, A153657.
Cf. A001029 (powers of 19).

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,8): 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,8], {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,8) 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 = 8.
Sum_{k=0..n} T(n, k, j) = 2*prime(j)^(n-1) for j=8 = 2*A001029(n-1).

Extensions

Edited by G. C. Greubel, Mar 03 2021

A153654 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 = 9, read by rows.

Original entry on oeis.org

2, 23, 23, 2, 1054, 2, 2, 12165, 12165, 2, 2, 13041, 484928, 13041, 2, 2, 13917, 5814074, 5814074, 13917, 2, 2, 14793, 11526908, 223541684, 11526908, 14793, 2, 2, 15669, 17623430, 2775818930, 2775818930, 17623430, 15669, 2, 2, 16545, 24103640, 7830701156, 103239353768, 7830701156, 24103640, 16545, 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, 13041,   484928,      13041,            2;
   2, 13917,  5814074,    5814074,        13917,          2;
   2, 14793, 11526908,  223541684,     11526908,      14793,        2;
   2, 15669, 17623430, 2775818930,   2775818930,   17623430,    15669,     2;
   2, 16545, 24103640, 7830701156, 103239353768, 7830701156, 24103640, 16545, 2;

Links

Crossrefs

Cf. A153652 (j=7), A153653 (j=8), this sequence (j=9), A153655 (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,9): 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,9], {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,9) 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 = 9.
From G. C. Greubel, Mar 04 2021: (Start)
Sum_{k=0..n} T(n, k, 10) = -(68/361)*[n=0] - (92/19)*[n=1] + 1058*(i*sqrt(437))^(n-2)*(ChebyshevU(n-2, -i/sqrt(437)) - (21*i/sqrt(437))*ChebyshevU(n-3, -i/sqrt(437) )).
Row sums satisfy the recurrence S(n) = 2*S(n-1) + 437*S(n-2) for n>4 with S(0) = 2, S(1) = 46, S(2) = 1058, S(3) = 24334. (End)

Extensions

Edited by G. C. Greubel, Mar 03 2021
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