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.

Previous Showing 11-18 of 18 results.

A193656 Q-residue of the triangle p(n,k)=(2^(n - k))*5^k, 0<=k<=n, where Q is the triangular array (t(i,j)) given by t(i,j)=1. (See Comments.)

Original entry on oeis.org

1, 7, 43, 247, 1363, 7327, 38683, 201607, 1040803, 5335087, 27199723, 138095767, 698867443, 3527891647, 17773675963, 89405250727, 449173737283, 2254458621007, 11306652843403, 56670703170487, 283903271666323
Offset: 0

Views

Author

Clark Kimberling, Aug 02 2011

Keywords

Comments

For the definition of Q-residue, see A193649.

Crossrefs

Cf. A193649.

Programs

  • Mathematica
    q[n_, k_] := 1;
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
p[n_, k_] := (2^(n - k))*5^k
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 20}]    (* A193656 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (*  *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]

Formula

Conjecture: a(n) = 3*5^n-2*4^n; G.f.: ( 1-2*x ) / ( (5*x-1)*(4*x-1) ). - R. J. Mathar, Feb 19 2015

A193658 Q-residue of the triangle A051162, where Q is the triangular array (t(i,j)) given by t(i,j)=1. (See Comments.)

Original entry on oeis.org

1, 1, 3, 15, 83, 503, 3403, 25807, 218451, 2049687, 21160667, 238690847, 2923054435, 38641535143, 548635554795, 8328494925615, 134634766604915, 2309386642312631, 41897258229334267, 801610384425038911, 16132033041827096451
Offset: 0

Views

Author

Clark Kimberling, Aug 02 2011

Keywords

Comments

For the definition of Q-residue, see A193649.

Crossrefs

Cf. A051162, A193649.

Programs

  • Mathematica
    q[n_, k_] := n + k; (* A051162 *)
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
p[n_, k_] := n!/(k! (n - k)!);
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 20}]    (* A193658 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 20}]  (* A001340 *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]

Formula

Conjecture: a(n) +(-n-4)*a(n-1) +(4*n-1)*a(n-2) +5*(-n+2)*a(n-3) +2*(n-3)*a(n-4)=0. - R. J. Mathar, Feb 19 2015

A193659 Q-residue of the triangle A094727, where Q=Pascal's triangle. (See Comments.)

Original entry on oeis.org

1, 2, 8, 43, 265, 1832, 14160, 121771, 1157557, 12080436, 137505288, 1696841395, 22578385961, 322377704664, 4917809053032, 79840791037379, 1374705370985669, 25024307510421060, 480230285880218992
Offset: 0

Views

Author

Clark Kimberling, Aug 02 2011

Keywords

Comments

For the definition of Q-residue, see A193649.

Crossrefs

Cf. A193668, A094727, A193649.

Programs

  • Mathematica
    q[n_, k_] := n + k + 1;(* A094727 *)
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
p[n_, k_] := n!/(k! (n - k)!); (* Pascal's triangle *)
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 18}]    (* A193659 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* A193668 *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]

Formula

Conjecture: a(n) +(-n-5)*a(n-1) +2*(2*n+1)*a(n-2) +(-5*n+8)*a(n-3) +2*(n-3)
*a(n-4)=0. - R. J. Mathar, Feb 19 2015

A193661 Q-residue of the triangle A193673, where Q is the triangular array (t(i,j)) given by t(i,j)=1. (See Comments.)

Original entry on oeis.org

1, 3, 15, 93, 621, 4263, 29595, 206433, 1442841, 10093323, 70633575, 494375973, 3460454661, 24222651183, 169556963955, 1186893964713, 8308243404081, 58157660781843, 407103496332735, 2849724086908653, 19948067446099101
Offset: 0

Views

Author

Clark Kimberling, Aug 02 2011

Keywords

Comments

See A193649 for the definition of Q-residue.

Crossrefs

Cf. A193649, A193673.

Programs

  • Mathematica
    q[n_, k_] := 1; r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
p[n_, k_] := Coefficient[(1/2) ((x + 3)^n + (x + 1)^n), x, k] (* A193673 *)
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 20}]    (* A193661 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* 2^k *)
TableForm[Table[p[n, k], {n, 0, 10}, {k, 0, n}]] (* A193673 as a triangle *)
Flatten[%] (* A193673 as a sequence *)

Formula

Conjecture: G.f.: ( -1+8*x-13*x^2 ) / ( (x-1)*(3*x-1)*(7*x-1) ). - R. J. Mathar, Feb 19 2015

A193662 Q-residue of the Lucas triangle A114525, where Q is the triangle given by t(i,j)=1 for 0<=i<=j. (See Comments.)

Original entry on oeis.org

2, 1, 5, 7, 25, 51, 149, 351, 945, 2347, 6125, 15511, 40009, 102051, 262085, 670287, 1718625, 4399771, 11274269, 28873351
Offset: 0

Views

Author

Clark Kimberling, Aug 02 2011

Keywords

Comments

For the definition of Q-residue, see A193649.

Crossrefs

Cf. A193649, A114525.

Programs

  • Mathematica
    q[n_, k_] := 1;
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}];
f[n_, x_] := LucasL[n, x];  (* A114525 *)
p[n_, k_] := Coefficient[f[n, x], x, k];
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 16}]    (* A193662 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, 4}]]

Formula

Conjecture: a(n) = 2*a(n-1) +3*a(n-2) -4*a(n-3) if n>3. - R. J. Mathar, Feb 19 2015

A193663 Q-residue of A049310 (triangle of coefficients of Fibonacci polynomials), where Q is the triangle given by t(n,k)=k+1 for 0<=k<=n. (See Comments.)

Original entry on oeis.org

0, 1, 1, 9, 17, 80, 198, 748, 2107, 7236, 21680, 71279, 219879, 708436, 2215513, 7071210, 22256567, 70723367, 223272153, 708017329, 2238347440, 7091170416, 22433032016
Offset: 0

Views

Author

Clark Kimberling, Aug 02 2011

Keywords

Comments

The definition of Q-residue is given at A193649.

Crossrefs

Cf. A049310, A193649.

Programs

  • Mathematica
    q[n_, k_] := k + 1;
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}];
f[n_, x_] := Fibonacci[n, x]; (* A049310 *)
p[n_, k_] := Coefficient[f[n, x], x, k];
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 22}]    (* A193663 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]
TableForm[Table[p[n, k], {n, 0, 6}, {k, 0, n}]]

Formula

Conjecture: G.f.: x*(1-x+x^2) / ( 1-2*x-6*x^2+7*x^3+x^4 ). - R. J. Mathar, Feb 19 2015

A193665 Q-residue of A075392, where Q=A075392. (See Comments.)

Original entry on oeis.org

1, 6, 33, 208, 1505, 12330, 112973, 1145568, 12742389, 154308350, 2021296189, 28480485024, 429565218277, 6905903216562, 117891260108985, 2129869055824000, 40600135597843817, 814383095809997142, 17147155400516728601, 378137512431282658800
Offset: 0

Views

Author

Clark Kimberling, Aug 02 2011

Keywords

Comments

The definition of Q-residue is given at A193649.

Crossrefs

Cf. A075362, A193649.

Programs

  • Mathematica
    q[n_, k_] := (k + 1) (n + 1);  (* A075362 *)
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}];
p[n_, k_] := (k + 1) (n + 1);  (* A075362 *)
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 20}]    (* A193665 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* A193665 *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]

Formula

Conjecture: a(n) +(-n-4)*a(n-1) +(n+1)*a(n-2) -a(n-3)=0. - R. J. Mathar, Feb 19 2015

A193660 Q-residue of the triangle A038207 of coefficients of (x+2)^n, where Q is the triangle given by t(i,j)=1 for 0<=i<=j. (See Comments.)

Original entry on oeis.org

1, 2, 5, 22, 201, 3690, 131149, 9004286, 1204317329, 316525415890, 164556516205461, 169974659148800742, 349799994417738642265, 1436618749673583674658362, 11785996128174350460348176861, 193254862258295280115072223316430
Offset: 0

Views

Author

Clark Kimberling, Aug 02 2011

Keywords

Comments

For the definition of Q-residue, see A193649.

Crossrefs

Cf. A038207, A126443, A193649.

Programs

  • Mathematica
    q[n_, k_] := Coefficient[(x + 2)^n, x, k]; (* A038207 *)
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
p[n_, k_] := 1
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 16}]    (* A038207 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* A126443 *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]
Previous Showing 11-18 of 18 results.

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