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 1-7 of 7 results.

A353094 a(1) = 2; for n > 1, a(n) = 3*a(n-1) + 3 - n.

Original entry on oeis.org

2, 7, 21, 62, 184, 549, 1643, 4924, 14766, 44291, 132865, 398586, 1195748, 3587233, 10761687, 32285048, 96855130, 290565375, 871696109, 2615088310, 7845264912, 23535794717, 70607384131, 211822152372, 635466457094, 1906399371259, 5719198113753, 17157594341234
Offset: 1

Views

Author

Seiichi Manyama, Apr 23 2022

Keywords

Links

Crossrefs

Cf. A064617, A353095, A353096, A353097, A353098, A353099, A353100.
Cf. A000340.

Programs

  • Mathematica
    LinearRecurrence[{5, -7, 3}, {2, 7, 21}, 28] (* Amiram Eldar, Apr 23 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(x*(2-3*x)/((1-x)^2*(1-3*x)))
  • PARI
    a(n) = (3^(n+1)+2*n-3)/4;
  • PARI
    b(n, k) = sum(j=0, n-1, (k-n+j)*k^j);
a(n) = b(n, 3);

Formula

G.f.: x * (2 - 3*x)/((1 - x)^2 * (1 - 3*x)).
a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3).
a(n) = A000340(n-1) + n.
a(n) = (3^(n+1) + 2*n - 3)/4.
a(n) = Sum_{k=0..n-1} (3 - n + k) * 3^k.
E.g.f.: exp(x)*(3*exp(2*x) + 2*x - 3)/4. - Stefano Spezia, May 28 2023

A353095 a(1) = 3; for n > 1, a(n) = 4*a(n-1) + 4 - n.

Original entry on oeis.org

3, 14, 57, 228, 911, 3642, 14565, 58256, 233019, 932070, 3728273, 14913084, 59652327, 238609298, 954437181, 3817748712, 15270994835, 61083979326, 244335917289, 977343669140, 3909374676543, 15637498706154, 62549994824597, 250199979298368, 1000799917193451
Offset: 1

Views

Author

Seiichi Manyama, Apr 23 2022

Keywords

Links

Crossrefs

Cf. A064617, A353094, A353096, A353097, A353098, A353099, A353100.
Cf. A014825.

Programs

  • Mathematica
    LinearRecurrence[{6, -9, 4}, {3, 14, 57}, 25] (* Amiram Eldar, Apr 23 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(x*(3-4*x)/((1-x)^2*(1-4*x)))
  • PARI
    a(n) = (2*4^(n+1)+3*n-8)/9;
  • PARI
    b(n, k) = sum(j=0, n-1, (k-n+j)*k^j);
a(n) = b(n, 4);

Formula

G.f.: x * (3 - 4*x)/((1 - x)^2 * (1 - 4*x)).
a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3).
a(n) = 2 * A014825(n) + n.
a(n) = (2*4^(n+1) + 3*n - 8)/9.
a(n) = Sum_{k=0..n-1} (4 - n + k) * 4^k.
E.g.f.: exp(x)*(8*exp(3*x) + 3*x - 8)/9. - Stefano Spezia, May 28 2023

A353096 a(1) = 4; for n > 1, a(n) = 5*a(n-1) + 5 - n.

Original entry on oeis.org

4, 23, 117, 586, 2930, 14649, 73243, 366212, 1831056, 9155275, 45776369, 228881838, 1144409182, 5722045901, 28610229495, 143051147464, 715255737308, 3576278686527, 17881393432621, 89406967163090, 447034835815434, 2235174179077153, 11175870895385747
Offset: 1

Views

Author

Seiichi Manyama, Apr 23 2022

Keywords

Links

Crossrefs

Cf. A064617, A353094, A353095, A353097, A353098, A353099, A353100.
Cf. A014827.

Programs

  • Mathematica
    LinearRecurrence[{7, -11, 5}, {4, 23, 117}, 23] (* Amiram Eldar, Apr 23 2022 *)
nxt[{n_, a_}] := {n + 1, 5 a + 4 - n}; NestList[nxt,{1,4},30][[;;,2]] (* Harvey P. Dale, Apr 28 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(x*(4-5*x)/((1-x)^2*(1-5*x)))
  • PARI
    a(n) = (3*5^(n+1)+4*n-15)/16;
  • PARI
    b(n, k) = sum(j=0, n-1, (k-n+j)*k^j);
a(n) = b(n, 5);

Formula

G.f.: x * (4 - 5*x)/((1 - x)^2 * (1 - 5*x)).
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3).
a(n) = 3*A014827(n) + n.
a(n) = (3*5^(n+1) + 4*n - 15)/16.
a(n) = Sum_{k=0..n-1} (5 - n + k) * 5^k.
E.g.f.: exp(x)*(15*exp(4*x) + 4*x - 15)/16. - Stefano Spezia, May 28 2023

A353097 a(1) = 5; for n > 1, a(n) = 6*a(n-1) + 6 - n.

Original entry on oeis.org

5, 34, 207, 1244, 7465, 44790, 268739, 1612432, 9674589, 58047530, 348285175, 2089711044, 12538266257, 75229597534, 451377585195, 2708265511160, 16249593066949, 97497558401682, 584985350410079, 3509912102460460, 21059472614762745, 126356835688576454
Offset: 1

Views

Author

Seiichi Manyama, Apr 23 2022

Keywords

Links

Crossrefs

Cf. A064617, A353094, A353095, A353096, A353098, A353099, A353100.
Cf. A014829.

Programs

  • Mathematica
    LinearRecurrence[{8, -13, 6}, {5, 34, 207}, 22] (* Amiram Eldar, Apr 23 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(x*(5-6*x)/((1-x)^2*(1-6*x)))
  • PARI
    a(n) = (4*6^(n+1)+5*n-24)/25;
  • PARI
    b(n, k) = sum(j=0, n-1, (k-n+j)*k^j);
a(n) = b(n, 6);

Formula

G.f.: x * (5 - 6*x)/((1 - x)^2 * (1 - 6*x)).
a(n) = 8*a(n-1) - 13*a(n-2) + 6*a(n-3).
a(n) = 4*A014829(n) + n.
a(n) = (4*6^(n+1) + 5*n - 24)/25.
a(n) = Sum_{k=0..n-1} (6 - n + k) * 6^k.
E.g.f.: exp(x)*(24*exp(5*x) + 5*x - 24)/25. - Stefano Spezia, May 28 2023

A353098 a(1) = 6; for n>1, a(n) = 7 * a(n-1) + 7 - n.

Original entry on oeis.org

6, 47, 333, 2334, 16340, 114381, 800667, 5604668, 39232674, 274628715, 1922401001, 13456807002, 94197649008, 659383543049, 4615684801335, 32309793609336, 226168555265342, 1583179886857383, 11082259208001669, 77575814456011670, 543030701192081676
Offset: 1

Views

Author

Seiichi Manyama, Apr 23 2022

Keywords

Links

Crossrefs

Cf. A064617, A353094, A353095, A353096, A353097, A353099, A353100.
Cf. A014830.

Programs

  • Mathematica
    LinearRecurrence[{9, -15, 7}, {6, 47, 333}, 21] (* Amiram Eldar, Apr 23 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(x*(6-7*x)/((1-x)^2*(1-7*x)))
  • PARI
    a(n) = (5*7^(n+1)+6*n-35)/36;
  • PARI
    b(n, k) = sum(j=0, n-1, (k-n+j)*k^j);
a(n) = b(n, 7);

Formula

G.f.: x * (6 - 7 * x)/((1 - x)^2 * (1 - 7 * x)).
a(n) = 9*a(n-1) - 15*a(n-2) + 7*a(n-3).
a(n) = 5 * A014830(n) + n.
a(n) = (5*7^(n+1) + 6*n - 35)/36.
a(n) = Sum_{k=0..n-1} (7 - n + k)*7^k.
E.g.f.: exp(x)*(35*(exp(6*x) - 1) + 6*x)/36. - Stefano Spezia, May 29 2023

A353100 a(1) = 8; for n>1, a(n) = 9 * a(n-1) + 9 - n.

Original entry on oeis.org

8, 79, 717, 6458, 58126, 523137, 4708235, 42374116, 381367044, 3432303395, 30890730553, 278016574974, 2502149174762, 22519342572853, 202674083155671, 1824066748401032, 16416600735609280, 147749406620483511, 1329744659584351589, 11967701936259164290
Offset: 1

Views

Author

Seiichi Manyama, Apr 23 2022

Keywords

Links

Crossrefs

Cf. A064617, A353094, A353095, A353096, A353097, A353098, A353099.
Cf. A014832.

Programs

  • Mathematica
    LinearRecurrence[{11, -19, 9}, {8, 79, 717}, 20] (* Amiram Eldar, Apr 23 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(x*(8-9*x)/((1-x)^2*(1-9*x)))
  • PARI
    a(n) = (7*9^(n+1)+8*n-63)/64;
  • PARI
    b(n, k) = sum(j=0, n-1, (k-n+j)*k^j);
a(n) = b(n, 9);

Formula

G.f.: x * (8 - 9 * x)/((1 - x)^2 * (1 - 9 * x)).
a(n) = 11*a(n-1) - 19*a(n-2) + 9*a(n-3).
a(n) = 7 * A014832(n) + n.
a(n) = (7*9^(n+1) + 8*n - 63)/64.
a(n) = Sum_{k=0..n-1} (9 - n + k)*9^k.
E.g.f.: exp(x)*(63*(exp(8*x) - 1) + 8*x)/64. - Stefano Spezia, May 29 2023

A363365 Array read by ascending antidiagonals: A(1, k) = k; for n > 1, A(n, k) = (k + 1)*A(n-1, k) + k + 1 - n, with k > 0.

Original entry on oeis.org

1, 2, 2, 3, 7, 3, 4, 21, 14, 4, 5, 62, 57, 23, 5, 6, 184, 228, 117, 34, 6, 7, 549, 911, 586, 207, 47, 7, 8, 1643, 3642, 2930, 1244, 333, 62, 8, 9, 4924, 14565, 14649, 7465, 2334, 501, 79, 9, 10, 14766, 58256, 73243, 44790, 16340, 4012, 717, 98, 10
Offset: 1

Views

Author

Stefano Spezia, May 29 2023

Keywords

Examples

			The array begins:
  1,   2,    3,     4,     5, ...
  2,   7,   14,    23,    34, ...
  3,  21,   57,   117,   207, ...
  4,  62,  228,   586,  1244, ...
  5, 184,  911,  2930,  7465, ...
  6, 549, 3642, 14649, 44790, ...
  ...

Crossrefs

Cf. A000027 (n=1 or k=1), A008865, A051846 (diagonal), A064017 (k=9), A353094 (k=2), A353095 (k=3), A353096 (k=4), A353097 (k=5), A353098 (k=6), A353099 (k=7), A353100 (k=8), A363366 (antidiagonal sums).

Programs

  • Mathematica
    A[n_,k_]:=((k-1)*(k+1)^(n+1)+k*n-k^2+1)/k^2; Table[A[n-k+1,k],{n,10},{k,n}]//Flatten (* or *)
A[n_,k_]:=SeriesCoefficient[x*(k-(k+1)*x)/((1-x)^2*(1-(k+1)*x)),{x,0,n}]; Table[A[n-k+1,k],{n,10},{k,n}]//Flatten (* or *)
A[n_,k_]:=n!SeriesCoefficient[Exp[x]((k^2-1)(Exp[k x]-1)+k x)/k^2,{x,0,n}]; Table[A[n-k+1,k],{n,10},{k,n}]//Flatten

Formula

A(n, k) = ((k - 1)*(k + 1)^(n+1) + k*n - k^2 + 1)/k^2.
O.g.f. of k-th column: x*(k - (k + 1)*x)/((1 - x)^2*(1 - (k + 1)*x)).
E.g.f. of k-th column: exp(x)*((k^2 - 1)*(exp(k*x) - 1) + k*x)/k^2.
A(2, n) = A008865(n+1).
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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