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 1-8 of 8 results.

A084061 Square number array read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 5, 27, 1, 1, 6, 36, 256, 1, 1, 7, 45, 353, 3125, 1, 1, 8, 54, 452, 4400, 46656, 1, 1, 9, 63, 553, 5725, 66637, 823543, 1, 1, 10, 72, 656, 7100, 87704, 1188544, 16777216, 1, 1, 11, 81, 761, 8525, 109863, 1577849, 24405761, 387420489, 1, 1, 12, 90, 868, 10000, 133120, 1991752, 32618512, 567108864, 10000000000
Offset: 0

Views

Author

Paul Barry, May 11 2003

Keywords

Examples

			Rows begin:
1 1 4 27 256 ...
1 1 5 36 353 ...
1 1 6 45 452 ...
1 1 7 54 553 ...
1 1 8 63 656 ...

Links

Crossrefs

Rows include A000312, A062024, A084063, A084064, A084065.
Diagonals include A084062, A084063, A084095.

Programs

  • GAP
    Flat(List([0..10], n-> List([0..n], k-> ((k+Sqrt(n-k))^k + (k-Sqrt(n-k))^k)/2 ))); # G. C. Greubel, Jan 11 2020
  • Magma
    [Round(((k+Sqrt(n-k))^k + (k-Sqrt(n-k))^k)/2): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 11 2020
  • Maple
    seq(seq( round(((k+sqrt(n-k))^k + (k-sqrt(n-k))^k)/2), k=0..n), n=0..10); # G. C. Greubel, Jan 11 2020
  • Mathematica
    Table[If[n==0 && k==0, 1, Round[((k-Sqrt[n-k])^k + (k+Sqrt[n-k])^k)/2]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 11 2020 *)
  • PARI
    T(n,k) = round( ((k+sqrt(n-k))^n + (k-sqrt(n-k))^k)/2 ); \\ G. C. Greubel, Jan 11 2020
  • Sage
    [[round(((k+sqrt(n-k))^k + (k-sqrt(n-k))^k)/2) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 11 2020

Formula

T(n, k) = ( (n - sqrt(k))^n + (n + sqrt(k))^n )/2.

A084063 First subdiagonal of number array A084061.

Original entry on oeis.org

1, 1, 7, 63, 761, 11525, 209539, 4440527, 107374753, 2915352729, 87771145551, 2900744369039, 104369641697881, 4060189444664093, 169777979925475531, 7592652139022106975, 361563242499379729537, 18263719440778358953457
Offset: 0

Views

Author

Paul Barry, May 11 2003

Keywords

Links

Crossrefs

Cf. A084061, A084062, A084095.

Programs

  • GAP
    List([0..20], n-> ((n-Sqrt(n+1))^n + (n+Sqrt(n+1))^n)/2); # G. C. Greubel, Jan 09 2020
  • Magma
    [Round(((n-Sqrt(n+1))^n + (n+Sqrt(n+1))^n)/2): n in [0..20]]; // G. C. Greubel, Jan 09 2020
  • Maple
    seq( round(((n-sqrt(n+1))^n + (n+sqrt(n+1))^n)/2), n=0..20); # G. C. Greubel, Jan 09 2020
  • Mathematica
    Table[Round[((n+Sqrt[n+1])^n + (n-Sqrt[n+1])^n)/2], {n,0,20}] (* G. C. Greubel, Jan 09 2020 *)
  • PARI
    vector(21, n, round(((n-1-sqrt(n))^(n-1) + (n-1+sqrt(n))^(n-1))/2) ) \\ G. C. Greubel, Jan 09 2020
  • Sage
    [round(((n-sqrt(n+1))^n + (n+sqrt(n+1))^n)/2) for n in (0..20)] # G. C. Greubel, Jan 09 2020

Formula

a(n) = ((n - sqrt(n+1))^n + (n + sqrt(n+1))^n)/2.

A084095 First superdiagonal of number array A084061.

Original entry on oeis.org

1, 1, 5, 45, 553, 8525, 157481, 3383989, 82823777, 2272771305, 69070483549, 2301873355661, 83445967372681, 3268307044050997, 137510640882447041, 6184402325475261525, 296032663549928711041, 15025296455500536616337
Offset: 0

Views

Author

Paul Barry, May 11 2003

Keywords

Links

Crossrefs

Cf. A084061, A084062, A084063, A084064, A084065, A084096.

Programs

  • GAP
    List([0..20], n-> ((n-Sqrt(n-1))^n + (n+Sqrt(n-1))^n)/2); # G. C. Greubel, Jan 11 2020
  • Magma
    [Round(((n-Sqrt(n-1))^n + (n+Sqrt(n-1))^n)/2): n in [0..20]]; // G. C. Greubel, Jan 11 2020
  • Maple
    seq( round(((n-sqrt(n-1))^n + (n+sqrt(n-1))^n)/2), n=0..20); # G. C. Greubel, Jan 11 2020
  • Mathematica
    Table[Round[((n+Sqrt[n-1])^n + (n-Sqrt[n-1])^n)/2], {n,0,20}] (* G. C. Greubel, Jan 11 2020 *)
  • PARI
    vector(21, n, round(((n-1-sqrt(n-2))^(n-1) + (n-1+sqrt(n-2))^(n-1))/2) ) \\ G. C. Greubel, Jan 11 2020
  • Sage
    [round(((n-sqrt(n-1))^n + (n+sqrt(n-1))^n)/2) for n in (0..20)] # G. C. Greubel, Jan 11 2020

Formula

a(n) = ((n - sqrt(n-1))^n + (n + sqrt(n-1))^n)/2.

A084064 Third row of number array A084061.

Original entry on oeis.org

1, 1, 6, 45, 452, 5725, 87704, 1577849, 32618512, 762046137, 19856872032, 571007744549, 17962793210944, 613650073693397, 22624291883495808, 895379458590349425, 37861032312753094912, 1703550488551604490353
Offset: 0

Views

Author

Paul Barry, May 11 2003

Keywords

Links

Crossrefs

Cf. A000312, A062024, A084061, A084062, A084063, A084065, A084095, A084096.

Programs

  • GAP
    List([0..20], n-> ((n-Sqrt(2))^n + (n+Sqrt(2))^n)/2); # G. C. Greubel, Jan 11 2020
  • Magma
    [Round(((n-Sqrt(2))^n + (n+Sqrt(2))^n)/2): n in [0..20]]; // G. C. Greubel, Jan 11 2020
  • Maple
    seq( round(((n-sqrt(2))^n + (n+sqrt(2))^n)/2), n=0..20); # G. C. Greubel, Jan 11 2020
  • Mathematica
    Table[Round[((n+Sqrt[2])^n + (n-Sqrt[2])^n)/2], {n,0,20}] (* G. C. Greubel, Jan 11 2020 *)
  • PARI
    vector(21, n, round(((n-1-sqrt(2))^(n-1) + (n-1+sqrt(2))^(n-1))/2) ) \\ G. C. Greubel, Jan 11 2020
  • Sage
    [round(((n-sqrt(2))^n + (n+sqrt(2))^n)/2) for n in (0..20)] # G. C. Greubel, Jan 11 2020

Formula

a(n) = ( (n - sqrt(2))^n + (n + sqrt(2))^n )/2.

A084065 Fourth row of number array A084061.

Original entry on oeis.org

1, 1, 7, 54, 553, 7100, 109863, 1991752, 41426257, 972602640, 25447064743, 734276705888, 23166635069241, 793426715543488, 29316839407495111, 1162492244159875200, 49240280161094287777, 2218952409252783579392
Offset: 0

Views

Author

Paul Barry, May 11 2003

Keywords

Links

Crossrefs

Cf. A084061, A084062, A084063, A084064, A084095, A084096.

Programs

  • GAP
    List([0..20], n-> ((n-Sqrt(3))^n + (n+Sqrt(3))^n)/2); # G. C. Greubel, Jan 11 2020
  • Magma
    [Round(((n-Sqrt(3))^n + (n+Sqrt(3))^n)/2): n in [0..20]]; // G. C. Greubel, Jan 11 2020
  • Maple
    seq( round(((n-sqrt(3))^n + (n+sqrt(3))^n)/2), n=0..20); # G. C. Greubel, Jan 11 2020
  • Mathematica
    Table[Round[((n+Sqrt[3])^n + (n-Sqrt[3])^n)/2], {n,0,20}] (* G. C. Greubel, Jan 11 2020 *)
  • PARI
    vector(21, n, round(((n-1-sqrt(3))^(n-1) + (n-1+sqrt(3))^(n-1))/2) ) \\ G. C. Greubel, Jan 11 2020
  • Sage
    [round(((n-sqrt(3))^n + (n+sqrt(3))^n)/2) for n in (0..20)] # G. C. Greubel, Jan 11 2020

Formula

a(n) = ( (n - sqrt(3))^n + (n + sqrt(3))^n )/2.

A084096 Fifth row of number array A084061.

Original entry on oeis.org

1, 1, 8, 63, 656, 8525, 133120, 2430547, 50839808, 1199150649, 31495553024, 911770726823, 28846956187648, 990358890251653, 36670756251238400, 1456804472261953275, 61808742217201811456, 2789456491560247772657
Offset: 0

Views

Author

Paul Barry, May 11 2003

Keywords

Links

Crossrefs

Cf. A084061, A084062, A084063, A084064, A084065, A084095.

Programs

  • GAP
    List([0..20], n-> ((n-2)^n + (n+2)^n)/2); # G. C. Greubel, Jan 11 2020
  • Magma
    [((n-2)^n + (n+2)^n)/2: n in [0..20]]; // G. C. Greubel, Jan 11 2020
  • Maple
    seq( ((n-2)^n + (n+2)^n)/2, n=0..20); # G. C. Greubel, Jan 11 2020
  • Mathematica
    Table[((n+2)^n + (n-2)^n)/2, {n,0,20}] (* G. C. Greubel, Jan 11 2020 *)
  • PARI
    vector(21, n, ((n-3)^(n-1) + (n+1)^(n-1))/2 ) \\ G. C. Greubel, Jan 11 2020
  • Sage
    [((n-2)^n + (n+2)^n)/2 for n in (0..20)] # G. C. Greubel, Jan 11 2020

Formula

a(n) = ((n - sqrt(4))^n + (n + sqrt(4))^n)/2 = ((n+2)^n + (n-2)^n)/2.

A361432 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,2*j).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 4, 0, 1, 4, 12, 20, 8, 0, 1, 5, 20, 54, 68, 16, 0, 1, 6, 30, 112, 252, 232, 32, 0, 1, 7, 42, 200, 656, 1188, 792, 64, 0, 1, 8, 56, 324, 1400, 3904, 5616, 2704, 128, 0, 1, 9, 72, 490, 2628, 10000, 23360, 26568, 9232, 256, 0
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2023

Keywords

Examples

			Square array begins:
  1,  1,   1,    1,    1,     1, ...
  0,  1,   2,    3,    4,     5, ...
  0,  2,   6,   12,   20,    30, ...
  0,  4,  20,   54,  112,   200, ...
  0,  8,  68,  252,  656,  1400, ...
  0, 16, 232, 1188, 3904, 10000, ...

Links

Crossrefs

Column k=0..11 give A000007, A011782, A006012, A083881, A081335, A090139, A145301, A145302, A145303, A143079, A289414, A289415.
Main diagonal gives A084062.
Cf. A084061, A084097.

Programs

  • PARI
    T(n,k) = sum(j=0, n\2, k^(n-j)*binomial(n, 2*j));
  • PARI
    T(n, k) = round(((k+sqrt(k))^n+(k-sqrt(k))^n))/2;

Formula

T(0,k) = 1, T(1,k) = k; T(n,k) = 2 * k * T(n-1,k) - (k-1) * k * T(n-2,k).
T(n,k) = ((k + sqrt(k))^n + (k - sqrt(k))^n)/2.
G.f. of column k: (1 - k * x)/(1 - 2 * k * x + (k-1) * k * x^2).
E.g.f. of column k: exp(k * x) * cosh(sqrt(k) * x).

A360766 a(0) = 0; a(n) = ( (n + sqrt(n))^n - (n - sqrt(n))^n )/(2 * sqrt(n)).

Original entry on oeis.org

0, 1, 4, 30, 320, 4400, 73872, 1462552, 33325056, 858283776, 24641000000, 779935205984, 26972930949120, 1011642325897216, 40890444454377728, 1771640957790000000, 81896889467638120448, 4022826671022707900416, 209224123984489179202560
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2023

Keywords

Crossrefs

Main diagonal of A361290.
Cf. A007070, A030192, A093145, A154237.
Cf. A084062.

Programs

  • PARI
    a(n) = polcoeff(lift(Mod('x, ('x-n)^2-n)^n), 1); \\ Kevin Ryde, Mar 16 2023

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} n^(n-1-k) * binomial(n,2*k+1).
a(n) = [x^n] x/(1 - 2*n*x + (n-1)*n*x^2).
a(n) = n! * [x^n] exp(n * x) * sinh(sqrt(n) * x) / sqrt(n) for n > 0.
Showing 1-8 of 8 results.

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