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 10 results.

A051846 Digits 1..n in strict descending order n..1 interpreted in base n+1.

Original entry on oeis.org

1, 7, 57, 586, 7465, 114381, 2054353, 42374116, 987654321, 25678050355, 736867805641, 23136292864686, 789018236134297, 29043982525261081, 1147797409030816545, 48471109094902544776, 2178347851919531492065, 103805969587115219182431
Offset: 1

Views

Author

Antti Karttunen, Dec 13 1999

Keywords

Comments

All odd-indexed (2n+1) terms are divisible by (2n+1). See A051847.
All even-indexed (2n) terms are divisible by n. - Alexander R. Povolotsky, Oct 20 2022

Examples

			a(1) = 1,
a(2) = 2*3 + 1 = 7,
a(3) = 3*(4^2) + 2*4 + 1 = 57,
a(4) = 4*(5^3) + 3*(5^2) + 2*5 + 1 = 586.

Links

Crossrefs

The right edge of A051845.
Cf. A023811, A051847, A058128, A062806, A199969.

Programs

  • Maple
    a(n) := proc(n) local i; add(i*((n+1)^(i-1)),i=1..n); end;
  • Mathematica
    Array[Sum[i*(# + 1)^(i - 1), {i, #}] &, 18] (* Michael De Vlieger, Apr 04 2024 *)
  • Maxima
    makelist(((n+1)^(n+1)*(n-1) + 1)/n^2,n,1,20); /* Martin Ettl, Jan 25 2013 */
  • PARI
    a(n)=((n+1)^(n+1)*(n-1)+1)/n^2
  • Python
    def a(n): return sum((i+1)*(n+1)**i for i in range(n))
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Apr 10 2022

Formula

a(n) = Sum_{i=1..n} i*(n+1)^(i-1).
a(n) = ((n+1)^(n+1)*(n-1) + 1)/n^2 = A062806(n+1)/(n+1) - (n+1)^(n+1). - Benoit Cloitre, Sep 28 2002
a(n) = A028310(n-1) * A023811(n+1) + A199969(n+1). - M. F. Hasler, Jan 22 2013
a(n) = (n-1) * A058128(n+1) + 1. - Seiichi Manyama, Apr 10 2022

Extensions

Minor edits in formulas by M. F. Hasler, Oct 11 2019

A068475 a(n) = Sum_{m=0..n} m*n^(m-1).

Original entry on oeis.org

0, 1, 5, 34, 313, 3711, 54121, 937924, 18831569, 429794605, 10987654321, 310989720966, 9652968253897, 326011399456939, 11901025061692313, 466937872906120456, 19594541482740368161, 875711370981239308953, 41524755927216069067489, 2082225625247428808306410
Offset: 0

Views

Author

Francois Jooste (phukraut(AT)hotmail.com), Mar 10 2002

Keywords

Comments

The closed form comes from taking the derivative of the closed form of A031972, for which each term of this sequence is a derivative. - Jonas Whidden, Oct 18 2011

Examples

			a(2) = Sum_{m = 1..2} m*2^(m-1) = 1 + 2*2 = 5.

Links

Crossrefs

Derivative sequence of A031972.
Cf. A023037, A062805, A062806, A368534.

Programs

  • Haskell
    a068475 n = sum $ zipWith (*) [1..n] $ iterate (* n) 1
-- Reinhard Zumkeller, Nov 22 2014
  • Magma
    [0] cat [(&+[m*n^(m-1): m in [0..n]]): n in [1..30]]; // G. C. Greubel, Oct 13 2018
  • Maple
    a := n->sum(m*n^(m-1),m=1..n);
  • Mathematica
    Join[{0}, Table[Sum[m*n^(m-1), {m,0,n}], {n,1,30}]] (* G. C. Greubel, Oct 13 2018 *)
  • PARI
    for(n=0,30, print1(if(n==0, 0, sum(m=0,n, m*n^(m-1))), ", ")) \\ G. C. Greubel, Oct 13 2018

Formula

a(1) = 1. For n > 1, a(n) = ((n-1)*(n+1)*n^n - n^(n+1) + 1)/(n-1)^2. - Jonas Whidden, Oct 18 2011
a(n) = A062806(n) / n for n>=1. - Reinhard Zumkeller, Nov 22 2014
a(n) = [x^(n-1)] 1/((1 - x)*(1 - n*x)^2). - Peter Bala, Feb 12 2024

A368526 a(n) = Sum_{k=1..n} k^2 * n^k.

Original entry on oeis.org

0, 1, 18, 282, 4740, 89355, 1896846, 45050852, 1186829064, 34391135205, 1087928669410, 37322190255966, 1380461544684300, 54772368958008975, 2320775754168090870, 104596636848116060040, 4996700995031905899536, 252208510175779038669321
Offset: 0

Views

Author

Seiichi Manyama, Dec 28 2023

Keywords

Crossrefs

Cf. A062806, A303991, A368527.
Cf. A368524.

Programs

  • PARI
    a(n) = sum(k=1, n, k^2*n^k);

Formula

a(n) = [x^n] n*x * (1+n*x)/((1-x) * (1-n*x)^3).
a(n) = n * (n+1) * (n^n * (n^3-3*n^2+2*n+1) - 1)/(n-1)^3 for n > 1.

A368527 a(n) = Sum_{k=1..n} k^3 * n^k.

Original entry on oeis.org

0, 1, 34, 804, 18244, 434205, 11138766, 310151632, 9370253320, 306232628625, 10783859167810, 407523041660196, 16461877678462668, 708207095198943613, 32338800248010936694, 1562509380160144645440, 79657105206246202521616
Offset: 0

Views

Author

Seiichi Manyama, Dec 28 2023

Keywords

Crossrefs

Cf. A062806, A303991, A368526.
Cf. A368525.

Programs

  • PARI
    a(n) = sum(k=1, n, k^3*n^k);

Formula

a(n) = [x^n] n*x * (1+4*n*x+(n*x)^2)/((1-x) * (1-n*x)^4).
a(n) = n * (n^n * (n^6-3*n^5+8*n^3-4*n^2-7*n-1) + n^2 + 4*n + 1)/(n-1)^4 for n > 1.

A368534 a(n) = Sum_{k=1..n} binomial(k+1,2) * n^(n-k).

Original entry on oeis.org

0, 1, 5, 24, 146, 1215, 13431, 186816, 3130436, 61291125, 1371742105, 34522712136, 964626945558, 29621465864627, 991330604373851, 35906022352657920, 1399219698628043016, 58367293868445147657, 2594796705962971336125, 122463905297217627859000
Offset: 0

Views

Author

Seiichi Manyama, Dec 29 2023

Keywords

Crossrefs

Cf. A062805, A368535.
Cf. A002662, A052150, A052161.
Cf. A023037, A062806, A068475.

Programs

  • Mathematica
    Table[Sum[Binomial[k+1,2]n^(n-k),{k,n}],{n,0,20}] (* Harvey P. Dale, May 14 2025 *)
  • PARI
    a(n) = sum(k=1, n, binomial(k+1, 2)*n^(n-k));

Formula

a(n) = [x^n] x/((1-n*x) * (1-x)^3).
a(n) = n * (2*n^(n+1) - n^3 - n^2 + n - 1)/(2 * (n-1)^3) for n > 1.

A189001 a(n) = Sum_{i=0..n} (i+1)*n^i.

Original entry on oeis.org

1, 3, 17, 142, 1593, 22461, 380713, 7526268, 169826513, 4303999495, 120987654321, 3734729768298, 125562274081225, 4566262891748481, 178581127445062553, 7473240118999870456, 333189190735802745633, 15766036084935301064139
Offset: 0

Views

Author

Bruno Berselli, Apr 15 2011

Keywords

Examples

			a(4) = 1593  because  1593 = 1+2*4+3*4^2+4*4^3+5*4^4.

Links

Crossrefs

Cf. A031973, A051846, A062806.
Cf. A189122: Sum_{i=0..n} (i+1)^2*n^i.

Programs

  • Magma
    [&+[(k+1)*n^k: k in [0..n]]: n in [0..17]];
  • Mathematica
    Join[{1, 3}, Table[(((n^2 - 2) n^(n + 1) + 1) / (n - 1)^2), {n, 2, 20}]] (* Vincenzo Librandi, Aug 19 2013 *)

Formula

a(n) = ((n^2-2)*n^(n+1)+1)/(n-1)^2 for n > 1; a(0)=1, a(1)=3.

A189122 a(n) = Sum_{i=0..n} (i+1)^2*n^i.

Original entry on oeis.org

1, 5, 45, 526, 7585, 130371, 2602285, 59142588, 1507308129, 42563286145, 1318792866941, 44477806954890, 1621859437812289, 63576780042697663, 2665971232476845805, 119073945060707737336, 5643402849491554535745
Offset: 0

Views

Author

Bruno Berselli, Apr 19 2011

Keywords

Examples

			a(4) = 7585  because  7585 = 1+2^2*4+3^2*4^2+4^2*4^3+5^2*4^4.

References

  • "Supplemento al Periodico di Matematica", Raffaello Giusti Editore (Livorno), Apr/May 1913 p. 99 (Problem 1277, case x=n).

Links

Crossrefs

Cf. A062806, A189001.

Programs

  • Magma
    [&+[(k+1)^2*n^k: k in [0..n]]: n in [0..17]];
  • Mathematica
    Join[{1, 5}, Table[(n + 1) (4 n^(n + 1) - 3 n^(n + 2) - n^(n + 3) + n^(n + 4) -1) / (n - 1)^3, {n, 2, 20}]] (* Vincenzo Librandi, Aug 19 2013 *)

Formula

a(n) = (n+1)*(4*n^(n+1)-3*n^(n+2)-n^(n+3)+n^(n+4)-1)/(n-1)^3 for n>1; a(0)=1, a(1)=5.

A068476 a(n) = Sum_{m=1..n} m*n^(m+(-1)^n).

Original entry on oeis.org

0, 1, 20, 34, 5008, 3711, 1948356, 937924, 1205220416, 429794605, 1098765432100, 310989720966, 1390027428561168, 326011399456939, 2332600912091693348, 466937872906120456, 5016202619581534249216, 875711370981239308953
Offset: 0

Views

Author

Francois Jooste (phukraut(AT)hotmail.com), Mar 10 2002

Keywords

Examples

			a(2) = 2^(1+1)+2*2^(2+1) = 4+16 = 20.

Links

Crossrefs

Cf. A062806 (without the (-1)^n).

Programs

  • Magma
    [0] cat [(&+[m*n^(m+(-1)^n): m in [1..n]]): n in [1..30]]; // G. C. Greubel, Oct 13 2018
  • Maple
    b := n->sum(m*n^(m+(-1)^n),m=1..n);
# Alternative:
f:= n -> n^(1+(-1)^n)*(n^n*(n^2-n-1)+1)/(n-1)^2:
f(0):= 0: f(1):= 1:
map(f, [$0..40]);
  • Mathematica
    Table[Sum[m*n^(m+(-1)^n), {m,1,n}], {n,0,30}] (* G. C. Greubel, Oct 13 2018 *)
  • PARI
    a(n) = sum(m=1, n, m*n^(m+(-1)^n)); \\ Michel Marcus, Nov 15 2017

Formula

If n >= 2, a(n) = n^(1+(-1)^n)*(n^n*(n^2-n-1)+1)/(n-1)^2. - Robert Israel, Nov 15 2017

A368536 a(n) = Sum_{k=1..n} binomial(k+1,2) * n^k.

Original entry on oeis.org

0, 1, 14, 192, 2996, 53955, 1110786, 25808160, 668740808, 19129643325, 598902606310, 20371538593296, 748148581865532, 29505258575474591, 1243695052515891626, 55800352470853933440, 2655106829377875895056, 133547801741230053460761
Offset: 0

Views

Author

Seiichi Manyama, Dec 29 2023

Keywords

Crossrefs

Cf. A062806, A368537.
Cf. A368526.

Programs

  • PARI
    a(n) = sum(k=1, n, binomial(k+1, 2)*n^k);

Formula

a(n) = [x^n] n*x/((1-x) * (1-n*x)^3).
a(n) = n * (n^n * (n^4-n^3-3*n^2+3*n+2) - 2)/(2 * (n-1)^3) for n > 1.

A368537 a(n) = Sum_{k=1..n} binomial(k+2,3) * n^k.

Original entry on oeis.org

0, 1, 18, 309, 5828, 123230, 2913126, 76405854, 2205340936, 69523722855, 2377899710410, 87721897714891, 3472488925101516, 146833416409808492, 6605726035373765678, 315051237815279406540, 15879038919798268666896, 843348814519524716426685
Offset: 0

Views

Author

Seiichi Manyama, Dec 29 2023

Keywords

Crossrefs

Cf. A062806, A368536.
Cf. A368527.

Programs

  • PARI
    a(n) = sum(k=1, n, binomial(k+2, 3)*n^k);

Formula

a(n) = [x^n] n*x /((1-x) * (1-n*x)^4).
a(n) = n * (n^n * (n^6-7*n^4+5*n^3+12*n^2-11*n-6) + 6)/(6 * (n-1)^4) for n > 1.
Showing 1-10 of 10 results.

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