A056579 -id:A056579 - cp's OEIS frontend

A056578 a(n) = 1 + 2n + 3n^2 + 4*n^3.

1, 10, 49, 142, 313, 586, 985, 1534, 2257, 3178, 4321, 5710, 7369, 9322, 11593, 14206, 17185, 20554, 24337, 28558, 33241, 38410, 44089, 50302, 57073, 64426, 72385, 80974, 90217, 100138, 110761, 122110, 134209, 147082, 160753, 175246, 190585, 206794, 223897, 241918

Offset: 0

Henry Bottomley, Jun 29 2000

			For n>4 this is 4321 translated from base n to base 10.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Note: 1 + 2*x + 3*x^2 + 4*x^3 is the first derivative of 1 + x + x^2 + x^3 + x^4, i.e., A053699.

Cf. A000012, A005408, A056109, A056579.

f[n_]:=1+2*n+3*n^2+4*n^3; lst={}; Do[AppendTo[lst,f[n]],{n,0,5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 12 2010 *)

a(n) = (A053699(n+1) - A053699(n-1))/2 - 4*n - 1.
G.f.: (1 + 6*x + 15*x^2 + 2*x^3)/(1-x)^4. - Colin Barker, Jan 10 2012
From Elmo R. Oliveira, Apr 20 2025: (Start)
E.g.f.: exp(x)*(1 + 9*x + 15*x^2 + 4*x^3).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

More terms from Elmo R. Oliveira, Apr 20 2025

A059045 Square array T(n,k) read by antidiagonals where T(0,k) = 0 and T(n,k) = 1 + 2k + 3k^2 + ... + n*k^(n-1).

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 5, 1, 0, 1, 10, 17, 7, 1, 0, 1, 15, 49, 34, 9, 1, 0, 1, 21, 129, 142, 57, 11, 1, 0, 1, 28, 321, 547, 313, 86, 13, 1, 0, 1, 36, 769, 2005, 1593, 586, 121, 15, 1, 0, 1, 45, 1793, 7108, 7737, 3711, 985, 162, 17, 1, 0, 1, 55, 4097, 24604, 36409

Offset: 0

Henry Bottomley, Dec 18 2000

			   0,   0,   0,    0,     0,      0,      0,      0,       0, ...
   1,   1,   1,    1,     1,      1,      1,      1,       1, ...
   1,   3,   5,    7,     9,     11,     13,     15,      17, ...
   1,   6,  17,   34,    57,     86,    121,    162,     209, ...
   1,  10,  49,  142,   313,    586,    985,   1534,    2257, ...
   1,  15, 129,  547,  1593,   3711,   7465,  13539,   22737, ...
   1,  21, 321, 2005,  7737,  22461,  54121, 114381,  219345, ...
   1,  28, 769, 7108, 36409, 131836, 380713, 937924, 2054353, ...

Cf. A000004, A000012, A005408, A056109, A056578, A056579 (rows 1-6).

Cf. A057427, A000217, A000337, A014915, A014916, A014917, A014918, A014920 (columns 1-7).

A059045 := proc(n,k)
    if k = 1 then
        n*(n+1) /2 ;
    else
        (1+n*k^(n+1)-k^n*(n+1))/(k-1)^2 ;
    end if;
end proc: # R. J. Mathar, Mar 29 2013

T(n,k) = n*k^(n-1)+T(n-1, k) = (n*k^(n+1)-(n+1)*k^n+1)/(k-1)^2.

A113630 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6 + 8n^7 + 9n^8.

Original entry on oeis.org

1, 45, 4097, 83653, 757305, 4272461, 17736745, 59409477, 169826513, 429794605, 987654321, 2098573445, 4178995657, 7879732173, 14181546905, 24517448581, 40926266145, 66242446637, 104327377633, 160347899205, 241108033241

Offset: 0

Jonathan Vos Post, Jan 14 2006

1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 + 7*x^6 + 8*x^7 + 9*x^8 is the derivative of 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 = (x^10 - 1)/(x-1).

			a(3) = 1 + 2*3 + 3*3^2 + 4*3^3 + 5*3^4 + 6*3^5 + 7*3^6 + 8*3^7 + 9*3^8 = 83653 is prime.
a(5) = 1 + 2*5 + 3*5^2 + 4*5^3 + 5*5^4 + 6*5^5 + 7*5^6 + 8*5^7 + 9*5^8 = 4272461 is prime.
a(8) = 1 + 2*8 + 3*8^2 + 4*8^3 + 5*8^4 + 6*8^5 + 7*8^6 + 8*8^7 + 9*8^8 = 169826513 is prime.
a(23) = 1 + 2*23 + 3*23^2 + 4*23^3 + 5*23^4 + 6*23^5 + 7*23^6 + 8*23^7 + 9*23^8 = 733113789893 is prime.

Chai Wah Wu, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000012, A005408, A056109, A056578, A056579.

Cf. A249951 (primes), A068475.

a113630 n = sum $ zipWith (*) [1..9] $ iterate (* n) 1
-- Reinhard Zumkeller, Nov 22 2014

[1+2*n+3*n^2+4*n^3+5*n^4+6*n^5+7*n^6+8*n^7+9*n^8: n in [0..20]]; // Vincenzo Librandi, Nov 09 2014

CoefficientList[Series[(5 x^8 + 1548 x^7 + 31360 x^6 + 129620 x^5 + 148266 x^4 + 48316 x^3 + 3728 x^2 + 36 x + 1) / (1 - x)^9, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 09 2014 *)
With[{c=Total[Table[k n^(k-1),{k,9}]]},Table[c,{n,0,30}]] (* or *) LinearRecurrence[ {9,-36,84,-126,126,-84,36,-9,1},{1,45,4097,83653,757305,4272461,17736745,59409477,169826513},30] (* Harvey P. Dale, Jul 18 2017 *)

vector(100,n,1 + 2*(n-1)+ 3*(n-1)^2 + 4*(n-1)^3 + 5*(n-1)^4 + 6*(n-1)^5 + 7*(n-1)^6 + 8*(n-1)^7 + 9*(n-1)^8) \\ Derek Orr, Nov 09 2014

A113630_list, m = [1], [362880, -1229760, 1607760, -1011480, 309816, -40752, 1584, -4, 1]
for _ in range(10**3):
    for i in range(8):
        m[i+1]+= m[i]
    A113630_list.append(m[-1]) # Chai Wah Wu, Nov 09 2014

a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6 + 8*n^7 + 9*n^8.

G.f.: -(5*x^8 +1548*x^7 +31360*x^6 +129620*x^5 +148266*x^4 +48316*x^3 +3728*x^2 +36*x +1) / (x -1)^9. - Colin Barker, May 08 2013

A113531 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6*n^5.

Original entry on oeis.org

1, 21, 321, 2005, 7737, 22461, 54121, 114381, 219345, 390277, 654321, 1045221, 1604041, 2379885, 3430617, 4823581, 6636321, 8957301, 11886625, 15536757, 20033241, 25515421, 32137161, 40067565, 49491697, 60611301, 73645521

Offset: 0

Jonathan Vos Post, Jan 12 2006

1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 is the derivative of 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 = (n^7 - 1)/(n-1). a(6) = 1 + 2*6 + 3*6^2 + 4*6^3 + 5*6^4 + 6*6^5 = 54121 is prime, the smallest prime in the sequence. The next is a(a(1)) = a(21) = 1 + 2*21 + 3*21^2 + 4*21^3 + 5*21^4 + 6*21^5 = 25515421. Then a(24) = 49491697.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Cf. A000012, A005408, A056109, A056578, A056579.

With[{eq=Total[# n^(#-1)&/@Range[6]]},Table[eq,{n,0,30}]] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1,21,321,2005,7737,22461},30] (* Harvey P. Dale, Nov 02 2011 *)

for(n=0,50, print1(1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5, ", ")) \\ G. C. Greubel, Mar 15 2017

a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5.
O.g.f.: 2064/(-1+x)^4+3/(-1+x)+2040/(-1+x)^5+132/(-1+x)^2+720/(-1+x)^6+872/(-1+x)^3 . - R. J. Mathar, Feb 26 2008
a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6), a(0)=1, a(1)=21, a(2)=321, a(3)=2005, a(4)=7737, a(5)=22461. - Harvey P. Dale, Nov 02 2011

Corrected by R. J. Mathar, Feb 26 2008

A113532 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6.

Original entry on oeis.org

1, 28, 769, 7108, 36409, 131836, 380713, 937924, 2054353, 4110364, 7654321, 13446148, 22505929, 36167548, 56137369, 84557956, 124076833, 177920284, 249972193, 344857924, 468033241, 625878268, 825796489, 1076318788

Offset: 0

Jonathan Vos Post, Jan 12 2006

1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 + 7*n^6 is the derivative of 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 = (x^8 - 1)/(x-1). a(2) = 1 + 2*2 + 3*2^2 + 4*2^3 + 5*2^4 + 6*2^5 + 7*2^6 = 769 is prime. Other primes begin a(6) = 380713, a(12) = 22505929, a(26) = 2236055953, a(38) = 21562615273, a(44) = 51802781449, a(52) = 140712620569.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000012, A005408, A056109, A056578, A056579.

Table[1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6, {n,0,50}] (* or *) LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {1, 28, 769, 7108, 36409, 131836, 380713}, 50] (* G. C. Greubel, Mar 15 2017 *)

for(n=0,50, print1(1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6, ", ")) \\ G. C. Greubel, Mar 15 2017

a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6.

O.g.f.: -12636/(-1+x)^4 -4/(-1+x) -21480/(-1+x)^5 -309/(-1+x)^2 -16920/(-1+x)^6 -3342/(-1+x)^3-5040/(-1+x)^7 . - R. J. Mathar, Feb 26 2008

A123099 Primes of the form 1 + 2k + 3k^2 + 4k^3 + 5k^4.

Original entry on oeis.org

547, 35983, 111049, 2738179, 6076687, 15860209, 53530639, 685318537, 1043755441, 1670649571, 2347515619, 9761226721, 10330521727, 12188475769, 15042514033, 25486958659, 30383211043, 40608270601, 45701408383

Offset: 1

Jonathan Vos Post, Sep 27 2006

Primes in A056579.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A056579.

[ a: n in [0..400] | IsPrime(a) where a is 1+2*n+3*n^2+4*n^3+5*n^4]; // Vincenzo Librandi, Nov 13 2010

Select[Table[1+2n+3n^2+4n^3+5n^4,{n,500}],PrimeQ] (* Harvey P. Dale, Oct 29 2022 *)

A131466 a(n) = 5n^4 - 4n^3 + 3n^2 - 2n + 1.

Original entry on oeis.org

1, 3, 57, 319, 1065, 2691, 5713, 10767, 18609, 30115, 46281, 68223, 97177, 134499, 181665, 240271, 312033, 398787, 502489, 625215, 769161, 936643, 1130097, 1352079, 1605265, 1892451, 2216553, 2580607, 2987769, 3441315

Offset: 0

Mohammad K. Azarian, Jul 26 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A084849, A056109, A056105, A056578, A056579.

Table[5n^4-4n^3+3n^2-2n+1,{n,0,30}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,3,57,319,1065},30] (* Harvey P. Dale, Nov 01 2020 *)

a(n)=5*n^4-4*n^3+3*n^2-2*n+1 \\ Charles R Greathouse IV, Oct 21 2022

From Chai Wah Wu, Nov 13 2018: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
G.f.: (-15*x^4 - 54*x^3 - 52*x^2 + 2*x - 1)/(x - 1)^5. (End)

A113618 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6 + 8*n^7.

Original entry on oeis.org

1, 36, 1793, 24604, 167481, 756836, 2620201, 7526268, 18831569, 42374116, 87654321, 169343516, 309160393, 538155684, 899445401, 1451432956, 2271560481, 3460629668, 5147732449, 7495831836, 10708033241, 15034586596, 20780659593

Offset: 0

Jonathan Vos Post, Jan 14 2006

1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 + 7*n^6 + 8*n^7 is the derivative of 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 = (x^9 - 1)/(x-1).

			1 + 2*8 + 3*8^2 + 4*8^3 + 5*8^4 + 6*8^5 + 7*8^6 + 8*8^7 = 18831569 = 173 * 199 * 547.
1 + 2*26 + 3*26^2 + 4*26^3 + 5*26^4 + 6*26^5 + 7*26^6 + 8*26^7 = 66490537361 is prime, the smallest prime in the sequence.

Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A000012, A005408, A056109, A056578, A056579.

[1+2*n+3*n^2+4*n^3+5*n^4+6*n^5+7*n^6+8*n^7: n in [1..43]] // Vincenzo Librandi, Dec 21 2010

Join[{1},Table[Total[Table[p*n^(p-1),{p,8}]],{n,30}]] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,36,1793,24604,167481,756836,2620201,7526268},30] (* Harvey P. Dale, Jul 16 2014 *)

G.f.: (1+28*x+1533*x^2+11212*x^3+18907*x^4+7956*x^5+679*x^6+4*x^7)/(x-1)^8. - R. J. Mathar, Dec 21 2010

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8), with a(0)=1, a(1)=36, a(2)=1793, a(3)=24604, a(4)=167481, a(5)=756836, a(6)=2620201, a(7)=7526268. - Harvey P. Dale, Jul 16 2014

A113632 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6 + 8n^7 + 9n^8 + 10*n^9.

Original entry on oeis.org

1, 55, 9217, 280483, 3378745, 23803711, 118513705, 462945547, 1512003793, 4303999495, 10987654321, 25678050355, 55776799177, 113924725903, 220792014745, 408951042331, 728121033505, 1252121211607, 2087920281313

Offset: 0

Jonathan Vos Post, Jan 14 2006

1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6 + 8*x^7 + 9*x^8 + 10*x^9 is the derivative of 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 = (x^11 - 1)/(x-1).

			a(5) = 1 + 2*5 + 3*5^2 + 4*5^3 + 5*5^4 + 6*5^5 + 7*5^6 + 8*5^7 + 9*5^8 + 10*5^9 = 23803711 is prime.
a(30) = 1 + 2*30 + 3*30^2 + 4*30^3 + 5*30^4 + 6*30^5 + 7*30^6 + 8*30^7 + 9*30^8 + 10*30^9 = 202915112960761 is prime.

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A056578, A056579.

With[{eq=Total[Range[10](n^Range[0,9])]},Table[eq,{n,0,20}]] (* Harvey P. Dale, Mar 14 2011 *)

a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6 + 8*n^7 + 9*n^8 + 10*n^9.

G.f.: (1+x*(45+x*(8712+x*(190668+x*(982290+x*(1543254+x*(784080+x*(116268+x*(3477+5*x)))))))))/(x-1)^10. - Harvey P. Dale, Mar 14 2011

cp's OEIS Frontend

A056578 a(n) = 1 + 2*n + 3*n^2 + 4*n^3.

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A059045 Square array T(n,k) read by antidiagonals where T(0,k) = 0 and T(n,k) = 1 + 2k + 3k^2 + ... + n*k^(n-1).

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A113630 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6 + 8*n^7 + 9*n^8.

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A113531 a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5.

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A113532 a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6.

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A123099 Primes of the form 1 + 2*k + 3*k^2 + 4*k^3 + 5*k^4.

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Magma

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A131466 a(n) = 5n^4 - 4n^3 + 3n^2 - 2n + 1.

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A113618 a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6 + 8*n^7.

A056578 a(n) = 1 + 2n + 3n^2 + 4*n^3.

A113630 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6 + 8n^7 + 9n^8.

A113531 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6*n^5.

A113532 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6.

A123099 Primes of the form 1 + 2k + 3k^2 + 4k^3 + 5k^4.

A113618 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6 + 8*n^7.

A113632 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6 + 8n^7 + 9n^8 + 10*n^9.