A219069 -id:A219069 - cp's OEIS frontend

A059826 a(n) = (n^2 - n + 1)*(n^2 + n + 1).

1, 3, 21, 91, 273, 651, 1333, 2451, 4161, 6643, 10101, 14763, 20881, 28731, 38613, 50851, 65793, 83811, 105301, 130683, 160401, 194923, 234741, 280371, 332353, 391251, 457653, 532171, 615441, 708123, 810901, 924483, 1049601, 1187011, 1337493, 1501851, 1680913

Offset: 0

N. J. A. Sloane, Feb 24 2001

Main diagonal of A082039. - Paul Barry, Apr 02 2003

The base of the natural logarithms e = 2*Sum_{n>=0} 1/(a(n)*n!) and zeta(2) = Pi^2/6 = 1 + 2*Sum_{n>=1} (-1)^(n+1)/(a(n)*n^2). - Peter Bala, Jan 20 2008

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

Cf. A002061, A062159.

Main diagonal of A082039.

[n^4+n^2+1 : n in [0..50]]; // Wesley Ivan Hurt, Jun 09 2014

with(combinat): seq(fibonacci(3,n)+n^4, n=0..40); # Zerinvary Lajos, May 25 2008

Table[n^4 + n^2 + 1, {n, 0, 50}] (* Wesley Ivan Hurt, Jun 09 2014 *)

a(n) = { my(f=n^2 + 1); (f - n)*(f + n) } \\ Harry J. Smith, Jun 29 2009

a(n) = n^4+n^2+1. - Paul Barry, Apr 02 2003
a(n) = (n^2-n+1) * (n^2+n+1) = A002061(n) * A002061(n+1), products of two consecutive central polygonal numbers. a(n) = (n^6-1)/(n^2-1), n>1. a(n) = (n^5-n^4+n^3-n^2+n-1)/(n-1) = A062159(n)/(n-1), n>1. - Alexander Adamchuk, Apr 12 2006
O.g.f.: (-1+2*x-16*x^2-6*x^3-3*x^4) / (x-1)^5. - R. J. Mathar, Feb 26 2008
a(n) = A219069(n,1), for n > 0. - Reinhard Zumkeller, Nov 11 2012
a(n+2) = (n^2+3n+3) * (n^2+5n+7) = (t(n)+t(n+2)) * (t(n+1)+t(n+3)), where t=A000217 are triangular numbers. For n>=1, a(n+2) = t(2*t(n+2)+t(n)) -t(t(n)-1). - J. M. Bergot, Nov 29 2012
4*a(n) = (n^2+n+1)^2+(n^2-n+1)^2+(n^2+n-1)^2+(n^2-n-1)^2. - Bruno Berselli, Jul 03 2014
a(n) = A002061(n^2). - Franklin T. Adams-Watters, Aug 01 2014
Sum_{n>=0} 1/a(n) = 1/2 + sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6. - Amiram Eldar, Feb 14 2021

A219056 a(n) = 3*n^4.

Original entry on oeis.org

0, 3, 48, 243, 768, 1875, 3888, 7203, 12288, 19683, 30000, 43923, 62208, 85683, 115248, 151875, 196608, 250563, 314928, 390963, 480000, 583443, 702768, 839523, 995328, 1171875, 1370928, 1594323, 1843968, 2121843, 2430000, 2770563, 3145728, 3557763, 4009008

Offset: 0

Reinhard Zumkeller, Nov 11 2012

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

Cf. A008585, A000583, A000290, A033428, A219069.

Haskell
```
a219056 = (* 3) . (^ 4)
```

LinearRecurrence[{5,-10,10,-5,1},{0, 3, 48, 243, 768},100] (* or *) Table[3*n^4, {n,0,50}] (* G. Greubel, Jun 22 2016 *)

makelist(3*n^4,n,0,30); /* Martin Ettl, Nov 12 2012 */

a(n) = 3*n^4; \\ Michel Marcus, Jan 26 2022

a(n) = A219069(n,n) for n > 0;
a(n) = A008585(A000583(n)) = A000290(n)*A033428(n).
From Chai Wah Wu, Jun 22 2016: (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.: 3*x*(1 + x)*(1 + 10*x + x^2)/(1 - x)^5. (End)
E.g.f.: 3*x*(1 + 7*x + 6*x^2 + x^3)*exp(x). - G. C. Greubel, Jun 22 2016

A243201 Odd octagonal numbers indexed by triangular numbers.

Original entry on oeis.org

1, 21, 133, 481, 1281, 2821, 5461, 9633, 15841, 24661, 36741, 52801, 73633, 100101, 133141, 173761, 223041, 282133, 352261, 434721, 530881, 642181, 770133, 916321, 1082401, 1270101, 1481221, 1717633, 1981281, 2274181, 2598421, 2956161, 3349633, 3781141, 4253061, 4767841, 5328001

Offset: 0

Mathew Englander, Jun 01 2014

			a(2) = 133 because the second triangular number is 3 and third odd octagonal number is 133.
a(3) = 481 because the third triangular number is 6 and the sixth odd octagonal number is 481.
a(4) = 1281 because the fourth triangular number is 10 and the tenth odd octagonal number is 1281.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Row 5 of A059259 (coefficients of 1 + 4*n + 7*n^2 + 6*n^3 + 3*n^4 + 0*n^5 which is a formula for the within sequence).
Column 5 of A081297.
Column 6 of A072024.
Diagonal T(n + 1, n) of A219069, n > 0.
Cf. A000217, A000567, A002061, A003215, A005408, A014641, A022522, A140676, A187156.

[3*n^4+6*n^3+7*n^2+4*n+1: n in [0..40]]; // Bruno Berselli, Jun 03 2014

Table[((3 n^2 + 3 n + 2)^2 - 1)/3, {n, 0, 39}] (* Alonso del Arte, Jun 01 2014 *)

[3*n^4+6*n^3+7*n^2+4*n+1 for n in (0..40)] # Bruno Berselli, Jun 03 2014

a(n) = 3*n^4 + 6*n^3 + 7*n^2 + 4*n + 1.
a(n) = (n^2 + n + 1)*(3*n^2 + 3*n + 1).
a(n) = ((3*n^2 + 3*n + 2)^2 - 1)/3.
a(n) = A003215(n) * A002061(n + 1).
a(n) = A022522(n) / A005408(n).
a(n) = A000567(n^2 + n + 1).
a(n) = A014641((n^2 + n)/2).
a(n) = 1 + A140676(n^2 + n).
a(n) = 1 + A187156((n^2 + n + 4)/2) (empirical).
G.f.: (1 + 16*x + 38*x^2 + 16*x^3 + x^4)/(1 - x)^5. - Bruno Berselli, Jun 03 2014
E.g.f.: exp(x)*(1 + 20*x + 46*x^2 + 24*x^3 + 3*x^4). - Stefano Spezia, Apr 16 2022

A239426 21n^4 - 36n^3 + 25n^2 - 8n + 1.

Original entry on oeis.org

1, 3, 133, 931, 3441, 9211, 20293, 39243, 69121, 113491, 176421, 262483, 376753, 524811, 712741, 947131, 1235073, 1584163, 2002501, 2498691, 3081841, 3761563, 4547973, 5451691, 6483841, 7656051, 8980453, 10469683, 12136881, 13995691, 16060261, 18345243

Offset: 0

Reinhard Zumkeller, Mar 19 2014

For n > 0: a(n) = A219069(2*n-1,n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

a239426 n = (((21 * n - 36) * n + 25) * n - 8) * n + 1

[21*n^4-36*n^3+25*n^2-8*n+1: n in [0..31]]; // Vincenzo Librandi, Mar 21 2014

Table[(21 n^4 - 36 n^3 + 25 n^2 - 8 n + 1), {n, 0, 40}] (* Vincenzo Librandi, Mar 21 2014 *)
LinearRecurrence[{5,-10,10,-5,1},{1,3,133,931,3441},40] (* Harvey P. Dale, May 04 2016 *)

a(n) = (1-3*n+3*n^2) * (1-5*n+7*n^2) = A003215(n-1) * A239449(n).

G.f.: ( -1+2*x-128*x^2-286*x^3-91*x^4 ) / (x-1)^5 . - R. J. Mathar, Mar 31 2014

A219070 a(n) = (46n^5 + 30n^4 + 15*n^3 - n) / 30.

Original entry on oeis.org

0, 3, 69, 467, 1858, 5479, 13327, 28343, 54596, 97467, 163833, 262251, 403142, 598975, 864451, 1216687, 1675400, 2263091, 3005229, 3930435, 5070666, 6461399, 8141815, 10154983, 12548044, 15372395, 18683873, 22542939, 27014862, 32169903, 38083499, 44836447

Offset: 0

Reinhard Zumkeller, Nov 11 2012

For n > 0: row sums of the triangle A219069.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A000290, A000330, A000538, A000584, A219069.

a219070 n = n * (n * (n * (n * (46 * n + 30) + 15)) - 1) `div` 30 -- Reinhard Zumkeller

Table[(46n^5 + 30n^4 + 15n^3 - n)/30, {n, 0, 39}] (* Alonso del Arte, Nov 12 2012 *)

A219070(n):=(46*n^5 + 30*n^4 + 15*n^3-n)/30$
makelist(A219070(n),n,0,30); /* Martin Ettl, Nov 12 2012 */

a(n) = A000584(n) + A000290(n)*A000330(n) + A000538(n).

cp's OEIS Frontend

A059826 a(n) = (n^2 - n + 1)*(n^2 + n + 1).

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A219056 a(n) = 3*n^4.

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A243201 Odd octagonal numbers indexed by triangular numbers.

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A239426 21*n^4 - 36*n^3 + 25*n^2 - 8*n + 1.

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A219070 a(n) = (46*n^5 + 30*n^4 + 15*n^3 - n) / 30.

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A239426 21n^4 - 36n^3 + 25n^2 - 8n + 1.

A219070 a(n) = (46n^5 + 30n^4 + 15*n^3 - n) / 30.