A081490 -id:A081490 - cp's OEIS frontend

A081489 a(n) = n(2n^2 -3n +7)/6 = C(n, 1) + C(n, 2) + 2C(n, 3).

1, 3, 8, 18, 35, 61, 98, 148, 213, 295, 396, 518, 663, 833, 1030, 1256, 1513, 1803, 2128, 2490, 2891, 3333, 3818, 4348, 4925, 5551, 6228, 6958, 7743, 8585, 9486, 10448, 11473, 12563, 13720, 14946, 16243, 17613, 19058, 20580, 22181, 23863, 25628

Offset: 1

Amarnath Murthy, Mar 25 2003

Diagonal of triangle in A081491.
First difference is given by A002522 = n^2 + 1. Second difference is odd numbers given by A005408.
With offset 0, this is the binomial transform of (0,1,1,2,0,0,0,...). - Paul Barry, Jul 03 2003
Equals row sums of triangle A144337. - Gary W. Adamson, Sep 18 2008
a(n) = sum of first (n-1) squares + n; e.g., a(5) = 35 = (1 + 4 + 9 + 16 + 5). - Gary W. Adamson, Aug 27 2010
Equals the number of functions from {1,2,...,n} to {1,2,...,n} that occur as compositions of U(x) = min{x+1,n} and D(x) = max{x-1,1}, including the identity function (the empty composition). Problem 11901 in The American Mathematical Monthly, volume 123, page 399, April 2016), proposed by Don Knuth, asked for the count of such functions (solution submitted to Monthly by Jerrold W. Grossman and Serge Kruk, August 21, 2016). - Jerrold Grossman, Aug 21 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jerrold W. Grossman and Serge Kruk, Solution to Problem 11901 in The American Mathematical Monthly, 2016
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002522, A005408, A081490, A081491, A081492, A144337.

List([1..50], n-> n*(2*n^2-3*n+7)/6); # G. C. Greubel, Aug 13 2019

I:=[1,3,8,18]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Feb 28 2014

with(combinat):a:=n->sum(fibonacci(3,i), i=0..n): seq(a(n), n=0..42); # Zerinvary Lajos, Mar 20 2008

Table[n*(2*n^2-3*n+7)/6, {n,50}] (* Vladimir Joseph Stephan Orlovsky, Nov 07 2008, modified by G. C. Greubel, Aug 13 2019 *)
LinearRecurrence[{4,-6,4,-1},{1,3,8,18},50] (* Harvey P. Dale, Aug 07 2025 *)

my(x='x+O(x^50)); Vec(serlaplace(exp(x)*(x+x^2/2+x^3/3)))

def A081489(n): return n*(n*((n<<1)-3)+7)//6 # Chai Wah Wu, Nov 05 2024

[n*(2*n^2-3*n+7)/6 for n in (1..50)] # G. C. Greubel, Aug 13 2019

a(n) = n*(2*n^2 -3*n +7)/6 = C(n, 1) + C(n, 2) + 2*C(n, 3). - Paul Barry, Jul 03 2003
E.g.f.: exp(x)*(x +x^2/2 +x^3/3).
O.g.f.: x*(1-x+2*x^2)/(1-x)^4. - Colin Barker, Jul 28 2012
a(n) = 2*n + Sum_{i=1..n} (i^2 - 2*i). - Wesley Ivan Hurt, Feb 25 2014

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 29 2003

New name, using the formulas of Paul Barry, Joerg Arndt, Feb 28 2014

A081491 Triangle read by rows in which the n-th row contains n terms of an arithmetic progression with a common difference of (n-1) and the first term of (n+1)-th row is 1 more than the last term of the n-th row.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 15, 18, 19, 23, 27, 31, 35, 36, 41, 46, 51, 56, 61, 62, 68, 74, 80, 86, 92, 98, 99, 106, 113, 120, 127, 134, 141, 148, 149, 157, 165, 173, 181, 189, 197, 205, 213, 214, 223, 232, 241, 250, 259, 268, 277, 286, 295, 296, 306, 316, 326, 336, 346

Offset: 1

Amarnath Murthy, Mar 25 2003

			1; 2,3; 4,6,8; 9,12,15,18; 19,23,27,31,35; 36,41,46,51,56,61; ...

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A002522, A005408, A081489, A081490, A081492.

Table[NestList[#+n-1&,(2n^3-9n^2+19n-6)/6,n-1],{n,11}]//Flatten (* Harvey P. Dale, Feb 12 2024 *)

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 29 2003

A081492 Sum of terms in n-th row of A081491.

Original entry on oeis.org

1, 5, 18, 54, 135, 291, 560, 988, 1629, 2545, 3806, 5490, 7683, 10479, 13980, 18296, 23545, 29853, 37354, 46190, 56511, 68475, 82248, 98004, 115925, 136201, 159030, 184618, 213179, 244935, 280116, 318960, 361713, 408629, 459970, 516006, 577015

Offset: 1

Amarnath Murthy, Mar 25 2003

For odd n a(n) is a multiple of n and a(n)/n is the middle term of the corresponding row.

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

Cf. A002522, A005408, A081489, A081490, A081491.

List([1..40], n-> n*(2*(n-1)^3+7*n-1)/6); # G. C. Greubel, Aug 13 2019

[n*(2*(n-1)^3+7*n-1)/6: n in [1..40]]; // G. C. Greubel, Aug 13 2019

seq(n*(2*(n-1)^3+7*n-1)/6, n=1..40); # G. C. Greubel, Aug 13 2019

LinearRecurrence[{5,-10,10,-5,1},{1,5,18,54,135},40] (* Harvey P. Dale, Jul 01 2018 *)

vector(40, n, n*(2*(n-1)^3+7*n-1)/6) \\ G. C. Greubel, Aug 13 2019

[n*(2*(n-1)^3+7*n-1)/6 for n in (1..40)] # G. C. Greubel, Aug 13 2019

a(n) = n*(2*n^3 - 6*n^2 + 13*n - 3)/6.
G.f.: x*(1+x)*(1-x+4*x^2)/(1-x)^5. - Colin Barker, Jul 28 2012
E.g.f.: x*(6 +9*x +6*x^2 +2*x^3)/6. - G. C. Greubel, Aug 13 2019

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 29 2003.

Formula corrected by Colin Barker, Jul 28 2012

cp's OEIS Frontend

A081489 a(n) = n*(2*n^2 -3*n +7)/6 = C(n, 1) + C(n, 2) + 2*C(n, 3).

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A081491 Triangle read by rows in which the n-th row contains n terms of an arithmetic progression with a common difference of (n-1) and the first term of (n+1)-th row is 1 more than the last term of the n-th row.

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A081492 Sum of terms in n-th row of A081491.

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GAP

Magma

Maple

Mathematica

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Sage

Formula

Extensions

A081489 a(n) = n(2n^2 -3n +7)/6 = C(n, 1) + C(n, 2) + 2C(n, 3).