A004188 -id:A004188 - cp's OEIS frontend

A085788 Partial sums of n 3-spaced triangular numbers beginning with t(3), e.g., a(2) = t(3)+t(6) = 6+21 = 27.

6, 27, 72, 150, 270, 441, 672, 972, 1350, 1815, 2376, 3042, 3822, 4725, 5760, 6936, 8262, 9747, 11400, 13230, 15246, 17457, 19872, 22500, 25350, 28431, 31752, 35322, 39150, 43245, 47616, 52272, 57222, 62475, 68040, 73926, 80142, 86697, 93600, 100860, 108486

Offset: 1

Jon Perry, Jul 23 2003

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

Cf. A004188, A006002.
Row sums of triangle A001283.
Cf. A254407. - Bruno Berselli, Jan 30 2015

a:=n->sum(sum(sum(j-k+1, j=1..n), k=0..n),m=0..n): seq(a(n), n=1..45); # Zerinvary Lajos, May 30 2007

LinearRecurrence[{4,-6,4,-1},{6,27,72,150},50] (* Harvey P. Dale, Dec 14 2017 *)

v=vector(40,i,i*(i+1)/2); s=0; forstep(i=3,40,3,s+=v[i]; print1(s","))

a(n) = (3/2)*n*(n+1)^2 = 3*A006002(n).
a(n) = Sum_{j=1..n} (j+n+1)*(n+1). - Zerinvary Lajos, Sep 10 2006
From Colin Barker, Mar 17 2014: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: 3*x*(x+2)/(x-1)^4. (End)
E.g.f.: 3*exp(x)*x*(1 + x)*(4 + x)/2. - Elmo R. Oliveira, Aug 14 2025

Edited and more terms from Michel Marcus, Mar 17 2014

A085789 Partial sums of n 3-spaced triangular numbers beginning with t(2), e.g., a(2) = t(2) + t(5) = 3 + 15 = 18.

Original entry on oeis.org

3, 18, 54, 120, 225, 378, 588, 864, 1215, 1650, 2178, 2808, 3549, 4410, 5400, 6528, 7803, 9234, 10830, 12600, 14553, 16698, 19044, 21600, 24375, 27378, 30618, 34104, 37845, 41850, 46128, 50688, 55539, 60690, 66150, 71928, 78033, 84474, 91260, 98400, 105903

Offset: 1

Jon Perry, Jul 23 2003

Sums of rows of triangle A100345 (n>0).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000217, A002411, A004188, A100345.

[3/2*n^2*(n+1): n in [1..40]]; // Vincenzo Librandi, Aug 14 2017

CoefficientList[Series[3 (1 + 2 x) / (1 - x)^4, {x, 0, 40}], x](* Vincenzo Librandi, Aug 14 2017 *)
LinearRecurrence[{4,-6,4,-1},{3,18,54,120},50] (* Harvey P. Dale, May 14 2023 *)

a(n) = 3/2 * n^2*(n+1).
a(n) = 3*n*binomial(n+1,2) = 3*n*A000217(n) = 3*A002411(n). - Arkadiusz Wesolowski, Feb 10 2012
G.f.: 3*(x + 2*x^2)/(1 - x)^4. - Arkadiusz Wesolowski, Feb 11 2012
From Amiram Eldar, Jun 29 2025: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/9 - 2/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/18 - 4*log(2)/3 + 2/3. (End)

More terms from Reinhard Zumkeller, Nov 18 2004

A195309 Row sums of an irregular triangle read by rows in which row n lists the next A026741(n+1) natural numbers A000027.

Original entry on oeis.org

1, 9, 11, 45, 39, 126, 94, 270, 185, 495, 321, 819, 511, 1260, 764, 1836, 1089, 2565, 1495, 3465, 1991, 4554, 2586, 5850, 3289, 7371, 4109, 9135, 5055, 11160, 6136, 13464, 7361, 16065, 8739, 18981, 10279, 22230, 11990, 25830, 13881

Offset: 1

Omar E. Pol, Sep 21 2011

nonn

The integers in same rows of the source triangle have a property related to Euler's Pentagonal Theorem.

Note that the column 1 of the mentioned triangle gives the positive terms of A001318 (see example).

			a(1) = 1
a(2) = 2+3+4 = 9
a(3) = 5+6 = 11
a(4) = 7+8+9+10+11 = 45
a(5) = 12+13+14 = 39
a(6) = 15+16+17+18+19+20+21 = 126
a(7) = 22+23+24+25 = 94
a(8) = 26+27+28+29+30+31+32+33+34 = 270
a(9) = 35+36+37+38+39 = 185

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Cf. A026741, A195310, A195311, A004188 (bisection).

A195309 := proc(n)
        (n+1)*(9*n^2+18*n-1+(3*n^2+6*n+1)*(-1)^n)/32
end proc:
seq(A195309(n),n=1..60) ; # R. J. Mathar, Oct 08 2011

LinearRecurrence[{0,4,0,-6,0,4,0,-1},{1,9,11,45,39,126,94,270},80] (* Harvey P. Dale, Jun 22 2015 *)

a(n) = (n+1)*(9*n^2+18*n-1+(3*n^2+6*n+1)*(-1)^n)/32 . - R. J. Mathar, Oct 08 2011

G.f. x*(1+9*x+7*x^2+9*x^3+x^4) / ( (x-1)^4*(1+x)^4 ). - R. J. Mathar, Oct 08 2011

A236311 Riordan array ((1-x)/(1-3x+3x^2), x/(1-3x+3x^2)).

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 3, 15, 8, 1, 0, 33, 36, 11, 1, -9, 54, 117, 66, 14, 1, -27, 54, 297, 282, 105, 17, 1, -54, -27, 594, 945, 555, 153, 20, 1, -81, -297, 864, 2583, 2295, 963, 210, 23, 1, -81, -891, 513, 5778, 7803, 4725, 1533, 276, 26, 1, 0, -1863, -1944, 10098

Offset: 0

Philippe Deléham, Jan 21 2014

Row sums are 3^n = A000244(n).
Diagonals sums are 2^n = A000079(n).
T(n,n) = A000012(n).
T(n+1,n) = A016789(n).
T(n+2,n) = A062741(n+1).
T(n+3,n) = 3*A004188(n+1).
T(n,0) = A057682(n+1).

			Triangle begins :
1;
2, 1;
3, 5, 1;
3, 15, 8, 1;
0, 33, 36, 11, 1;
-9, 54, 117, 66, 14, 1;
-27, 54, 297, 282, 105, 17, 1;

Cf. A057083, A057682, A000012, A016789, A062741

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) -3*T(n-2,k), T(0,0) = T(1,1) = 1, T(1,0) = 2, T(n,k) = 0 if k<0 or if k>n.

cp's OEIS Frontend

A085788 Partial sums of n 3-spaced triangular numbers beginning with t(3), e.g., a(2) = t(3)+t(6) = 6+21 = 27.

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A085789 Partial sums of n 3-spaced triangular numbers beginning with t(2), e.g., a(2) = t(2) + t(5) = 3 + 15 = 18.

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A195309 Row sums of an irregular triangle read by rows in which row n lists the next A026741(n+1) natural numbers A000027.

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A236311 Riordan array ((1-x)/(1-3*x+3*x^2), x/(1-3*x+3*x^2)).

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A236311 Riordan array ((1-x)/(1-3x+3x^2), x/(1-3x+3x^2)).