A011779 -id:A011779

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A139600

Square array T(n,k) = n*(k-1)*k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards.

+10
49

0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 6, 4, 0, 1, 5, 9, 10, 5, 0, 1, 6, 12, 16, 15, 6, 0, 1, 7, 15, 22, 25, 21, 7, 0, 1, 8, 18, 28, 35, 36, 28, 8, 0, 1, 9, 21, 34, 45, 51, 49, 36, 9, 0, 1, 10, 24, 40, 55, 66, 70, 64, 45, 10, 0, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 11

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

A general formula for polygonal numbers is P(n,k) = (n-2)*(k-1)*k/2 + k, where P(n,k) is the k-th n-gonal number.

The triangle sums, see A180662 for their definitions, link this square array read by antidiagonals with twelve different sequences, see the crossrefs. Most triangle sums are linear sums of shifted combinations of a sequence, see e.g. A189374. - Johannes W. Meijer, Apr 29 2011

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.

Omar E. Pol, Polygonal numbers, An alternative illustration of initial terms.

Index to sequences related to polygonal numbers

FORMULA

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

T(n,k) = A057145(n+2,k). - R. J. Mathar, Jul 28 2016

From Stefano Spezia, Apr 12 2024: (Start)

G.f.: y*(1 - x - y + 2*x*y)/((1 - x)^2*(1 - y)^3).

E.g.f.: exp(x+y)*y*(2 + x*y)/2. (End)

EXAMPLE

The square array of nonnegatives together with polygonal numbers begins:

=========================================================

....................... A A . . A A A A

....................... 0 0 . . 0 0 1 1

....................... 0 0 . . 1 1 3 3

....................... 0 0 . . 6 7 9 9

....................... 0 0 . . 9 3 6 6

....................... 0 1 . . 5 2 0 0

....................... 4 2 . . 7 9 6 7

=========================================================

Nonnegatives . A001477: 0, 1, 2, 3, 4, 5, 6, 7, ...

Triangulars .. A000217: 0, 1, 3, 6, 10, 15, 21, 28, ...

Squares ...... A000290: 0, 1, 4, 9, 16, 25, 36, 49, ...

Pentagonals .. A000326: 0, 1, 5, 12, 22, 35, 51, 70, ...

Hexagonals ... A000384: 0, 1, 6, 15, 28, 45, 66, 91, ...

Heptagonals .. A000566: 0, 1, 7, 18, 34, 55, 81, 112, ...

Octagonals ... A000567: 0, 1, 8, 21, 40, 65, 96, 133, ...

9-gonals ..... A001106: 0, 1, 9, 24, 46, 75, 111, 154, ...

10-gonals .... A001107: 0, 1, 10, 27, 52, 85, 126, 175, ...

11-gonals .... A051682: 0, 1, 11, 30, 58, 95, 141, 196, ...

12-gonals .... A051624: 0, 1, 12, 33, 64, 105, 156, 217, ...

...

=========================================================

The column with the numbers 2, 3, 4, 5, 6, ... is formed by the numbers > 1 of A000027. The column with the numbers 3, 6, 9, 12, 15, ... is formed by the positive members of A008585.

MAPLE

T:= (n, k)-> n*(k-1)*k/2+k:

seq(seq(T(d-k, k), k=0..d), d=0..14); # Alois P. Heinz, Oct 14 2018

MATHEMATICA

T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[T[n - k - 1, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 12 2009 *)

PROG

(Python)

def A139600Row(n):

x, y = 1, 1

yield 0

while True:

yield x

x, y = x + y + n, y + n

for n in range(8):

R = A139600Row(n)

print([next(R) for _ in range(11)]) # Peter Luschny, Aug 04 2019

(Magma)

T:= func< n, k | k*(n*(k-1)+2)/2 >;

A139600:= func< n, k | T(n-k, k) >;

[A139600(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2024

(SageMath)

def T(n, k): return k*(n*(k-1)+2)/2

def A139600(n, k): return T(n-k, k)

flatten([[A139600(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 12 2024

CROSSREFS

Cf. A001477, A057145, A139601, A139617, A139618, A139620, A180662.

A formal extension negative n is in A326728.

Triangle sums (see the comments): A055795 (Row1), A080956 (Row2; terms doubled), A096338 (Kn11, Kn12, Kn13, Fi1, Ze1), A002624 (Kn21, Kn22, Kn23, Fi2, Ze2), A000332 (Kn3, Ca3, Gi3), A134393 (Kn4), A189374 (Ca1, Ze3), A011779 (Ca2, Ze4), A101357 (Ca4), A189375 (Gi1), A189376 (Gi2), A006484 (Gi4). - Johannes W. Meijer, Apr 29 2011

Sequences of m-gonal numbers: A000217 (m=3), A000290 (m=4), A000326 (m=5), A000384 (m=6), A000566 (m=7), A000567 (m=8), A001106 (m=9), A001107 (m=10), A051682 (m=11), A051624 (m=12), A051865 (m=13), A051866 (m=14), A051867 (m=15), A051868 (m=16), A051869 (m=17), A051870 (m=18), A051871 (m=19), A051872 (m=20), A051873 (m=21), A051874 (m=22), A051875 (m=23), A051876 (m=24), A255184 (m=25), A255185 (m=26), A255186 (m=27), A161935 (m=28), A255187 (m=29), A254474 (m=30).

KEYWORD

nonn,tabl,easy

AUTHOR

Omar E. Pol, Apr 27 2008

EXTENSIONS

Edited by Omar E. Pol, Jan 05 2009

STATUS

approved

A236770

a(n) = n*(n + 1)*(3*n^2 + 3*n - 2)/8.

+10
11

0, 1, 12, 51, 145, 330, 651, 1162, 1926, 3015, 4510, 6501, 9087, 12376, 16485, 21540, 27676, 35037, 43776, 54055, 66045, 79926, 95887, 114126, 134850, 158275, 184626, 214137, 247051, 283620, 324105, 368776, 417912, 471801, 530740, 595035, 665001, 740962

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

After 0, first trisection of A011779 and right border of A177708.

LINKS

Bruno Berselli, Table of n, a(n) for n = 0..1000

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

FORMULA

G.f.: x*(1 + 7*x + x^2)/(1 - x)^5.

a(n) = a(-n-1) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

a(n) = A000326(A000217(n)).

a(n) = A000217(n) + 9*A000332(n+2).

Sum_{n>=1} 1/a(n) = 2 + 4*sqrt(3/11)*Pi*tan(sqrt(11/3)*Pi/2) = 1.11700627139319... . - Vaclav Kotesovec, Apr 27 2016

MATHEMATICA

Table[n (n + 1) (3 n^2 + 3 n - 2)/8, {n, 0, 40}]

LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 12, 51, 145}, 40] (* Harvey P. Dale, Aug 22 2016 *)

PROG

(PARI) for(n=0, 40, print1(n*(n+1)*(3*n^2+3*n-2)/8", "));

(Magma) [n*(n+1)*(3*n^2+3*n-2)/8: n in [0..40]];

CROSSREFS

Partial sums of A004188.

Cf. A000217, A000332, A011779, A177708.

Cf. similar sequences on the polygonal numbers: A002817(n) = A000217(A000217(n)); A000537(n) = A000290(A000217(n)); A037270(n) = A000217(A000290(n)); A062392(n) = A000384(A000217(n)).

Cf. sequences of the form A000217(m)+k*A000332(m+2): A062392 (k=12); A264854 (k=11); A264853 (k=10); this sequence (k=9); A006324 (k=8); A006323 (k=7); A000537 (k=6); A006322 (k=5); A006325 (k=4), A002817 (k=3), A006007 (k=2), A006522 (k=1).

Cf. A232713, A260810.

KEYWORD

nonn,easy

AUTHOR

Bruno Berselli, Jan 31 2014

STATUS

approved

A189374

Expansion of 1/((1-x)^5*(x^2+x+1)^3).

+10
4

1, 2, 3, 7, 11, 15, 25, 35, 45, 65, 85, 105, 140, 175, 210, 266, 322, 378, 462, 546, 630, 750, 870, 990, 1155, 1320, 1485, 1705, 1925, 2145, 2431, 2717, 3003, 3367, 3731, 4095, 4550, 5005, 5460, 6020, 6580, 7140, 7820

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The Ca1(n) and Ze3(n) triangle sums of A139600 lead to the sequence given above, see the formulas. For the definitions of these triangle sums see A180662.

LINKS

Table of n, a(n) for n=0..42.

Index entries for linear recurrences with constant coefficients, signature (2,-1,3,-6,3,-3,6,-3,1,-2,1)

FORMULA

a(n) = (2*a(n-1) + 2*a(n-2) + (8+n)*a(n-3))/n with a(0)=1, a(1)=2, a(2)=3 and a(3)=7.

a(n) = sum(A011779(n-k)*A049347(k), k=0..n).

Ca1(n) = A189374(n-3) - A189374(n-4) - A189374(n-6) + 2*A189374(n-7).

Ze3(n) = 2*A189374(n-3) - A189374(n-4) - 2*A189374(n-6) + 5*A189374(n-7) with A189374(n)=0 for n <= -1.

a(n) = (floor(n/3)+1)*(floor(n/3)+2)*(floor(n/3)+3)*(3*floor(n/3)+4*(4-(3*floor((n+3)/3)-n)))/24. - Luce ETIENNE, Jun 29 2015

MAPLE

a:= proc(n) option remember; `if` (n<4, [1, 2, 3, 7][n+1], (2*a(n-1) +2*a(n-2) +(8+n) *a(n-3))/n) end: seq (a(n), n=0..50);

CROSSREFS

Cf. A139600, A189375, A189376.

KEYWORD

easy,nonn

AUTHOR

Johannes W. Meijer, Apr 29 2011

STATUS

approved

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