A162607 -id:A162607 - cp's OEIS frontend

A159798 Triangle read by rows in which row n lists n terms, starting with 1, such that the difference between successive terms is equal to n-3.

1, 1, 0, 1, 1, 1, 1, 2, 3, 4, 1, 3, 5, 7, 9, 1, 4, 7, 10, 13, 16, 1, 5, 9, 13, 17, 21, 25, 1, 6, 11, 16, 21, 26, 31, 36, 1, 7, 13, 19, 25, 31, 37, 43, 49, 1, 8, 15, 22, 29, 36, 43, 50, 57, 64, 1, 9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 1, 11

Offset: 1

Omar E. Pol, Jul 09 2009

Note that for n>1 the last term of the n-th row is the square A000290(n-2).
Row sums are n*(n^2-4*n+5)/2 = 1, 1, 3, 10, 25, 51, 91, 148, 225, ... - R. J. Mathar, Jul 17 2009, Jul 20 2009
Row sums are the positive terms of A162607. - Omar E. Pol, Jul 24 2009

			Triangle begins:
  1;
  1,  0;
  1,  1,  1;
  1,  2,  3,  4;
  1,  3,  5,  7,  9;
  1,  4,  7, 10, 13, 16;
  1,  5,  9, 13, 17, 21, 25;
  1,  6, 11, 16, 21, 26, 31, 36;
  1,  7, 13, 19, 25, 31, 37, 43, 49;
  1,  8, 15, 22, 29, 36, 43, 50, 57, 64;
  1,  9, 17, 25, 33, 41, 49, 57, 65, 73, 81;
  1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100;

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A000290, A081493, A159797, A162607, A162609, A162610, A162611.

/* As triangle */ [[1 + k*(n-3): k in [0..n-1]]: n in [1..15]]; // G. C. Greubel, Apr 21 2018

Table[1 + k*(n-3), {n, 1, 20}, {k, 0, n-1}]// Flatten (* G. C. Greubel, Apr 21 2018 *)

for(n=1, 20, for(k=0,n-1, print1(1 + k*(n-3), ", "))) \\ G. C. Greubel, Apr 21 2018

T(n,k) = 1 + k*(n-3), 0<=kR. J. Mathar, Jul 17 2009

More terms from R. J. Mathar, Jul 17 2009
Typo in row sums corrected by R. J. Mathar, Jul 20 2009
Edited by Omar E. Pol, Jul 24 2009

A188947 a(n) = n^3 - 2n^2 + 2n + 1.

Original entry on oeis.org

2, 5, 16, 41, 86, 157, 260, 401, 586, 821, 1112, 1465, 1886, 2381, 2956, 3617, 4370, 5221, 6176, 7241, 8422, 9725, 11156, 12721, 14426, 16277, 18280, 20441, 22766, 25261, 27932, 30785, 33826, 37061, 40496, 44137, 47990, 52061, 56356, 60881, 65642, 70645

Offset: 1

Adeniji, Adenike, Apr 14 2011

The original definition was "Identity difference partial one - one transformation semigroup is a semigroup having the property that the difference between max im(alpha) and min im(alpha) is not greater than 1. This is denoted by S = IDI_n for each n." [Needs editing.]

For all n >= 3, a(n) expressed in base n has the three digits n-2, 2, and 1; for example, a(16) in hexadecimal is "E21". For all n >= 3, a(n+1) expressed in base n is "1112". For all n >= 7, a(n+2) expressed in base n is "1465". - Mathew Englander, Jan 07 2021

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. A027444, A053698, A056106 (first differences), A060354, A162607, A188377, A188716.

[n^3 - 2*n^2 + 2*n + 1: n in [1..30]]; // Vincenzo Librandi, May 01 2011

A188947[n_] := n^3 - 2*n^2 + 2*n + 1; Table[A188947[n], {n, 1, 42}] (* Robert P. P. McKone, Jan 31 2021 *)

a(n)=n^3-2*n^2+2*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (n+1) + n*(n-1)^2 = n^3 - 2*n^2 + 2*n + 1 = 1 + A053698(n-1).
G.f.: ( -x*(-2 + 3*x - 8*x^2 + x^3) ) / ( (x-1)^4 ). - R. J. Mathar, Apr 14 2011
a(n) = A060354(n) + A162607(n+1). - Lechoslaw Ratajczak, Sep 24 2020
E.g.f.: exp(x)*(1 + x)*(1 + x^2) - 1. - Stefano Spezia, Apr 10 2022

Edited by N. J. A. Sloane, Apr 23 2011

A164845 a(n) = (6 + 10n + 5n^2 + n^3)/2.

Original entry on oeis.org

3, 11, 27, 54, 95, 153, 231, 332, 459, 615, 803, 1026, 1287, 1589, 1935, 2328, 2771, 3267, 3819, 4430, 5103, 5841, 6647, 7524, 8475, 9503, 10611, 11802, 13079, 14445, 15903, 17456, 19107, 20859, 22715, 24678, 26751, 28937, 31239, 33660, 36203, 38871

Offset: 0

Paul Curtz, Aug 28 2009

Row sums of the triangle defined by non-interrupted runs in A080036.
If the sequence of integers is split at positions defined by A000124 we obtain A080036. Its runs of consecutive integers can be placed into rows of a triangle:
    3;
    5,  6;
    8,  9, 10;
   12, 13, 14, 15;
   17, 18, 19, 20, 21;
   ...
The a(n) are the row sums of this triangle.
The a(n) are also the binomial transform of the quasi-finite sequence 3, 8, 8, 3, 0 (0 continued).
An associated integer sequence could be defined by a(n)/A026741(n+1) = 3, 11, 9, 27, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A135278.

[3+5*n+5*n^2/2+n^3/2: n in [0..50]]; // Vincenzo Librandi, Aug 07 2011

Table[(6 + 10*n + 5*n^2 + n^3)/2, {n,0,50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {3, 11, 27, 54}, 50] (* G. C. Greubel, Apr 21 2018 *)

for(n=0, 50, print1((6+10*n+5*n^2+n^3)/2, ", ")) \\ G. C. Greubel, Apr 21 2018

a(n) = A162607(n+3) + n.
First differences: a(n+1) - a(n) = A104249(n+2), i.e., a(n) = a(n-1) + 3*n^2/2 + 7*n/2 +3.
Second differences: a(n+2) - 2*a(n+1) + a(n) = A016789(n+2).
a(n) = 2*a(n-1) - a(n-2) + 3*n + 5, n>1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 3, n>2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>3.
G.f.: (3-x+x^2)/(x-1)^4.
E.g.f.: (6 + 16*x + 8*x^2 + x^3)*exp(x)/2. - G. C. Greubel, Apr 21 2018

Edited and extended by R. J. Mathar, Aug 31 2009

Corrected typo in recurrence, observed by Paul Curtz - R. J. Mathar, Sep 25 2009

A072277 Smallest integer > 1 which is both n-gonal and centered n-gonal.

Original entry on oeis.org

10, 25, 51, 91, 148, 225, 325, 451, 606, 793, 1015, 1275, 1576, 1921, 2313, 2755, 3250, 3801, 4411, 5083, 5820, 6625, 7501, 8451, 9478, 10585, 11775, 13051, 14416, 15873, 17425, 19075, 20826, 22681, 24643, 26715, 28900, 31201, 33621, 36163

Offset: 3

David W. Wilson, Jul 09 2002

a(n) is the (n-1)-th centered n-gonal number. The n-th centered n-gonal number is A100119(n) and the (n+1)-th centered n-gonal number is A158842(n). - Mohammed Yaseen, Jun 06 2021

			a(4) = 25 is both square and centered square.

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

Cf. A100119, A158842, A162607.

LinearRecurrence[{4,-6,4,-1},{10,25,51,91},50] (* or *) Table[(n^3-n^2+ 2)/2,{n,3,50}] (* Harvey P. Dale, Aug 19 2011 *)

a(n) = (n^3 - n^2 + 2)/2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), with a(3)=10, a(4)=25, a(5)=51, a(6)=91. - Harvey P. Dale, Aug 19 2011
G.f.: x^3*(-3*x^3 + 11*x^2 - 15*x + 10)/(x-1)^4. - Harvey P. Dale, Aug 19 2011

A330892 Square array of polygonal numbers read by descending antidiagonals (the transpose of A317302).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, -3, 1, 1, 0, -8, 0, 2, 1, 0, -15, -2, 3, 3, 1, 0, -24, -5, 4, 6, 4, 1, 0, -35, -9, 5, 10, 9, 5, 1, 0, -48, -14, 6, 15, 16, 12, 6, 1, 0, -63, -20, 7, 21, 25, 22, 15, 7, 1, 0, -80, -27, 8, 28, 36, 35, 28, 18, 8, 1, 0, -99, -35, 9, 36, 49, 51, 45, 34, 21, 9, 1, 0

Offset: 1

Robert G. Wilson v, Apr 27 2020

\c  0 1  2  3  4   5   6   7   8   9  10  11   12   13    14  15  ...
r\
_0  0 1  0 -3 -8 -15 -24 -35 -48 -63 -80 -99 -120 -143 -168 -195  A067998
_1  0 1  1  0 -2  -5  -9 -14 -20 -27 -35 -44  -54  -65  -77  -90  A080956
_2  0 1  2  3  4   5   6   7   8   9  10  11   12   13   14   15  A001477
_3  0 1  3  6 10  15  21  28  36  45  55  66   78   91  105  120  A000217
_4  0 1  4  9 16  25  36  49  64  81 100 121  144  169  196  225  A000290
_5  0 1  5 12 22  35  51  70  92 117 145 176  210  247  287  330  A000326
_6  0 1  6 15 28  45  66  91 120 153 190 231  276  325  378  435  A000384
_7  0 1  7 18 34  55  81 112 148 189 235 286  342  403  469  540  A000566
_8  0 1  8 21 40  65  96 133 176 225 280 341  408  481  560  645  A000567
_9  0 1  9 24 46  75 111 154 204 261 325 396  474  559  651  750  A001106
10  0 1 10 27 52  85 126 175 232 297 370 451  540  637  742  855  A001107
11  0 1 11 30 58  95 141 196 260 333 415 506  606  715  833  960  A051682
12  0 1 12 33 64 105 156 217 288 369 460 561  672  793  924 1065  A051624
13  0 1 13 36 70 115 171 238 316 405 505 616  738  871 1015 1170  A051865
14  0 1 14 39 76 125 186 259 344 441 550 671  804  949 1106 1275  A051866
15  0 1 15 42 82 135 201 280 372 477 595 726  870 1027 1197 1380  A051867
...
Each row has a second forward difference of (r-2) and each column has a forward difference of c(c-1)/2.

E. Deza and M. Deza, Figurate Numbers, World Scientific, 2012; see p. 45.
Eric Weisstein's World of Mathematics, Polygonal Number.
Wikipedia, Polygonal number.
Index to sequences related to polygonal numbers

Cf. A317302 (the same array) but read by ascending antidiagonals.
Sub-arrays: A089000, A139600, A206735;
Rows: A067998, A080956, A001477, A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A051682, A051624, A051865, A051866, A051867, A051868, A051869, A051870, A051871, A051872, A051873, A051874, A051875, A051876, A255184, A255185, A255186, A161935, A255187, A254474, ..., ;
Columns (maybe missing some leading terms): A000004, A000012, A001477, A008585, A016957, A017329, A139606, A139607, A139608, A139609, A139610, A139611, A139612, A139613, A139614, A139615, A139616, A139617, A139618, A139619, A139620;
Diagonals: A256857, A127736, A002411, A006003, A006000, A064808, A060354, A162607, A077414,
Number of times k>1 appears: A129654, First occurrence of k: A063778.

Table[ PolygonalNumber[r - c, c], {r, 0, 11}, {c, r, 0, -1}] // Flatten

P(r, c) = (r - 2)(c(c-1)/2) + c.

cp's OEIS Frontend

A159798 Triangle read by rows in which row n lists n terms, starting with 1, such that the difference between successive terms is equal to n-3.

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A188947 a(n) = n^3 - 2*n^2 + 2*n + 1.

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A164845 a(n) = (6 + 10*n + 5*n^2 + n^3)/2.

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A072277 Smallest integer > 1 which is both n-gonal and centered n-gonal.

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Mathematica

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A330892 Square array of polygonal numbers read by descending antidiagonals (the transpose of A317302).

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Author

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Programs

Mathematica

Formula

A188947 a(n) = n^3 - 2n^2 + 2n + 1.

A164845 a(n) = (6 + 10n + 5n^2 + n^3)/2.