A008595 -id:A008595 - cp's OEIS frontend

A269100 a(n) = 13*n + 11.

11, 24, 37, 50, 63, 76, 89, 102, 115, 128, 141, 154, 167, 180, 193, 206, 219, 232, 245, 258, 271, 284, 297, 310, 323, 336, 349, 362, 375, 388, 401, 414, 427, 440, 453, 466, 479, 492, 505, 518, 531, 544, 557, 570, 583, 596, 609, 622, 635, 648, 661, 674, 687, 700, 713, 726, 739

Offset: 0

Bruno Berselli, Feb 19 2016

Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 (A010376) but not 11, and for this reason there are no squares in the sequence. Likewise, any cube mod 13 is one of 0, 1, 5, 8 or 12, therefore no a(k) is a cube.
Sequences of the type 13*n + k, for k = 0..12, without squares and cubes:
k =  2: A153080,
k =  6: A186113,
k =  7: A269044,
k = 11: this case.
The sum of the sixth powers of any two terms of the sequence is also a term of the sequence. Example: a(3)^6 + a(8)^6 = a(179129674278) = 2328685765625.
The primes of the sequence are listed in A140373.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A094784, A106389.
Cf. A140373.
Similar sequences of the type k*n+k-2: A023443 (k=1), A005843 (k=2), A016777 (k=3), A016825 (k=4), A016885 (k=5), A016957 (k=6), A017041 (k=7), A017137 (k=8), A017245 (k=9), A017365 (k=10), A017497 (k=11), A017641 (k=12).
Sequences of the form 13*n+q: A008595 (q=0), A190991 (q=1), A153080 (q=2), A127547 (q=4), A154609 (q=5), A186113 (q=6), A269044 (q=7), this sequence (q=11).

Magma
```
[13*n+11: n in [0..60]];
```

13 Range[0,60] + 11
Range[11, 800, 13]
Table[13 n + 11, {n, 0, 60}] (* Bruno Berselli, Feb 22 2016 *)
LinearRecurrence[{2,-1},{11,24},60] (* Harvey P. Dale, Jun 14 2023 *)

Maxima
```
makelist(13*n+11, n, 0, 60);
```
PARI
```
vector(60, n, n--; 13*n+11)
```
Python
```
[13*n+11 for n in range(61)]
```
Sage
```
[13*n+11 for n in range(61)]
```

G.f.: (11 + 2*x)/(1 - x)^2.
a(n) = -A153080(-n-1).
Sum_{i = h..h+13*k} a(i) = a(h*(13*k + 1) + k*(169*k + 35)/2).
Sum_{i >= 0} 1/a(i)^2 = .012486605016510955990... = polygamma(1, 11/13)/13^2.
E.g.f.: (11 + 13*x)*exp(x). - G. C. Greubel, May 31 2024

A186113 a(n) = 13*n + 6.

Original entry on oeis.org

6, 19, 32, 45, 58, 71, 84, 97, 110, 123, 136, 149, 162, 175, 188, 201, 214, 227, 240, 253, 266, 279, 292, 305, 318, 331, 344, 357, 370, 383, 396, 409, 422, 435, 448, 461, 474, 487, 500, 513, 526, 539, 552, 565, 578, 591, 604, 617, 630, 643, 656, 669, 682

Offset: 0

Omar E. Pol, Feb 12 2011

These numbers appear in the G. E. Andrews paper, for example: see the abstract, formula (1.7), etc. Also "13n + 6" appears in the Folsom-Ono paper (see links).
Row 6 of triangle A151890 lists the first seven terms of this sequence.
Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 (A010376) but not 6, and for this reason there are no squares in the sequence. Likewise, any cube mod 13 is one of 0, 1, 5, 8 or 12, therefore no a(k) is a cube. - Bruno Berselli, Feb 19 2016

G. C. Greubel, Table of n, a(n) for n = 0..5000
G. E. Andrews, The number of smallest parts in the partitions of n, J. Reine Angew. Math. 624 (2008) 133
Amanda Folsom and Ken Ono, The spt-function of Andrews, Proc. Natl. Acad. Sci. USA 105 (51) (2008) 20152-20156.
Ken Ono, Congruences for the Andrews spt-function, Proc. Natl. Acad. Sci. USA 108 (2011) 473-476.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A092269, A151890.
Sequences of the form 13*n+q: A008595 (q=0), A190991 (q=1), A153080 (q=2),
A127547 (q=4), A154609 (q=5), this sequence (q=6), A269044 (q=7), A269100 (q=11).

[13*n+6: n in [0..60]]; // G. C. Greubel, May 31 2024

Range[6, 1000, 13] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
LinearRecurrence[{2,-1},{6,19},60] (* Harvey P. Dale, May 12 2023 *)

[13*n+6 for n in range(61)] # G. C. Greubel, May 31 2024

G.f.: (6+7*x)/(1-x)^2.

E.g.f.: (6 + 13*x)*exp(x). - G. C. Greubel, May 31 2024

A076310 a(n) = floor(n/10) + 4*(n mod 10).

Original entry on oeis.org

0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 7, 11, 15, 19, 23

Offset: 0

Reinhard Zumkeller, Oct 06 2002

nonn

(n==0 modulo 13) iff (a(n)==0 modulo 13); applied recursively, this property provides a divisibility test for numbers given in base 10 notation.

			435598 is not a multiple of 13, as 435598 -> 43559+4*8=43591 -> 4359+4*1=4363 -> 436+4*3=448 -> 44+4*8=76 -> 7+4*6=29=13*2+3, therefore the answer is NO.
Is 8424 divisible by 13? 8424 -> 842+4*4=858 -> 85+4*8=117 -> 11+4*7=39=13*3, therefore the answer is YES.

Karl Menninger, Rechenkniffe, Vandenhoeck & Ruprecht in Goettingen (1961), 79A.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Divisibility Tests.
Wikipedia, Divisibility rule
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A008595, A076309, A076311, A076312.

a076310 n =  n' + 4 * m where (n', m) = divMod n 10
-- Reinhard Zumkeller, Jun 01 2013

[Floor(n/10)+4*(n mod 10): n in [0..75]]; // Vincenzo Librandi, Feb 27 2016

A076310:=n->floor(n/10) + 4*(n mod 10); seq(A076310(n), n=0..100); # Wesley Ivan Hurt, Jan 30 2014

Table[Floor[n/10] + 4*Mod[n, 10], {n, 0, 100}] (* Wesley Ivan Hurt, Jan 30 2014 *)
LinearRecurrence[{1,0,0,0,0,0,0,0,0,1,-1},{0,4,8,12,16,20,24,28,32,36,1},80] (* Harvey P. Dale, Sep 30 2015 *)

a(n) = n\10 + 4*(n % 10); \\ Michel Marcus, Jan 31 2014

a(n) = +a(n-1) +a(n-10) -a(n-11). G.f.: -x*(-4-4*x-4*x^2-4*x^3-4*x^4-4*x^5-4*x^6-4*x^7-4*x^8+35*x^9) / ( (1+x) *(x^4+x^3+x^2+x+1) *(x^4-x^3+x^2-x+1) *(x-1)^2 ). - R. J. Mathar, Feb 20 2011

A127547 a(n) = 13*n + 4.

Original entry on oeis.org

4, 17, 30, 43, 56, 69, 82, 95, 108, 121, 134, 147, 160, 173, 186, 199, 212, 225, 238, 251, 264, 277, 290, 303, 316, 329, 342, 355, 368, 381, 394, 407, 420, 433, 446, 459, 472, 485, 498, 511, 524, 537, 550, 563, 576, 589, 602, 615, 628, 641, 654, 667, 680, 693, 706, 719

Offset: 0

Robert H Barbour, Apr 01 2007

Superhighway created by 'LQTL Ant' L90R90L45R45 from iteration 4 where the Ant moves in a 'Moore neighborhood' (nine cells), the L indicates a left turn, the R a right turn, and the numerical value is the size of the turn (in degrees) at each iteration.

Ant Farm algorithm available from Robert H Barbour.

P. Sakar, "A Brief History of Cellular Automata," ACM Computing Surveys, vol. 32, pp. 80-107, 2000.

G. C. Greubel, Table of n, a(n) for n = 0..5000
C. Langton, Studying Artificial Life with Cellular Automata, Physica D: Nonlinear Phenomena, vol. 22, pp. 120-149, 1986.
James Propp, Further Ant-ics, Mathematical Intelligencer, 16 pp. 37-42, 1994.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

A subsequence of A092464.
Cf. A126979, A126980.
Sequences of the form 13*n+q: A008595 (q=0), A190991 (q=1), A153080 (q=2), this sequence (q=4), A154609 (q=5), A186113 (q=6), A269044 (q=7), A269100 (q=11).

[13*n+4: n in [0..60]]; // G. C. Greubel, May 31 2024

Range[4,1000,13] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)

[13*n+4 for n in range(61)] # G. C. Greubel, May 31 2024

From Elmo R. Oliveira, Mar 21 2024: (Start)
G.f.: (4+9*x)/(1-x)^2.
E.g.f.: (4 + 13*x)*exp(x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

Edited by N. J. A. Sloane, May 10 2007

A195032 Vertex number of a square spiral in which the length of the first two edges are the legs of the primitive Pythagorean triple [5, 12, 13]. The edges of the spiral have length A195031.

Original entry on oeis.org

0, 5, 17, 27, 51, 66, 102, 122, 170, 195, 255, 285, 357, 392, 476, 516, 612, 657, 765, 815, 935, 990, 1122, 1182, 1326, 1391, 1547, 1617, 1785, 1860, 2040, 2120, 2312, 2397, 2601, 2691, 2907, 3002, 3230, 3330, 3570, 3675, 3927, 4037, 4301, 4416, 4692

Offset: 0

Omar E. Pol, Sep 12 2011

Zero together with partial sums of A195031.

The spiral contains infinitely many Pythagorean triples in which the hypotenuses on the main diagonal are the positives multiples of 13 (cf. A008595). The vertices on the main diagonal are the numbers A195037 = (5+12)*A000217 = 17*A000217, where both 5 and 12 are the first two edges in the spiral. The distance "a" between nearest edges that are perpendicular to the initial edge of the spiral is 5, while the distance "b" between nearest edges that are parallel to the initial edge is 12, so the distance "c" between nearest vertices on the same axis is 13 because from the Pythagorean theorem we can write c = (a^2 + b^2)^(1/2) = sqrt(5^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13. - Omar E. Pol, Oct 12 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Ron Knott, Pythagorean triangles and Triples
Eric Weisstein's World of Mathematics, Pythagorean Triple
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A195020, A195031, A195034, A195036, A195037.

[(2*n*(17*n+27)+(14*n-3)*(-1)^n+3)/16: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

a[n_] := (2 n (17 n + 27) + (14 n - 3)*(-1)^n + 3)/16; Array[a, 50, 0] (* Amiram Eldar, Nov 23 2018 *)

vector(50, n, n--; (2*n*(17*n+27)+(14*n-3)*(-1)^n+3)/16) \\ G. C. Greubel, Nov 23 2018

[(2*n*(17*n+27)+(14*n-3)*(-1)^n+3)/16 for n in range(50)] # G. C. Greubel, Nov 23 2018

From Bruno Berselli, Oct 13 2011: (Start)
G.f.: x*(5 + 12*x)/((1 + x)^2*(1 - x)^3).
a(n) = (1/2)*((2*n + (-1)^n + 3)/4)*((34*n - 3*(-1)^n+3)/4) = (2*n*(17*n + 27) + (14*n - 3)*(-1)^n + 3)/16.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). (End)
E.g.f.: (1/16)*((3 + 88*x + 34*x^2)*exp(x) - (3 + 14*x)*exp(-x)). - Franck Maminirina Ramaharo, Nov 23 2018

A244633 a(n) = 26*n^2.

Original entry on oeis.org

0, 26, 104, 234, 416, 650, 936, 1274, 1664, 2106, 2600, 3146, 3744, 4394, 5096, 5850, 6656, 7514, 8424, 9386, 10400, 11466, 12584, 13754, 14976, 16250, 17576, 18954, 20384, 21866, 23400, 24986, 26624, 28314, 30056, 31850, 33696, 35594, 37544, 39546, 41600, 43706

Offset: 0

Vincenzo Librandi, Jul 03 2014

Sequence found by reading the line from 0, in the direction 0, 26, ..., in the square spiral whose vertices are the generalized 15-gonal numbers. - Omar E. Pol, Jul 03 2014

Norms of purely imaginary numbers in Z[sqrt(-26)]. - Alonso del Arte, Dec 25 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. similar sequences listed in A244630.

Cf. A000290, A001105, A005843, A008595, A152742, A252994, A277082.

Magma
```
[26*n^2: n in [0..40]];
```

A244633:=n->26*n^2: seq(A244633(n), n=0..50); # Wesley Ivan Hurt, Jul 04 2014

Table[26 n^2, {n, 0, 40}]
26 Range[0, 50]^2 (* Wesley Ivan Hurt, Jul 04 2014 *)

a(n)=26*n^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 26*x*(1 + x)/(1 - x)^3. [corrected by Bruno Berselli, Jul 03 2014]
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = 26*A000290(n) = 13*A001105(n) = 2*A152742(n). - Omar E. Pol, Jul 03 2014
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 26*x*(1 + x)*exp(x).
a(n) = n*A252994(n) = A005843(n)*A008595(n). (End)

A317297 a(n) = (n - 1)(4n^2 - 8*n + 5).

Original entry on oeis.org

0, 5, 34, 111, 260, 505, 870, 1379, 2056, 2925, 4010, 5335, 6924, 8801, 10990, 13515, 16400, 19669, 23346, 27455, 32020, 37065, 42614, 48691, 55320, 62525, 70330, 78759, 87836, 97585, 108030, 119195, 131104, 143781, 157250, 171535, 186660, 202649, 219526, 237315, 256040, 275725, 296394, 318071

Offset: 1

Omar E. Pol, Sep 01 2018

Conjecture: For n > 1, a(n) is the maximum eigenvalue of a 2*(n-1) X 2*(n-1) square matrix M defined as M[i,j,n] = j + n*(i-1) if i is odd and M[i,j,n] = n*i - j + 1 if i is even (see A317614). - Stefano Spezia, Dec 27 2018

Connections can be made to A022144 and A010014. Namely, a formula for A022144 is (2*n+1)^2 - (2*n-1)^2. A formula for A010014 is (2*n+1)^3 - (2*n-1)^3. The general form can be represented by (2*n+1)^d - (2*n-1)^d, where d designates the number of dimensions. When d is 4, a(n) = ((2*(n-1)+1)^4 - (2*(n-1)-1)^4)/16, namely the general form shifted by 1 and divided by 16 is a(n). - Yigit Oktar, Aug 16 2024

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

First bisection of A006003.
Nonzero terms give the row sums of A007607.
Conjecture: 0 together with a bisection of A246697.
Cf. A219086 (partial sums).
Cf. A008595, A033430, A135453, A317614.
Cf. A010014, A022144 (see comments)

Table[(n - 1) (4 n^2 - 8 n + 5), {n, 1, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 5, 34, 111}, 50] (* or *) CoefficientList[Series[x (5 + 14 x + 5 x^2)/(1 - x)^4, {x, 0, 50}], x] (* Stefano Spezia, Sep 01 2018 *)

PARI
```
a(n) = (n - 1)*(4*n^2 - 8*n + 5)
```

concat(0, Vec(x^2*(5 + 14*x + 5*x^2)/(1 - x)^4 + O(x^50))) \\ Colin Barker, Sep 01 2018

a(n) = 4*n^3 - 12*n^2 + 13*n - 5 = A033430(n) - A135453(n) + A008595(n) - 5.
G.f.: x^2*(5 + 14*x + 5*x^2)/(1 - x)^4. - Colin Barker, Sep 01 2018
a(n) = 4*a(n - 1) - 6*a(n - 2) + 4*a(n - 3) - a(n - 4) for n > 4. - Stefano Spezia, Sep 01 2018
E.g.f.: exp(x)*(5*x + 12*x^2 + 4*x^3). - Stefano Spezia, Jan 15 2019
a(n) = ((2*(n-1)+1)^4 - (2*(n-1)-1)^4)/16. - Yigit Oktar, Aug 16 2024

A113763 Non-multiples of 13, or numbers not divisible by 13.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Zak Seidov, Jan 18 2006

Cf. A011760 (Elevator buttons in U.S.A).
Cf. A023806 (all digits in base 12 are different).
Complement of A008595. - Reinhard Zumkeller, Apr 26 2011

Eric Weisstein's World of Mathematics, Triskaidekaphobia
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A011760, A023806.

Table[n+Quotient[2n-1, 24], {n, 1, 85}];(* or *)Table[n+Floor[(2n-1)/24], {n, 1, 85}]

a(n)=(26*n-1)\24 \\ Charles R Greathouse IV, Jun 08 2015

[i for i in range(80) if not i%13] # Zerinvary Lajos, Apr 21 2009

a(n) = n+floor((2n-1)/24) = n+quotient(2n-1, 24)

A252994 Multiples of 26.

Original entry on oeis.org

0, 26, 52, 78, 104, 130, 156, 182, 208, 234, 260, 286, 312, 338, 364, 390, 416, 442, 468, 494, 520, 546, 572, 598, 624, 650, 676, 702, 728, 754, 780, 806, 832, 858, 884, 910, 936, 962, 988, 1014, 1040, 1066, 1092, 1118, 1144, 1170, 1196, 1222, 1248, 1274, 1300, 1326, 1352, 1378

Offset: 0

Alonso del Arte, Dec 25 2014

Richard A. Mollin, Codes: The Guide to Secrecy From Ancient to Modern Times. Chapman and Hall/CRC (2005): 12.
Anne L. Young, Mathematical Ciphers: From Caesar to RSA. American Mathematical Society (2006): 23.

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

Cf. A005843, A008595, A008607, A244633.

Mathematica
```
26 Range[0, 51]
```
PARI
```
a(n)=26*n \\ Georg Fischer, Feb 25 2019
```

a(n) = 26*n.
a(n) = 2*a(n-1) - a(n-2).
G.f.: 26*x/(1-x)^2. - Georg Fischer, Feb 25 2019
From Elmo R. Oliveira, Apr 08 2025: (Start)
E.g.f.: 26*x*exp(x).
a(n) = A005843(A008595(n)). (End)

A274824 Triangle read by rows: T(n,k) = (n-k+1)*sigma(k), n>=1, 1<=k<=n.

Original entry on oeis.org

1, 2, 3, 3, 6, 4, 4, 9, 8, 7, 5, 12, 12, 14, 6, 6, 15, 16, 21, 12, 12, 7, 18, 20, 28, 18, 24, 8, 8, 21, 24, 35, 24, 36, 16, 15, 9, 24, 28, 42, 30, 48, 24, 30, 13, 10, 27, 32, 49, 36, 60, 32, 45, 26, 18, 11, 30, 36, 56, 42, 72, 40, 60, 39, 36, 12, 12, 33, 40, 63, 48, 84, 48, 75, 52, 54, 24, 28, 13, 36, 44, 70, 54, 96, 56, 90, 65, 72, 36, 56, 14

Offset: 1

Omar E. Pol, Oct 02 2016

Theorem: for any sequence S the partial sums of the partial sums are also the antidiagonal sums of the square array in which the n-th row gives n times the sequence S. Therefore they are also the row sums of the triangular array in which the n-th diagonal gives n times the sequence S.
In this case the sequence S is A000203.
The n-th diagonal of this triangle gives n times A000203.
The row sums give A175254 which gives the partial sums of A024916 which gives the partial sums of A000203.
T(n,k) is also the total number of unit cubes that are exactly below the terraces of the k-th level (starting from the top) up the base of the stepped pyramid with n levels described in A245092. This fact is because the mentioned terraces have the same shape as the symmetric representation of sigma(k). For more information see A237593 and A237270.
In the definition of this sequence the value n-k+1 is also the height of the terraces associated to sigma(k) in the mentioned pyramid with n levels, or in other words, the distance between the mentioned terraces and the base of the pyramid.
The sum of the n-th row of triangle equals the volume (also the number of cubes) of the mentioned pyramid with n levels.
For an illustration of the pyramid, see the Links section.
The sum of the n-th row is also 1/4 of the volume of the stepped pyramid described in A244050 with n levels.
Column k lists the positive multiples of sigma(k).
The k-th term in the n-th diagonal is equal to n*sigma(k).
Note that this is also a square array read by antidiagonals upwards: T(i,j) = i*sigma(j), i>=1, j>=1. The first row of the array is A000203. So consider that the pyramid is upside down. The value of "i" is the distance between the base of the pyramid and the terraces associated to sigma(j). The antidiagonal sums give the partial sums of the partial sums of A000203.

			Triangle begins:
1;
2,  3;
3,  6,  4;
4,  9,  8,  7;
5,  12, 12, 14, 6;
6,  15, 16, 21, 12, 12;
7,  18, 20, 28, 18, 24,  8;
8,  21, 24, 35, 24, 36,  16, 15;
9,  24, 28, 42, 30, 48,  24, 30,  13;
10, 27, 32, 49, 36, 60,  32, 45,  26,  18;
11, 30, 36, 56, 42, 72,  40, 60,  39,  36,  12;
12, 33, 40, 63, 48, 84,  48, 75,  52,  54,  24, 28;
13, 36, 44, 70, 54, 96,  56, 90,  65,  72,  36, 56,  14;
14, 39, 48, 77, 60, 108, 64, 105, 78,  90,  48, 84,  28, 24;
15, 42, 52, 84, 66, 120, 72, 120, 91,  108, 60, 112, 42, 48, 24;
16, 45, 56, 91, 72, 132, 80, 135, 104, 126, 72, 140, 56, 72, 48, 31;
...
For n = 16 and k = 10 the sum of the divisors of 10 is 1 + 2 + 5 + 10 = 18, and 16 - 10 + 1 = 7, and 7*18 = 126, so T(16,10) = 126.
On the other hand, the symmetric representation of sigma(10) has two parts of 9 cells, giving a total of 18 cells. In the stepped pyramid described in A245092, with 16 levels, there are 16 - 10 + 1 = 7 cubes exactly below every cell of the symmetric representation of sigma(10) up the base of pyramid; hence the total numbers of cubes exactly below the terraces of the 10th level (starting from the top) up the base of the pyramid is equal to 7*18 = 126. So T(16,10) = 126.
The sum of the 16th row of the triangle is 16 + 45 + 56 + 91 + 72 + 132 + 80 + 135 + 104 + 126 + 72 + 140 + 56 + 72 + 48 + 31 = A175254(16) = 1276, equaling the volume (also the number of cubes) of the stepped pyramid with 16 levels described in A245092 (see Links section).

Indranil Ghosh, Rows 1..100 of triangle, flattened
Omar E. Pol, Illustration of the stepped pyramid with 16 levels

Row sums of triangle give A175254.
Diagonals 1-6: A000203, A074400, A272027, A239050, A274535, A274536.
Column 1 is A000027.
Initial zeros should be omitted in the following sequences related to the columns of triangle:
Columns 2-5: A008585, A008586, A008589, A008588.
Columns 6 and 11: A008594.
Columns 7-9: A008590, A008597, A008595.
Columns 10 and 17: A008600.
Columns 12-13: A135628, A008596.
Columns 14, 15 and 23: A008606.
Columns 16 and 25: A135631.
(There are many other OEIS sequences that are also columns of this triangle.)
Cf. A004736, A024916, A054973, A196020, A236104, A235791, A237591, A237593, A237270, A237271, A244050, A245092, A245093, A262612.

T(n,k) = (n-k+1) * A000203(k).

T(n,k) = A004736(n,k) * A245093(n,k).

cp's OEIS Frontend

A269100 a(n) = 13*n + 11.

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A186113 a(n) = 13*n + 6.

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A076310 a(n) = floor(n/10) + 4*(n mod 10).

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A127547 a(n) = 13*n + 4.

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A195032 Vertex number of a square spiral in which the length of the first two edges are the legs of the primitive Pythagorean triple [5, 12, 13]. The edges of the spiral have length A195031.

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A244633 a(n) = 26*n^2.

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A317297 a(n) = (n - 1)(4n^2 - 8*n + 5).