A016993 -id:A016993 - cp's OEIS frontend

A081581 Pascal-(1,6,1) array.

1, 1, 1, 1, 8, 1, 1, 15, 15, 1, 1, 22, 78, 22, 1, 1, 29, 190, 190, 29, 1, 1, 36, 351, 848, 351, 36, 1, 1, 43, 561, 2339, 2339, 561, 43, 1, 1, 50, 820, 5006, 9766, 5006, 820, 50, 1, 1, 57, 1128, 9192, 28806, 28806, 9192, 1128, 57, 1, 1, 64, 1485, 15240, 68034, 116208, 68034, 15240, 1485, 64, 1

Offset: 0

Paul Barry, Mar 23 2003

One of a family of Pascal-like arrays. A007318 is equivalent to the (1,0,1)-array. A008288 is equivalent to the (1,1,1)-array. Rows include A016993, A081591, A081592. Coefficients of the row polynomials in the Newton basis are given by A013614.

			Rows start as:
  1,  1,   1,    1,    1, ... A000012;
  1,  8,  15,   22,   29, ... A016993;
  1, 15,  78,  190,  351, ... A081591;
  1, 22, 190,  848, 2339, ...
  1, 29, 351, 2339, 9766, ...
The triangle starts as:
  1;
  1,  1;
  1,  8,   1;
  1, 15,  15,    1;
  1, 22,  78,   22,    1;
  1, 29, 190,  190,   29,   1;
  1, 36, 351,  848,  351,  36,  1;
  1, 43, 561, 2339, 2339, 561, 43, 1;

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081580 (m = 5), A081582 (m = 7), A143683 (m = 8).

Cf. A016993, A081591, A083099.

A081581:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A081581(n,k,6): k in [0..n], n in [0..12]]; // G. C. Greubel, May 26 2021

Table[Hypergeometric2F1[-k, k-n, 1, 7], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

t(n, k) = sum(j=0, n-k, binomial(n-k, j)*binomial(k, j)*7^j) \\ Michel Marcus, May 24 2013

flatten([[hypergeometric([-k, k-n], [1], 7).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2021

Square array T(n, k) defined by T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 6*T(n-1, k-1) + T(n-1, k).
Rows are the expansions of (1+6*x)^k/(1-x)^(k+1).
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 7). - Jean-François Alcover, May 24 2013
E.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(7*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 14*x + 49*x^2/2) = 1 + 15*x + 78*x^2/2! + 190*x^3/3! + 351*x^4/4! + 561*x^5/5! + .... - Peter Bala, Mar 05 2017
From G. C. Greubel, May 26 2021: (Start)
T(n, k, m) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*m^j, for m = 6.
Sum_{k=0..n} T(n, k, 6) = A083099(n+1). (End)

A171890 Octonomial coefficient array.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 42, 46, 48, 48, 46, 42, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 161, 204, 246, 284, 315, 336, 344, 336, 315, 284, 246, 204, 161, 120, 84, 56, 35

Offset: 0

N. J. A. Sloane, Oct 19 2010

Row lengths are 1,8,15,22,... = 1+7n = A016993(n). Row sums are 1,8,64,... = 8^n = A001018(n). M. F. Hasler, Jun 17 2012

			Array begins:
[1]
[1, 1, 1, 1, 1, 1, 1, 1]
[1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1]
...

T. D. Noe, Rows n = 0..25, flattened

The q-nomial arrays are for q=2..10: A007318 (Pascal), A027907, A008287,A035343, A063260, A063265, A171890, A213652, A213651.

#Define the r-nomial coefficients for r = 1, 2, 3, ...
rnomial := (r,n,k) -> add((-1)^i*binomial(n,i)*binomial(n+k-1-r*i,n-1), i = 0..floor(k/r)):
#Display the 8-nomials as a table
r := 8:  rows := 10:
for n from 0 to rows do
seq(rnomial(r,n,k), k = 0..(r-1)*n)
end do;
# Peter Bala, Sep 07 2013

Flatten[Table[CoefficientList[(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7)^n, x], {n, 0, 10}]] (* T. D. Noe, Apr 04 2011 *)

concat(vector(5, k, Vec(sum(j=0, 7, x^j)^k)))  \\ M. F. Hasler, Jun 17 2012

Row n has g.f. (1+x+...+x^7)^n.

T(n,k) = sum {i = 0..floor(k/8)} (-1)^i*binomial(n,i)*binomial(n+k-1-8*i,n-1) for n >= 0 and 0 <= k <= 7*n. - Peter Bala, Sep 07 2013

A227095 Numbers whose base-8 sum of digits is 8.

Original entry on oeis.org

15, 22, 29, 36, 43, 50, 57, 71, 78, 85, 92, 99, 106, 113, 120, 134, 141, 148, 155, 162, 169, 176, 197, 204, 211, 218, 225, 232, 260, 267, 274, 281, 288, 323, 330, 337, 344, 386, 393, 400, 449, 456, 519, 526, 533, 540, 547, 554, 561, 568, 582, 589, 596, 603

Offset: 1

Tom Edgar, Sep 01 2013

Subsequence of A016993. - Michel Marcus, Sep 03 2013

In general, the set of numbers with sum of base-b digits equal to b is a subset of { (b-1)*k + 1; k = 2, 3, 4, ... }. - M. F. Hasler, Dec 23 2016

			The 8-ary expansion of 15 is (1,7), which has sum of digits 8.
The 8-ary expansion of 78 is (1,1,6), which has sum of digits 8.
10 is not on the list since the 8-ary expansion of 10 is (1,2), which has sum of digits 3 not 8.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A018900, A187813.

Cf. A226636 (b = 3), A226969 (b = 4), A227062 (b = 5), A227080 (b = 6), A227092 (b = 7), A227095 (b = 8), A227238 (b = 9), A052224 (b = 10).

Select[Range@ 603, Total@ IntegerDigits[#, 8] == 8 &] (* Michael De Vlieger, Dec 23 2016 *)

select( is(n)=sumdigits(n,8)==8, [1..999]) \\ M. F. Hasler, Dec 23 2016

agen = A226636gen(sod=8, base=8) # generator of terms using code in A226636
print([next(agen) for n in range(1, 55)]) # Michael S. Branicky, Jul 10 2022

[i for i in [0..1000] if sum(Integer(i).digits(base=8))==8]

A245301 a(n) = n(7n^2 + 15*n + 8)/6.

Original entry on oeis.org

0, 5, 22, 58, 120, 215, 350, 532, 768, 1065, 1430, 1870, 2392, 3003, 3710, 4520, 5440, 6477, 7638, 8930, 10360, 11935, 13662, 15548, 17600, 19825, 22230, 24822, 27608, 30595, 33790, 37200, 40832, 44693, 48790, 53130, 57720, 62567, 67678, 73060, 78720, 84665

Offset: 0

Reinhard Zumkeller, Jul 17 2014

Row sums of the triangle in A245300.

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

Cf. A002378, A016993, A245300, A254407.

a245301 n = n * (n * (7 * n + 15) + 8) `div` 6

[n*(7*n^2+15*n+8)/6: n in [0..60]]; // Vincenzo Librandi, Feb 01 2016

A245301:= n-> n*(n+1)*(7*n+8)/6; seq(A245301(n), n=0..50); # G. C. Greubel, Mar 31 2021

Table[n (7 n^2 + 15 n + 8)/6, {n, 0, 50}] (* Vincenzo Librandi, Feb 01 2016 *)
LinearRecurrence[{4,-6,4,-1},{0,5,22,58},50] (* Harvey P. Dale, Sep 21 2019 *)

a(n)=n*(7*n^2+15*n+8)/6 \\ Charles R Greathouse IV, Feb 01 2016

[n*(n+1)*(7*n+8)/6 for n in (0..50)] # G. C. Greubel, Mar 31 2021

a(n) = n*(n+1)*(7*n+8)/6 = A002378(n)*A016993(n+1)/6.
a(n) = Sum_{j=0..n} A000217(2n-j)+j. - Manfred Arens, Dec 26 2015
G.f.: x*(5 + 2*x)/(1-x)^4. - Vincenzo Librandi, Feb 01 2016
E.g.f.: x*(30 + 36*x + 7*x^2)*exp(x)/6. - G. C. Greubel, Mar 31 2021

A081591 Third row of Pascal-(1,6,1) array A081581.

Original entry on oeis.org

1, 15, 78, 190, 351, 561, 820, 1128, 1485, 1891, 2346, 2850, 3403, 4005, 4656, 5356, 6105, 6903, 7750, 8646, 9591, 10585, 11628, 12720, 13861, 15051, 16290, 17578, 18915, 20301, 21736, 23220, 24753, 26335, 27966, 29646, 31375, 33153, 34980, 36856, 38781, 40755

Offset: 0

Paul Barry, Mar 23 2003

1. Smallest triangular number T(k) (other than the trivial adjacent ones) such that T(n) + T(k) is a square. T(n-1) and T(n+1) are trivial triangular numbers such that T(n) + T(n-1) and T(n) + T(n+1) both are squares. 0+1 = 1, 1+15 = 16, 3+78 = 81, 6+190 = 196 etc. 2. (7n+5)-th triangular number. - Amarnath Murthy, Jun 20 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016993, A081581, A081592.

[(2-21*n+49*n^2)/2: n in [0..50]]; // Vincenzo Librandi, Jun 18 2011

Table[(2-21n+49n^2)/2,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{1,15,78},40] (* Harvey P. Dale, Aug 03 2012 *)

a(n)=(2-21*n+49*n^2)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (2 - 21*n + 49*n^2)/2.
G.f.: (1+6*x)^2/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=15, a(2)=78. - Harvey P. Dale, Aug 03 2012
E.g.f.: exp(x)*(2 + 28*x + 49*x^2)/2. - Elmo R. Oliveira, Jun 09 2025

A154680 Triangle read by rows where T(m,n)=2mn + m + n - 2.

Original entry on oeis.org

2, 5, 10, 8, 15, 22, 11, 20, 29, 38, 14, 25, 36, 47, 58, 17, 30, 43, 56, 69, 82, 20, 35, 50, 65, 80, 95, 110, 23, 40, 57, 74, 91, 108, 125, 142, 26, 45, 64, 83, 102, 121, 140, 159, 178, 29, 50, 71, 92, 113, 134, 155, 176, 197, 218, 32, 55, 78, 101, 124, 147, 170, 193, 216, 239, 262

Offset: 1

Vincenzo Librandi, Jan 18 2009

All terms are in A153052.

First column: A016789; second column: 5*A000027; third column: A016993; fourth column: A017185. - Vincenzo Librandi, Nov 18 2012

			Triangle begins:
2;
5,  10;
8,  15, 22;
11, 20, 29, 38;
14, 25, 36, 47, 58;
17, 30, 43, 56, 69,  82;
20, 35, 50, 65, 80,  95,  110;
23, 40, 57, 74, 91,  108, 125, 142;
26, 45, 64, 83, 102, 121, 140, 159, 178;
29, 50, 71, 92, 113, 134, 155, 176, 197, 218; etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A153052, A089038, A016789, A000027, A017185.

[2*n*k+n+k-2: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 18 2012

Flatten[Table[Floor[2 n m + m + n - 2], {n, 1, 16}, {m, n}]] (* Vincenzo Librandi, May 14 2012 *)

A155551 Triangle read by rows where T(m,n)=2mn + m + n - 9.

Original entry on oeis.org

-5, -2, 3, 1, 8, 15, 4, 13, 22, 31, 7, 18, 29, 40, 51, 10, 23, 36, 49, 62, 75, 13, 28, 43, 58, 73, 88, 103, 16, 33, 50, 67, 84, 101, 118, 135, 19, 38, 57, 76, 95, 114, 133, 152, 171, 22, 43, 64, 85, 106, 127, 148, 169, 190, 211, 25, 48, 71, 94, 117, 140, 163, 186, 209

Offset: 1

Vincenzo Librandi, Jan 24 2009

The numbers 2*T(m,n)+19 =(2*n+1)*(2*m+1) are not prime.

First column: A016777, second column: A016885, third column: A016993, fourth column: A017209. - Vincenzo Librandi, Nov 20 2012

			Triangle begins:
-5;
-2, 3;
1,  8,  15;
4,  13, 22, 31;
7,  18, 29, 40, 51;
10, 23, 36, 49, 62,  75;
13, 28, 43, 58, 73,  88,  103;
16, 33, 50, 67, 84,  101, 118, 135;
19, 38, 57, 76, 95,  114, 133, 152, 171;
22, 43, 64, 85, 106, 127, 148, 169, 190, 211; etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A153143, A153144, A016777, A016885, A016993, A017209.

[2*n*k + n + k - 9: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

t[n_,k_]:=2 n*k + n + k - 9; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

A176972 a(n) = 7^n + 7*n + 1.

Original entry on oeis.org

2, 15, 64, 365, 2430, 16843, 117692, 823593, 5764858, 40353671, 282475320, 1977326821, 13841287286, 96889010499, 678223072948, 4747561510049, 33232930569714, 232630513987327, 1628413597910576, 11398895185373277, 79792266297612142, 558545864083284155, 3909821048582988204

Offset: 0

Jonathan Vos Post, Apr 29 2010

			a(5) = 7^5 + 7*5 + 1 = 16843 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (9,-15,7).

Cf. A000420, A008589, A016993, A176691, A176805, A176916.

Cf. A370664.

[7^n + 7*n + 1: n in [0..25]]; // Vincenzo Librandi, May 06 2011

Table[7^n+7n+1,{n,0,20}] (* or *) LinearRecurrence[{9,-15,7},{2,15,64},20] (* Harvey P. Dale, Apr 17 2014 *)

a(n) = A000420(n) + A008589(n) + 1 = A000420(n) + A016993(n).
a(n) = 7*a(n-1) - 42*(n-1) + 1, with n > 0. For n=5, a(5) = 7*2430 - 42*4 + 1 = 16843. - Bruno Berselli, May 18 2010
From R. J. Mathar, May 22 2010: (Start)
a(n) = 9*a(n-1) - 15*a(n-2) + 7*a(n-3).
G.f.: (-2 + 3*x + 41*x^2) / ((7*x-1)*(x-1)^2). (End)
E.g.f.: exp(x)*(1 + exp(6*x) + 7*x). - Stefano Spezia, Aug 19 2024

A131844 3*A131821 - 2.

Original entry on oeis.org

1, 4, 4, 7, 1, 7, 10, 1, 1, 10, 13, 1, 1, 1, 13, 16, 1, 1, 1, 1, 16, 19, 1, 1, 1, 1, 1, 19, 22, 1, 1, 1, 1, 1, 1, 22, 25, 1, 1, 1, 1, 1, 1, 1, 25, 28, 1, 1, 1, 1, 1, 1, 1, 1, 28, 31, 1, 1, 1, 1, 1, 1, 1, 1, 1, 31, 34, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 34, 37, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 37, 40, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 40

Offset: 0

Gary W. Adamson, Jul 22 2007

Row sums = A016993, (7n + 1): (1, 8, 15, 22, 29, ...).

			First few rows of the triangle:
   1;
   4,  4;
   7,  1,  7;
  10,  1,  1, 10;
  13,  1,  1,  1, 13;
  16,  1,  1,  1,  1, 16;
  ...

Cf. A131821, A016993.

3*A131821 - 2 as infinite lower triangular matrices.

Infinite lower triangular matrix, (3n + 1): (1, 4, 7, 10, ...) as right and left borders and the rest 1's.

More terms from Russ Cox, Apr 18 2024

A139615 a(n) = 105*n + 15.

Original entry on oeis.org

15, 120, 225, 330, 435, 540, 645, 750, 855, 960, 1065, 1170, 1275, 1380, 1485, 1590, 1695, 1800, 1905, 2010, 2115, 2220, 2325, 2430, 2535, 2640, 2745, 2850, 2955, 3060, 3165, 3270, 3375, 3480, 3585, 3690, 3795, 3900, 4005, 4110, 4215

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 15th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 15th column in the square array A057145.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016993, A057145, A139600.

[15*(7*n + 1): n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

105*Range[0,40]+15 (* or *) LinearRecurrence[{2,-1},{15,120},50] (* Harvey P. Dale, May 07 2016 *)

a(n)=105*n+15 \\ Charles R Greathouse IV, Oct 05 2011

From Elmo R. Oliveira, Apr 04 2024: (Start)
G.f.: 15*(1+6*x)/(x-1)^2.
E.g.f.: 15*exp(x)*(1 + 7*x).
a(n) = 15*A016993(n).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

cp's OEIS Frontend

A081581 Pascal-(1,6,1) array.

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A171890 Octonomial coefficient array.

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A227095 Numbers whose base-8 sum of digits is 8.

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A245301 a(n) = n*(7*n^2 + 15*n + 8)/6.

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A081591 Third row of Pascal-(1,6,1) array A081581.

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A154680 Triangle read by rows where T(m,n)=2*m*n + m + n - 2.

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A155551 Triangle read by rows where T(m,n)=2*m*n + m + n - 9.

A245301 a(n) = n(7n^2 + 15*n + 8)/6.

A154680 Triangle read by rows where T(m,n)=2mn + m + n - 2.

A155551 Triangle read by rows where T(m,n)=2mn + m + n - 9.